Society for Industrial and Applied Mathematics: Multiscale Modeling & Simulation: Table of Contents
Table of Contents for Multiscale Modeling & Simulation. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/mmsubt?ai=s9&mi=3csssi&af=R
Society for Industrial and Applied Mathematics: Multiscale Modeling & Simulation: Table of Contents
Society for Industrial and Applied Mathematics
en-US
Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
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https://epubs.siam.org/loi/mmsubt?ai=s9&mi=3csssi&af=R
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A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles
https://epubs.siam.org/doi/abs/10.1137/22M1495512?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 1-38, March 2024. <br/> Abstract. In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem that is valid when the size of the particles tends to zero. We present a systematic approach to constructing such models based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. We prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 1-38, March 2024. <br/> Abstract. In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem that is valid when the size of the particles tends to zero. We present a systematic approach to constructing such models based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. We prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments.
A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles
10.1137/22M1495512
Multiscale Modeling & Simulation
2024-01-03T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Maryna Kachanovska
A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles
22
1
1
38
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1495512
https://epubs.siam.org/doi/abs/10.1137/22M1495512?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Three-Dimensional Random Wave Coupling Along a Boundary and an Associated Inverse Problem
https://epubs.siam.org/doi/abs/10.1137/23M1544842?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 39-65, March 2024. <br/> Abstract. We consider random wave coupling along a flat boundary in dimension three, where the coupling is between surface and body modes and is induced by scattering by a randomly heterogeneous medium. In an appropriate scaling regime we obtain a system of radiative transfer equations which are satisfied by the mean Wigner transform of the mode amplitudes. We provide a rigorous probabilistic framework for describing solutions to this system using that it has the form of a Kolmogorov equation for some Markov process. We then prove statistical stability of the smoothed Wigner transform under the Gaussian approximation. We conclude with analyzing the nonlinear inverse problem for the radiative transfer equations and establish the unique recovery of phase and group velocities as well as power spectral information for the medium fluctuations from the observed smoothed Wigner transform. The mentioned statistical stability is essential in monitoring applications where the realization of the random medium may change.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 39-65, March 2024. <br/> Abstract. We consider random wave coupling along a flat boundary in dimension three, where the coupling is between surface and body modes and is induced by scattering by a randomly heterogeneous medium. In an appropriate scaling regime we obtain a system of radiative transfer equations which are satisfied by the mean Wigner transform of the mode amplitudes. We provide a rigorous probabilistic framework for describing solutions to this system using that it has the form of a Kolmogorov equation for some Markov process. We then prove statistical stability of the smoothed Wigner transform under the Gaussian approximation. We conclude with analyzing the nonlinear inverse problem for the radiative transfer equations and establish the unique recovery of phase and group velocities as well as power spectral information for the medium fluctuations from the observed smoothed Wigner transform. The mentioned statistical stability is essential in monitoring applications where the realization of the random medium may change.
Three-Dimensional Random Wave Coupling Along a Boundary and an Associated Inverse Problem
10.1137/23M1544842
Multiscale Modeling & Simulation
2024-01-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Maarten V. de Hoop
Josselin Garnier
Knut Sølna
Three-Dimensional Random Wave Coupling Along a Boundary and an Associated Inverse Problem
22
1
39
65
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1544842
https://epubs.siam.org/doi/abs/10.1137/23M1544842?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Semiclassical Propagation Along Curved Domain Walls
https://epubs.siam.org/doi/abs/10.1137/23M1545872?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 66-105, March 2024. <br/> Abstract. We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set [math] of a slowly varying domain wall modeling the interface separating two insulating media. We propose a semiclassical oscillatory representation of the propagating wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along [math] and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary smooth localized initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets. The theory finds applications for both Dirac systems of equations modeling topologically nontrivial systems as well as Klein–Gordon equations, which are topologically trivial.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 66-105, March 2024. <br/> Abstract. We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set [math] of a slowly varying domain wall modeling the interface separating two insulating media. We propose a semiclassical oscillatory representation of the propagating wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along [math] and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary smooth localized initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets. The theory finds applications for both Dirac systems of equations modeling topologically nontrivial systems as well as Klein–Gordon equations, which are topologically trivial.
Semiclassical Propagation Along Curved Domain Walls
10.1137/23M1545872
Multiscale Modeling & Simulation
2024-01-10T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Guillaume Bal
Semiclassical Propagation Along Curved Domain Walls
22
1
66
105
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1545872
https://epubs.siam.org/doi/abs/10.1137/23M1545872?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited
https://epubs.siam.org/doi/abs/10.1137/22M1511060?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 106-124, March 2024. <br/> Abstract. We derive the optimal energy error estimate for a multiscale finite element method with oversampling technique applied to an elliptic system with rapidly oscillating periodic coefficients under the assumption that the coefficients are bounded and measurable, which may admit rough microstructures. As a byproduct of the energy error estimate, we derive the rate of convergence in the [math]-norm.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 106-124, March 2024. <br/> Abstract. We derive the optimal energy error estimate for a multiscale finite element method with oversampling technique applied to an elliptic system with rapidly oscillating periodic coefficients under the assumption that the coefficients are bounded and measurable, which may admit rough microstructures. As a byproduct of the energy error estimate, we derive the rate of convergence in the [math]-norm.
Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited
10.1137/22M1511060
Multiscale Modeling & Simulation
2024-01-10T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Pingbing Ming
Siqi Song
Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited
22
1
106
124
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1511060
https://epubs.siam.org/doi/abs/10.1137/22M1511060?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Metadynamics for Transition Paths in Irreversible Dynamics
https://epubs.siam.org/doi/abs/10.1137/23M1563025?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 125-141, March 2024. <br/> Abstract. Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows rigorous estimation of the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 125-141, March 2024. <br/> Abstract. Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows rigorous estimation of the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.
Metadynamics for Transition Paths in Irreversible Dynamics
10.1137/23M1563025
Multiscale Modeling & Simulation
2024-01-12T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Tobias Grafke
Alessandro Laio
Metadynamics for Transition Paths in Irreversible Dynamics
22
1
125
141
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1563025
https://epubs.siam.org/doi/abs/10.1137/23M1563025?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Improved Self-consistent Field Iteration for Kohn–Sham Density Functional Theory
https://epubs.siam.org/doi/abs/10.1137/23M1558215?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 142-154, March 2024. <br/> Abstract. In this article, an improved self-consistent field iteration scheme is introduced. The proposed method has essential applications in Kohn–Sham density functional theory and relies on an extrapolation scheme and the least squares method. Moreover, the proposed solution is easy to implement and can accelerate the convergence of self-consistent field iteration. The main idea is to fit out a polynomial based on the errors of the derived approximate solutions and then extrapolate the errors into zero to obtain a new approximation. The developed scheme can be applied not only to the Kohn–Sham density functional theory but also to accelerate the self-consistent field iterations of other nonlinear equations. Some numerical results for the Kohn–Sham equation and general nonlinear equations are presented to validate the efficiency of the new method.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 142-154, March 2024. <br/> Abstract. In this article, an improved self-consistent field iteration scheme is introduced. The proposed method has essential applications in Kohn–Sham density functional theory and relies on an extrapolation scheme and the least squares method. Moreover, the proposed solution is easy to implement and can accelerate the convergence of self-consistent field iteration. The main idea is to fit out a polynomial based on the errors of the derived approximate solutions and then extrapolate the errors into zero to obtain a new approximation. The developed scheme can be applied not only to the Kohn–Sham density functional theory but also to accelerate the self-consistent field iterations of other nonlinear equations. Some numerical results for the Kohn–Sham equation and general nonlinear equations are presented to validate the efficiency of the new method.
Improved Self-consistent Field Iteration for Kohn–Sham Density Functional Theory
10.1137/23M1558215
Multiscale Modeling & Simulation
2024-01-12T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Fei Xu
Qiumei Huang
Improved Self-consistent Field Iteration for Kohn–Sham Density Functional Theory
22
1
142
154
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1558215
https://epubs.siam.org/doi/abs/10.1137/23M1558215?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media
https://epubs.siam.org/doi/abs/10.1137/22M1529075?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 155-177, March 2024. <br/> Abstract. In this paper, we study two-grid preconditioners for three-dimensional single-phase nonlinear compressible flow in highly heterogeneous porous media arising from reservoir simulation. Our goal is to develop robust and efficient preconditioners that converge independently of the contrast of the media and types of boundary conditions and source term. This is accomplished by constructing coarse space that can capture important features of the local heterogeneous media. To detect these features, local eigenvalue problems are defined and eigenvectors are adaptively selected to form the coarse space. The coarse space just needs to be constructed only once with parallel computing, although the compressible flow is a time-dependent problem and the permeability field changes in different time steps. Smoothers such as Gauss–Seidel iteration and ILU(0) are used to remove high-frequency errors. We analyze this preconditioner by proving the smoothing property and approximation property. In particular, a new coarse interpolation operator is defined to facilitate the analysis. Extensive numerical experiments with different types of large-scale heterogeneous permeability fields and boundary conditions are provided to show the impressive performance of the proposed preconditioner.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 155-177, March 2024. <br/> Abstract. In this paper, we study two-grid preconditioners for three-dimensional single-phase nonlinear compressible flow in highly heterogeneous porous media arising from reservoir simulation. Our goal is to develop robust and efficient preconditioners that converge independently of the contrast of the media and types of boundary conditions and source term. This is accomplished by constructing coarse space that can capture important features of the local heterogeneous media. To detect these features, local eigenvalue problems are defined and eigenvectors are adaptively selected to form the coarse space. The coarse space just needs to be constructed only once with parallel computing, although the compressible flow is a time-dependent problem and the permeability field changes in different time steps. Smoothers such as Gauss–Seidel iteration and ILU(0) are used to remove high-frequency errors. We analyze this preconditioner by proving the smoothing property and approximation property. In particular, a new coarse interpolation operator is defined to facilitate the analysis. Extensive numerical experiments with different types of large-scale heterogeneous permeability fields and boundary conditions are provided to show the impressive performance of the proposed preconditioner.
An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media
10.1137/22M1529075
Multiscale Modeling & Simulation
2024-01-17T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Shubin Fu
Eric Chung
Lina Zhao
An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media
22
1
155
177
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1529075
https://epubs.siam.org/doi/abs/10.1137/22M1529075?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity
https://epubs.siam.org/doi/abs/10.1137/23M1582655?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 178-203, March 2024. <br/> Abstract. The distinct timescales of synaptic plasticity and neural activity dynamics play an important role in the brain’s learning and memory systems. Activity-dependent plasticity reshapes neural circuit architecture, determining spontaneous and stimulus-encoding spatiotemporal patterns of neural activity. Neural activity bumps maintain short term memories of continuous parameter values, emerging in spatially organized models with short-range excitation and long-range inhibition. Previously, we demonstrated nonlinear Langevin equations derived using an interface method which accurately describe the dynamics of bumps in continuum neural fields with separate excitatory/inhibitory populations. Here we extend this analysis to incorporate effects of short term plasticity that dynamically modifies connectivity described by an integral kernel. Linear stability analysis adapted to these piecewise smooth models with Heaviside firing rates further indicates how plasticity shapes the bumps’ local dynamics. Facilitation (depression), which strengthens (weakens) synaptic connectivity originating from active neurons, tends to increase (decrease) stability of bumps when acting on excitatory synapses. The relationship is inverted when plasticity acts on inhibitory synapses. Multiscale approximations of the stochastic dynamics of bumps perturbed by weak noise reveal that the plasticity variables evolve to slowly diffusing and blurred versions of their stationary profiles. Nonlinear Langevin equations associated with bump positions or interfaces coupled to slowly evolving projections of plasticity variables accurately describe how these smoothed synaptic efficacy profiles can tether or repel wandering bumps.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 178-203, March 2024. <br/> Abstract. The distinct timescales of synaptic plasticity and neural activity dynamics play an important role in the brain’s learning and memory systems. Activity-dependent plasticity reshapes neural circuit architecture, determining spontaneous and stimulus-encoding spatiotemporal patterns of neural activity. Neural activity bumps maintain short term memories of continuous parameter values, emerging in spatially organized models with short-range excitation and long-range inhibition. Previously, we demonstrated nonlinear Langevin equations derived using an interface method which accurately describe the dynamics of bumps in continuum neural fields with separate excitatory/inhibitory populations. Here we extend this analysis to incorporate effects of short term plasticity that dynamically modifies connectivity described by an integral kernel. Linear stability analysis adapted to these piecewise smooth models with Heaviside firing rates further indicates how plasticity shapes the bumps’ local dynamics. Facilitation (depression), which strengthens (weakens) synaptic connectivity originating from active neurons, tends to increase (decrease) stability of bumps when acting on excitatory synapses. The relationship is inverted when plasticity acts on inhibitory synapses. Multiscale approximations of the stochastic dynamics of bumps perturbed by weak noise reveal that the plasticity variables evolve to slowly diffusing and blurred versions of their stationary profiles. Nonlinear Langevin equations associated with bump positions or interfaces coupled to slowly evolving projections of plasticity variables accurately describe how these smoothed synaptic efficacy profiles can tether or repel wandering bumps.
Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity
10.1137/23M1582655
Multiscale Modeling & Simulation
2024-01-17T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Heather L. Cihak
Zachary P. Kilpatrick
Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity
22
1
178
203
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1582655
https://epubs.siam.org/doi/abs/10.1137/23M1582655?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation
https://epubs.siam.org/doi/abs/10.1137/23M1569216?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 204-229, March 2024. <br/> Abstract. A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse-scale representation of the elliptic operator, enriched by fine-scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 204-229, March 2024. <br/> Abstract. A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse-scale representation of the elliptic operator, enriched by fine-scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.
Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation
10.1137/23M1569216
Multiscale Modeling & Simulation
2024-01-18T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Annika Lang
Per Ljung
Axel Målqvist
Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation
22
1
204
229
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1569216
https://epubs.siam.org/doi/abs/10.1137/23M1569216?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Particle-Continuum Multiscale Modeling of Sea Ice Floes
https://epubs.siam.org/doi/abs/10.1137/23M155904X?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 230-255, March 2024. <br/> Abstract. Sea ice profoundly influences the polar environment and the global climate. Traditionally, sea ice has been modeled as a continuum under Eulerian coordinates to describe its large-scale features, using, for instance, viscous-plastic rheology. Recently, Lagrangian particle models, also known as the discrete element method models, have been utilized for characterizing the motion of individual sea ice fragments (called floes) at scales of 10 km and smaller, especially in marginal ice zones. This paper develops a multiscale model that couples the particle and the continuum systems to facilitate an effective representation of the dynamical and statistical features of sea ice across different scales. The multiscale model exploits a Boltzmann-type system that links the particle movement with the continuum equations. For the small-scale dynamics, it describes the motion of each sea ice floe. Then, as the large-scale continuum component, it treats the statistical moments of mass density and linear and angular velocities. The evolution of these statistics affects the motion of individual floes, which in turn provides bulk feedback that adjusts the large-scale dynamics. Notably, the particle model characterizing the sea ice floes is localized and fully parallelized in a framework that is sometimes called superparameterization, which significantly improves computational efficiency. Numerical examples demonstrate the effective performance of the multiscale model. Additionally, the study demonstrates that the multiscale model has a linear-order approximation to the truth model.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 230-255, March 2024. <br/> Abstract. Sea ice profoundly influences the polar environment and the global climate. Traditionally, sea ice has been modeled as a continuum under Eulerian coordinates to describe its large-scale features, using, for instance, viscous-plastic rheology. Recently, Lagrangian particle models, also known as the discrete element method models, have been utilized for characterizing the motion of individual sea ice fragments (called floes) at scales of 10 km and smaller, especially in marginal ice zones. This paper develops a multiscale model that couples the particle and the continuum systems to facilitate an effective representation of the dynamical and statistical features of sea ice across different scales. The multiscale model exploits a Boltzmann-type system that links the particle movement with the continuum equations. For the small-scale dynamics, it describes the motion of each sea ice floe. Then, as the large-scale continuum component, it treats the statistical moments of mass density and linear and angular velocities. The evolution of these statistics affects the motion of individual floes, which in turn provides bulk feedback that adjusts the large-scale dynamics. Notably, the particle model characterizing the sea ice floes is localized and fully parallelized in a framework that is sometimes called superparameterization, which significantly improves computational efficiency. Numerical examples demonstrate the effective performance of the multiscale model. Additionally, the study demonstrates that the multiscale model has a linear-order approximation to the truth model.
Particle-Continuum Multiscale Modeling of Sea Ice Floes
10.1137/23M155904X
Multiscale Modeling & Simulation
2024-01-30T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Quanling Deng
Samuel N. Stechmann
Nan Chen
Particle-Continuum Multiscale Modeling of Sea Ice Floes
22
1
230
255
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M155904X
https://epubs.siam.org/doi/abs/10.1137/23M155904X?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Exponential Convergence of a Generalized FEM for Heterogeneous Reaction-Diffusion Equations
https://epubs.siam.org/doi/abs/10.1137/22M1522231?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 256-282, March 2024. <br/> Abstract. A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter [math], based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size [math]. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to [math] and [math] (at the discrete level with [math] denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to [math] in the standard [math] norm, and that if the oversampling size is relatively large with respect to [math] and [math] (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 256-282, March 2024. <br/> Abstract. A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter [math], based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size [math]. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to [math] and [math] (at the discrete level with [math] denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to [math] in the standard [math] norm, and that if the oversampling size is relatively large with respect to [math] and [math] (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.
Exponential Convergence of a Generalized FEM for Heterogeneous Reaction-Diffusion Equations
10.1137/22M1522231
Multiscale Modeling & Simulation
2024-02-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chupeng Ma
J. M. Melenk
Exponential Convergence of a Generalized FEM for Heterogeneous Reaction-Diffusion Equations
22
1
256
282
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1522231
https://epubs.siam.org/doi/abs/10.1137/22M1522231?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Quantum Mechanics for Closure of Dynamical Systems
https://epubs.siam.org/doi/abs/10.1137/22M1514246?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 283-333, March 2024. <br/> Abstract. We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes’ law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems and show how this approach preserves important statistical and qualitative properties of the underlying chaotic dynamics.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 283-333, March 2024. <br/> Abstract. We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes’ law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems and show how this approach preserves important statistical and qualitative properties of the underlying chaotic dynamics.
Quantum Mechanics for Closure of Dynamical Systems
10.1137/22M1514246
Multiscale Modeling & Simulation
2024-02-05T08:00:00Z
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
David C. Freeman
Dimitrios Giannakis
Joanna Slawinska
Quantum Mechanics for Closure of Dynamical Systems
22
1
283
333
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1514246
https://epubs.siam.org/doi/abs/10.1137/22M1514246?ai=s9&mi=3csssi&af=R
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
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Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model
https://epubs.siam.org/doi/abs/10.1137/22M1514179?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 334-368, March 2024. <br/> Abstract. In this work, we consider a nonlinear strain-limiting elastic model in heterogeneous domains. We investigate heterogeneous material with soft and stiff inclusions and perforations that are important to understand an elastic solid’s behavior and crack-tip fields. Numerical solutions of problems in computational domains with inclusions and perforations require the construction of a sufficiently fine grid that resolves heterogeneity on the grid level. Approximations on such grids lead to a large system of equations with large computational costs. To reduce the size of the system and provide an accurate solution, we present a generalized multiscale finite element approximation on the coarse grid. In this method, we construct multiscale basis functions in each local domain associated with the coarse-grid cell and based on the construction of the snapshot space and solution of the local spectral problem reduce the size of the snapshot space. Two types of multiscale basis function construction are presented. The first type is a general case that can handle any boundary conditions on the global boundary of the heterogeneous domain. The considered problem requires an accurate approximation of the crack-surface boundary. In the second type of multiscale basis functions, we incorporate global boundary conditions in the basis construction process which provide an accurate approximation of the stress and strain on the crack boundary. We present numerical results for three cases of heterogeneity: soft inclusions, stiff inclusions, and perforations. A numerical investigation is presented for the two examples of loading on the domain with and without crack boundary conditions. The presented generalized multiscale finite element solver provides an accurate solution with a large reduction of the discrete system size. Our results illustrate the significant error reduction on the crack surface when we use the second type of basis functions.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 334-368, March 2024. <br/> Abstract. In this work, we consider a nonlinear strain-limiting elastic model in heterogeneous domains. We investigate heterogeneous material with soft and stiff inclusions and perforations that are important to understand an elastic solid’s behavior and crack-tip fields. Numerical solutions of problems in computational domains with inclusions and perforations require the construction of a sufficiently fine grid that resolves heterogeneity on the grid level. Approximations on such grids lead to a large system of equations with large computational costs. To reduce the size of the system and provide an accurate solution, we present a generalized multiscale finite element approximation on the coarse grid. In this method, we construct multiscale basis functions in each local domain associated with the coarse-grid cell and based on the construction of the snapshot space and solution of the local spectral problem reduce the size of the snapshot space. Two types of multiscale basis function construction are presented. The first type is a general case that can handle any boundary conditions on the global boundary of the heterogeneous domain. The considered problem requires an accurate approximation of the crack-surface boundary. In the second type of multiscale basis functions, we incorporate global boundary conditions in the basis construction process which provide an accurate approximation of the stress and strain on the crack boundary. We present numerical results for three cases of heterogeneity: soft inclusions, stiff inclusions, and perforations. A numerical investigation is presented for the two examples of loading on the domain with and without crack boundary conditions. The presented generalized multiscale finite element solver provides an accurate solution with a large reduction of the discrete system size. Our results illustrate the significant error reduction on the crack surface when we use the second type of basis functions.
Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model
10.1137/22M1514179
Multiscale Modeling & Simulation
2024-02-06T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Maria Vasilyeva
S. M. Mallikarjunaiah
Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model
22
1
334
368
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1514179
https://epubs.siam.org/doi/abs/10.1137/22M1514179?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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On the Compatibility of Sharp and Diffuse Interfaces Out of Equilibrium
https://epubs.siam.org/doi/abs/10.1137/22M1529294?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 369-405, March 2024. <br/> Abstract. There are two main approaches to modelling interfaces within nonequilibrium thermodynamics, the so-called sharp and diffuse interface models. Both of them are based on the local equilibrium assumption (LEA) in the bulk, but the latter additionally assumes the validity of this concept also within the interface itself (as the thermodynamic description is available and smoothly varying even within the interface), that is, on a finer length scale, which we call super-LEA. Instead of testing the two approaches against molecular dynamic simulations, we explore the mutual compatibility of these two descriptions of an interface in a nonequilibrium situation. Based on the level of detail in the two frameworks, one naturally cannot reconstruct a diffuse interface model from a sharp interface counterpart. One can test, however, whether diffuse interface models are indeed a more detailed description of the interface. Namely, assuming that both approaches are valid, we use the diffuse interface model (van der Waals entropy together with the Cahn–Hilliard type energy with the mass density as the order parameter) and its sharp interface counterpart (with the additional set of interfacial state variables subjected to known thermodynamic constraints) to test their mutual compatibility and indirectly verify the correctness of the additional super-LEA of the diffuse models. That is, thanks to super-LEA, we define five interfacial temperatures that should be equal. However, when we analyze diffuse interface results like experimental or simulation data in terms of sharp interfaces, we show that, contrary to molecular simulation data, they do not yield equal interfacial temperatures. We argue that the culprit is the super-LEA which is most prominently expressed in the accessibility of the entropy density profile. Nevertheless, it is observed that there is an inconsistency between diffuse and sharp interface descriptions; they cannot both be correct. The sharp interface framework has been recently tested against molecular dynamics and the obtained results suggest that super-LEA is the potential weakness of the diffuse framework. In this sense, sharp interfaces are found to be superior to diffuse interfaces in their general ability to model physical systems with interfaces.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 369-405, March 2024. <br/> Abstract. There are two main approaches to modelling interfaces within nonequilibrium thermodynamics, the so-called sharp and diffuse interface models. Both of them are based on the local equilibrium assumption (LEA) in the bulk, but the latter additionally assumes the validity of this concept also within the interface itself (as the thermodynamic description is available and smoothly varying even within the interface), that is, on a finer length scale, which we call super-LEA. Instead of testing the two approaches against molecular dynamic simulations, we explore the mutual compatibility of these two descriptions of an interface in a nonequilibrium situation. Based on the level of detail in the two frameworks, one naturally cannot reconstruct a diffuse interface model from a sharp interface counterpart. One can test, however, whether diffuse interface models are indeed a more detailed description of the interface. Namely, assuming that both approaches are valid, we use the diffuse interface model (van der Waals entropy together with the Cahn–Hilliard type energy with the mass density as the order parameter) and its sharp interface counterpart (with the additional set of interfacial state variables subjected to known thermodynamic constraints) to test their mutual compatibility and indirectly verify the correctness of the additional super-LEA of the diffuse models. That is, thanks to super-LEA, we define five interfacial temperatures that should be equal. However, when we analyze diffuse interface results like experimental or simulation data in terms of sharp interfaces, we show that, contrary to molecular simulation data, they do not yield equal interfacial temperatures. We argue that the culprit is the super-LEA which is most prominently expressed in the accessibility of the entropy density profile. Nevertheless, it is observed that there is an inconsistency between diffuse and sharp interface descriptions; they cannot both be correct. The sharp interface framework has been recently tested against molecular dynamics and the obtained results suggest that super-LEA is the potential weakness of the diffuse framework. In this sense, sharp interfaces are found to be superior to diffuse interfaces in their general ability to model physical systems with interfaces.
On the Compatibility of Sharp and Diffuse Interfaces Out of Equilibrium
10.1137/22M1529294
Multiscale Modeling & Simulation
2024-02-26T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Václav Klika
Hans Christian Öttinger
On the Compatibility of Sharp and Diffuse Interfaces Out of Equilibrium
22
1
369
405
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1529294
https://epubs.siam.org/doi/abs/10.1137/22M1529294?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Dynamical Properties of Coarse-Grained Linear SDEs
https://epubs.siam.org/doi/abs/10.1137/23M1549249?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 406-435, March 2024. <br/> Abstract. Coarse-graining or model reduction is a term describing a range of approaches used to extend the timescale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analyzed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics, such as the mean-squared displacement, display systematic error, even when a system exhibits large timescale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 406-435, March 2024. <br/> Abstract. Coarse-graining or model reduction is a term describing a range of approaches used to extend the timescale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analyzed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics, such as the mean-squared displacement, display systematic error, even when a system exhibits large timescale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.
Dynamical Properties of Coarse-Grained Linear SDEs
10.1137/23M1549249
Multiscale Modeling & Simulation
2024-02-27T08:00:00Z
© 2024 Thomas Hudson and Xingjie Helen Li
Thomas Hudson
Xingjie Helen Li
Dynamical Properties of Coarse-Grained Linear SDEs
22
1
406
435
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1549249
https://epubs.siam.org/doi/abs/10.1137/23M1549249?ai=s9&mi=3csssi&af=R
© 2024 Thomas Hudson and Xingjie Helen Li
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Upscaling an Extended Heterogeneous Stefan Problem from the Pore-Scale to the Darcy Scale in Permafrost
https://epubs.siam.org/doi/abs/10.1137/23M1552000?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 436-475, March 2024. <br/> Abstract. In this paper we upscale thermal models from the pore–scale to the Darcy scale for applications in permafrost. We incorporate thawing and freezing of water at the pore-scale and adapt rigorous homogenization theory from [A. Visintin, SIAM J. Math. Anal., 39 (2007), pp. 987–1017] to the original nonlinear multivalued relationship to derive the effective properties. To obtain agreement of the effective model with the known Darcy scale empirical models, we revisit and extend the pore-scale model to include the delicate microscale physics in small pores. We also propose a practical reduced model for the nonlinear effective conductivity. We illustrate with simulations.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 436-475, March 2024. <br/> Abstract. In this paper we upscale thermal models from the pore–scale to the Darcy scale for applications in permafrost. We incorporate thawing and freezing of water at the pore-scale and adapt rigorous homogenization theory from [A. Visintin, SIAM J. Math. Anal., 39 (2007), pp. 987–1017] to the original nonlinear multivalued relationship to derive the effective properties. To obtain agreement of the effective model with the known Darcy scale empirical models, we revisit and extend the pore-scale model to include the delicate microscale physics in small pores. We also propose a practical reduced model for the nonlinear effective conductivity. We illustrate with simulations.
Upscaling an Extended Heterogeneous Stefan Problem from the Pore-Scale to the Darcy Scale in Permafrost
10.1137/23M1552000
Multiscale Modeling & Simulation
2024-03-06T08:00:00Z
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
Malgorzata Peszynska
Naren Vohra
Lisa Bigler
Upscaling an Extended Heterogeneous Stefan Problem from the Pore-Scale to the Darcy Scale in Permafrost
22
1
436
475
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1552000
https://epubs.siam.org/doi/abs/10.1137/23M1552000?ai=s9&mi=3csssi&af=R
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
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Fano Resonances in All-Dielectric Electromagnetic Metasurfaces
https://epubs.siam.org/doi/abs/10.1137/23M1554825?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 476-526, March 2024. <br/> Abstract. We are interested in the resonant electromagnetic scattering by all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari, Li, and Zou, Trans. Amer. Math. Soc., 376 (2023), pp. 39–90], it has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single-particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all-dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 476-526, March 2024. <br/> Abstract. We are interested in the resonant electromagnetic scattering by all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari, Li, and Zou, Trans. Amer. Math. Soc., 376 (2023), pp. 39–90], it has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single-particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all-dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.
Fano Resonances in All-Dielectric Electromagnetic Metasurfaces
10.1137/23M1554825
Multiscale Modeling & Simulation
2024-03-06T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Habib Ammari
Bowen Li
Hongjie Li
Jun Zou
Fano Resonances in All-Dielectric Electromagnetic Metasurfaces
22
1
476
526
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1554825
https://epubs.siam.org/doi/abs/10.1137/23M1554825?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
-
Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs
https://epubs.siam.org/doi/abs/10.1137/22M1509205?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 527-560, March 2024. <br/> Abstract. Nonlinear Fokker–Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker–Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker–Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 527-560, March 2024. <br/> Abstract. Nonlinear Fokker–Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker–Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker–Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.
Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs
10.1137/22M1509205
Multiscale Modeling & Simulation
2024-03-13T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Giacomo Dimarco
Lorenzo Pareschi
Mattia Zanella
Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs
22
1
527
560
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1509205
https://epubs.siam.org/doi/abs/10.1137/22M1509205?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Synchronization-Capturing Multiscale Solver to the Noisy Integrate-and-Fire Neuron Networks
https://epubs.siam.org/doi/abs/10.1137/23M1573276?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 561-587, March 2024. <br/> Abstract. The noisy leaky integrate-and-fire (NLIF) model describes the voltage configurations of neuron networks with an interacting many-particles system at a microscopic level. When simulating neuron networks of large sizes, computing a coarse-grained mean-field Fokker–Planck equation solving the voltage densities of the networks at a macroscopic level practically serves as a feasible alternative in its high efficiency and credible accuracy when the interaction within the network remains relatively low. However, the macroscopic model fails to yield valid results of the networks when simulating considerably synchronous networks with active firing events. In this paper, we propose a multiscale solver for the NLIF networks, inheriting the macroscopic solver’s low cost and the microscopic solver’s high reliability. For each temporal step, the multiscale solver uses the macroscopic solver when the firing rate of the simulated network is low, while it switches to the microscopic solver when the firing rate tends to blow up. Moreover, the macroscopic and microscopic solvers are integrated with a high-precision switching algorithm to ensure the accuracy of the multiscale solver. The validity of the multiscale solver is analyzed from two perspectives: first, we provide practically sufficient conditions that guarantee the mean-field approximation of the macroscopic model and present rigorous numerical analysis on simulation errors when coupling the two solvers; second, the numerical performance of the multiscale solver is validated through simulating several large neuron networks, including networks with either instantaneous or periodic input currents which prompt active firing events over some time.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 561-587, March 2024. <br/> Abstract. The noisy leaky integrate-and-fire (NLIF) model describes the voltage configurations of neuron networks with an interacting many-particles system at a microscopic level. When simulating neuron networks of large sizes, computing a coarse-grained mean-field Fokker–Planck equation solving the voltage densities of the networks at a macroscopic level practically serves as a feasible alternative in its high efficiency and credible accuracy when the interaction within the network remains relatively low. However, the macroscopic model fails to yield valid results of the networks when simulating considerably synchronous networks with active firing events. In this paper, we propose a multiscale solver for the NLIF networks, inheriting the macroscopic solver’s low cost and the microscopic solver’s high reliability. For each temporal step, the multiscale solver uses the macroscopic solver when the firing rate of the simulated network is low, while it switches to the microscopic solver when the firing rate tends to blow up. Moreover, the macroscopic and microscopic solvers are integrated with a high-precision switching algorithm to ensure the accuracy of the multiscale solver. The validity of the multiscale solver is analyzed from two perspectives: first, we provide practically sufficient conditions that guarantee the mean-field approximation of the macroscopic model and present rigorous numerical analysis on simulation errors when coupling the two solvers; second, the numerical performance of the multiscale solver is validated through simulating several large neuron networks, including networks with either instantaneous or periodic input currents which prompt active firing events over some time.
A Synchronization-Capturing Multiscale Solver to the Noisy Integrate-and-Fire Neuron Networks
10.1137/23M1573276
Multiscale Modeling & Simulation
2024-03-19T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ziyu Du
Yantong Xie
Zhennan Zhou
A Synchronization-Capturing Multiscale Solver to the Noisy Integrate-and-Fire Neuron Networks
22
1
561
587
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M1573276
https://epubs.siam.org/doi/abs/10.1137/23M1573276?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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Representative Volume Element Approximations in Elastoplastic Spring Networks
https://epubs.siam.org/doi/abs/10.1137/23M156656X?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 588-638, March 2024. <br/> Abstract. We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary [math]-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl–Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 588-638, March 2024. <br/> Abstract. We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary [math]-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl–Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.
Representative Volume Element Approximations in Elastoplastic Spring Networks
10.1137/23M156656X
Multiscale Modeling & Simulation
2024-03-20T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Sabine Haberland
Patrick Jaap
Stefan Neukamm
Oliver Sander
Mario Varga
Representative Volume Element Approximations in Elastoplastic Spring Networks
22
1
588
638
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/23M156656X
https://epubs.siam.org/doi/abs/10.1137/23M156656X?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation
https://epubs.siam.org/doi/abs/10.1137/22M1533487?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/22/1">Volume 22, Issue 1</a>, Page 639-666, March 2024. <br/> Abstract.Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high-dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there has been a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work toward that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput., 91 (2022), pp. 1–27] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed RBM (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: (1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and (2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux. We perform benchmark test problems in 2D2V, and the numerical tests show that the MMD-RBM can capture the low-rank structure effectively when it exists. A careful study in the computational cost shows that the offline stage of the MMD-RBM is more efficient than the proper orthogonal decomposition method, and in the low-rank case, it even outperforms a standard full-order solve. Therefore, the proposed MMD-RBM can be seen both as a surrogate builder and a low-rank solver at the same time. Furthermore, it can be readily incorporated into multiquery scenarios to accelerate problems arising from uncertainty quantification, control, inverse problems, and optimization.
Multiscale Modeling & Simulation, Volume 22, Issue 1, Page 639-666, March 2024. <br/> Abstract.Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high-dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there has been a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work toward that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput., 91 (2022), pp. 1–27] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed RBM (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: (1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and (2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux. We perform benchmark test problems in 2D2V, and the numerical tests show that the MMD-RBM can capture the low-rank structure effectively when it exists. A careful study in the computational cost shows that the offline stage of the MMD-RBM is more efficient than the proper orthogonal decomposition method, and in the low-rank case, it even outperforms a standard full-order solve. Therefore, the proposed MMD-RBM can be seen both as a surrogate builder and a low-rank solver at the same time. Furthermore, it can be readily incorporated into multiquery scenarios to accelerate problems arising from uncertainty quantification, control, inverse problems, and optimization.
A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation
10.1137/22M1533487
Multiscale Modeling & Simulation
2024-03-21T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Zhichao Peng
Yanlai Chen
Yingda Cheng
Fengyan Li
A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation
22
1
639
666
2024-03-31T07:00:00Z
2024-03-31T07:00:00Z
10.1137/22M1533487
https://epubs.siam.org/doi/abs/10.1137/22M1533487?ai=s9&mi=3csssi&af=R
© 2024 Society for Industrial and Applied Mathematics
-
Stochastic Continuum Models for High-Entropy Alloys with Short-range Order
https://epubs.siam.org/doi/abs/10.1137/22M1496335?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1323-1343, December 2023. <br/> Abstract. High entropy alloys (HEAs) are a class of novel materials that exhibit superb engineering properties. It has been demonstrated by extensive experiments and first principles/atomistic simulations that short-range order in the atomic level randomness strongly influences the properties of HEAs. In this paper, we derive stochastic continuum models for HEAs with short-range order from atomistic models. A proper continuum limit is obtained such that the mean and variance of the atomic level randomness together with the short-range order described by a characteristic length are kept in the process from the atomistic interaction model to the continuum equation. The obtained continuum model with short-range order is in the form of an Ornstein–Uhlenbeck (OU) process. This validates the continuum model based on the OU process adopted phenomenologically by Zhang et al. [Acta Mater., 166 (2019), pp. 424–434] for HEAs with short-range order. We derive such stochastic continuum models with short-range order for both (i) the elastic deformation in HEAs without defects and (ii) HEAs with dislocations (line defects). The obtained stochastic continuum models are based on the energy formulations, whose variations lead to stochastic partial differential equations.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1323-1343, December 2023. <br/> Abstract. High entropy alloys (HEAs) are a class of novel materials that exhibit superb engineering properties. It has been demonstrated by extensive experiments and first principles/atomistic simulations that short-range order in the atomic level randomness strongly influences the properties of HEAs. In this paper, we derive stochastic continuum models for HEAs with short-range order from atomistic models. A proper continuum limit is obtained such that the mean and variance of the atomic level randomness together with the short-range order described by a characteristic length are kept in the process from the atomistic interaction model to the continuum equation. The obtained continuum model with short-range order is in the form of an Ornstein–Uhlenbeck (OU) process. This validates the continuum model based on the OU process adopted phenomenologically by Zhang et al. [Acta Mater., 166 (2019), pp. 424–434] for HEAs with short-range order. We derive such stochastic continuum models with short-range order for both (i) the elastic deformation in HEAs without defects and (ii) HEAs with dislocations (line defects). The obtained stochastic continuum models are based on the energy formulations, whose variations lead to stochastic partial differential equations.
Stochastic Continuum Models for High-Entropy Alloys with Short-range Order
10.1137/22M1496335
Multiscale Modeling & Simulation
2023-10-10T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Yahong Yang
Luchan Zhang
Yang Xiang
Stochastic Continuum Models for High-Entropy Alloys with Short-range Order
21
4
1323
1343
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1496335
https://epubs.siam.org/doi/abs/10.1137/22M1496335?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Electronic Observables for Relaxed Bilayer Two-Dimensional Heterostructures in Momentum Space
https://epubs.siam.org/doi/abs/10.1137/21M1451208?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1344-1378, December 2023. <br/> Abstract. Momentum space transformations for incommensurate two-dimensional electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer–MacDonald model [Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 12233–12237]. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. First, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straightforward algorithm for writing the complex momentum space model from the original real space model. Second, we implement this for twisted bilayer graphene with mechanical relaxation effects included. We also analyze the convergence rates of the approximations and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer–MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1344-1378, December 2023. <br/> Abstract. Momentum space transformations for incommensurate two-dimensional electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer–MacDonald model [Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 12233–12237]. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. First, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straightforward algorithm for writing the complex momentum space model from the original real space model. Second, we implement this for twisted bilayer graphene with mechanical relaxation effects included. We also analyze the convergence rates of the approximations and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer–MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.
Electronic Observables for Relaxed Bilayer Two-Dimensional Heterostructures in Momentum Space
10.1137/21M1451208
Multiscale Modeling & Simulation
2023-10-11T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Daniel Massatt
Stephen Carr
Mitchell Luskin
Electronic Observables for Relaxed Bilayer Two-Dimensional Heterostructures in Momentum Space
21
4
1344
1378
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/21M1451208
https://epubs.siam.org/doi/abs/10.1137/21M1451208?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
-
Elastic Far-Field Decay from Dislocations in Multilattices
https://epubs.siam.org/doi/abs/10.1137/22M1502021?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1379-1409, December 2023. <br/> Abstract. We precisely and rigorously characterize the decay of elastic fields generated by dislocations in crystalline materials, focusing specifically on the role of multilattices. Concretely, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by a linearized Cauchy–Born elasticity theory, and a discrete and nonlinear core corrector representing the defect core. We demonstrate both analytically and numerically the consequences of this result for cell size effects in numerical simulations.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1379-1409, December 2023. <br/> Abstract. We precisely and rigorously characterize the decay of elastic fields generated by dislocations in crystalline materials, focusing specifically on the role of multilattices. Concretely, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by a linearized Cauchy–Born elasticity theory, and a discrete and nonlinear core corrector representing the defect core. We demonstrate both analytically and numerically the consequences of this result for cell size effects in numerical simulations.
Elastic Far-Field Decay from Dislocations in Multilattices
10.1137/22M1502021
Multiscale Modeling & Simulation
2023-10-12T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Derek Olson
Christoph Ortner
Yangshuai Wang
Lei Zhang
Elastic Far-Field Decay from Dislocations in Multilattices
21
4
1379
1409
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1502021
https://epubs.siam.org/doi/abs/10.1137/22M1502021?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
-
An Effective Fractional Paraxial Wave Equation for Wave-Fronts in Randomly Layered Media with Long-Range Correlations
https://epubs.siam.org/doi/abs/10.1137/22M1525594?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1410-1456, December 2023. <br/> Abstract. This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O’Doherty–Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers–Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1410-1456, December 2023. <br/> Abstract. This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O’Doherty–Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers–Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations.
An Effective Fractional Paraxial Wave Equation for Wave-Fronts in Randomly Layered Media with Long-Range Correlations
10.1137/22M1525594
Multiscale Modeling & Simulation
2023-10-18T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Christophe Gomez
An Effective Fractional Paraxial Wave Equation for Wave-Fronts in Randomly Layered Media with Long-Range Correlations
21
4
1410
1456
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1525594
https://epubs.siam.org/doi/abs/10.1137/22M1525594?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Neural Network Approximation of Coarse-Scale Surrogates in Numerical Homogenization
https://epubs.siam.org/doi/abs/10.1137/22M1524278?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1457-1485, December 2023. <br/> Abstract. Coarse-scale surrogate models in the context of numerical homogenization of linear elliptic problems with arbitrary rough diffusion coefficients rely on the efficient solution of fine-scale subproblems on local subdomains whose solutions are then employed to deduce appropriate coarse contributions to the surrogate model. However, in the absence of periodicity and scale separation, the reliability of such models requires the local subdomains to cover the whole domain which may result in high offline costs, in particular for parameter-dependent and stochastic problems. This paper justifies the use of neural networks for the approximation of coarse-scale surrogate models by analyzing their approximation properties. For a prototypical and representative numerical homogenization technique, the Localized Orthogonal Decomposition method, we show that one single neural network is sufficient to approximate the coarse contributions of all occurring coefficient-dependent local subproblems for a nontrivial class of diffusion coefficients up to arbitrary accuracy. We present rigorous upper bounds on the depth and number of nonzero parameters for such a network to achieve a given accuracy. Further, we analyze the overall error of the resulting neural network enhanced numerical homogenization surrogate model.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1457-1485, December 2023. <br/> Abstract. Coarse-scale surrogate models in the context of numerical homogenization of linear elliptic problems with arbitrary rough diffusion coefficients rely on the efficient solution of fine-scale subproblems on local subdomains whose solutions are then employed to deduce appropriate coarse contributions to the surrogate model. However, in the absence of periodicity and scale separation, the reliability of such models requires the local subdomains to cover the whole domain which may result in high offline costs, in particular for parameter-dependent and stochastic problems. This paper justifies the use of neural networks for the approximation of coarse-scale surrogate models by analyzing their approximation properties. For a prototypical and representative numerical homogenization technique, the Localized Orthogonal Decomposition method, we show that one single neural network is sufficient to approximate the coarse contributions of all occurring coefficient-dependent local subproblems for a nontrivial class of diffusion coefficients up to arbitrary accuracy. We present rigorous upper bounds on the depth and number of nonzero parameters for such a network to achieve a given accuracy. Further, we analyze the overall error of the resulting neural network enhanced numerical homogenization surrogate model.
Neural Network Approximation of Coarse-Scale Surrogates in Numerical Homogenization
10.1137/22M1524278
Multiscale Modeling & Simulation
2023-10-20T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Fabian Kröpfl
Roland Maier
Daniel Peterseim
Neural Network Approximation of Coarse-Scale Surrogates in Numerical Homogenization
21
4
1457
1485
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1524278
https://epubs.siam.org/doi/abs/10.1137/22M1524278?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts
https://epubs.siam.org/doi/abs/10.1137/23M1550906?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1486-1501, December 2023. <br/> Abstract. We study the quantitative homogenization of linear second order elliptic equations in nondivergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called centered setting where homogenization occurs and the large drifts contribute to the effective diffusivity. Using the centering condition and the invariant measures associated with the underlying diffusion process, we transform the equation into divergence form with modified diffusion coefficients but without drift. The latter is in the standard setting for which quantitative homogenization results have been developed systematically. An application of those results then yields quantitative estimates, such as the convergence rates and uniform Lipschitz regularity, for equations in nondivergence form with large drifts.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1486-1501, December 2023. <br/> Abstract. We study the quantitative homogenization of linear second order elliptic equations in nondivergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called centered setting where homogenization occurs and the large drifts contribute to the effective diffusivity. Using the centering condition and the invariant measures associated with the underlying diffusion process, we transform the equation into divergence form with modified diffusion coefficients but without drift. The latter is in the standard setting for which quantitative homogenization results have been developed systematically. An application of those results then yields quantitative estimates, such as the convergence rates and uniform Lipschitz regularity, for equations in nondivergence form with large drifts.
On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts
10.1137/23M1550906
Multiscale Modeling & Simulation
2023-10-20T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Wenjia Jing
Yiping Zhang
On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts
21
4
1486
1501
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1550906
https://epubs.siam.org/doi/abs/10.1137/23M1550906?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
-
Homogenization and Dimension Reduction of the Stokes Problem with Navier-Slip Condition in Thin Perforated Layers
https://epubs.siam.org/doi/abs/10.1137/22M1528860?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1502-1533, December 2023. <br/> Abstract. We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: (1) dimensional reduction of the layer and (2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter [math], which is related to the thickness of the layer as well as the size of the periodic structure. By letting [math] tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit velocity and limit pressure satisfy a two pressure Stokes model, from which a Darcy law for thin layers can be derived. Due to nonstandard boundary conditions, some additional terms appear in Darcy’s law.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1502-1533, December 2023. <br/> Abstract. We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: (1) dimensional reduction of the layer and (2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter [math], which is related to the thickness of the layer as well as the size of the periodic structure. By letting [math] tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit velocity and limit pressure satisfy a two pressure Stokes model, from which a Darcy law for thin layers can be derived. Due to nonstandard boundary conditions, some additional terms appear in Darcy’s law.
Homogenization and Dimension Reduction of the Stokes Problem with Navier-Slip Condition in Thin Perforated Layers
10.1137/22M1528860
Multiscale Modeling & Simulation
2023-10-24T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
John Fabricius
Markus Gahn
Homogenization and Dimension Reduction of the Stokes Problem with Navier-Slip Condition in Thin Perforated Layers
21
4
1502
1533
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1528860
https://epubs.siam.org/doi/abs/10.1137/22M1528860?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Large Deviation Principle and Thermodynamic Limit of Chemical Master Equation via Nonlinear Semigroup
https://epubs.siam.org/doi/abs/10.1137/22M1505633?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1534-1569, December 2023. <br/> Abstract. Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan’s discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton–Jacobi equations (HJE). The discrete Hamiltonian is an m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation and the backward equation with “no reaction” boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan’s discrete nonlinear semigroup to the continuous Lax–Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1534-1569, December 2023. <br/> Abstract. Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan’s discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton–Jacobi equations (HJE). The discrete Hamiltonian is an m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation and the backward equation with “no reaction” boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan’s discrete nonlinear semigroup to the continuous Lax–Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.
Large Deviation Principle and Thermodynamic Limit of Chemical Master Equation via Nonlinear Semigroup
10.1137/22M1505633
Multiscale Modeling & Simulation
2023-10-24T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Yuan Gao
Jian-Guo Liu
Large Deviation Principle and Thermodynamic Limit of Chemical Master Equation via Nonlinear Semigroup
21
4
1534
1569
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1505633
https://epubs.siam.org/doi/abs/10.1137/22M1505633?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
-
FMM-LU: A Fast Direct Solver for Multiscale Boundary Integral Equations in Three Dimensions
https://epubs.siam.org/doi/abs/10.1137/22M1514040?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1570-1601, December 2023. <br/> Abstract. We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an [math]-like hierarchical factorization of the dense system matrix, permitting application of the inverse in [math] time, where [math] is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore, it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green’s functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies, and provide rigorous justification for compression of submatrices via proxy surfaces.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1570-1601, December 2023. <br/> Abstract. We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an [math]-like hierarchical factorization of the dense system matrix, permitting application of the inverse in [math] time, where [math] is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore, it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green’s functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies, and provide rigorous justification for compression of submatrices via proxy surfaces.
FMM-LU: A Fast Direct Solver for Multiscale Boundary Integral Equations in Three Dimensions
10.1137/22M1514040
Multiscale Modeling & Simulation
2023-11-03T07:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Daria Sushnikova
Leslie Greengard
Michael O’Neil
Manas Rachh
FMM-LU: A Fast Direct Solver for Multiscale Boundary Integral Equations in Three Dimensions
21
4
1570
1601
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1514040
https://epubs.siam.org/doi/abs/10.1137/22M1514040?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Extending the Regime of Linear Response with Synthetic Forcings
https://epubs.siam.org/doi/abs/10.1137/23M1557611?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1602-1643, December 2023. <br/> Abstract. Transport coefficients, such as the mobility, thermal conductivity, and shear viscosity, are quantities of prime interest in statistical physics. At the macroscopic level, transport coefficients relate an external forcing of magnitude [math], with [math], acting on the system to an average response expressed through some steady-state flux. In practice, steady-state averages involved in the linear response are computed as time averages over a realization of some stochastic differential equation. Variance reduction techniques are of paramount interest in this context, as the linear response is scaled by a factor of [math], leading to large statistical error. One way to limit the increase in the variance is to allow for larger values of [math] by increasing the range of values of the forcing for which the nonlinear part of the response is sufficiently small. In theory, one can add an extra forcing to the physical perturbation of the system, called synthetic forcing, as long as this extra forcing preserves the invariant measure of the reference system. The aim is to find synthetic perturbations allowing one to reduce the nonlinear part of the response as much as possible. We present a mathematical framework for quantifying the quality of synthetic forcings, in the context of linear response theory, and discuss various possible choices for them. Our findings are illustrated with numerical results in low-dimensional systems.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1602-1643, December 2023. <br/> Abstract. Transport coefficients, such as the mobility, thermal conductivity, and shear viscosity, are quantities of prime interest in statistical physics. At the macroscopic level, transport coefficients relate an external forcing of magnitude [math], with [math], acting on the system to an average response expressed through some steady-state flux. In practice, steady-state averages involved in the linear response are computed as time averages over a realization of some stochastic differential equation. Variance reduction techniques are of paramount interest in this context, as the linear response is scaled by a factor of [math], leading to large statistical error. One way to limit the increase in the variance is to allow for larger values of [math] by increasing the range of values of the forcing for which the nonlinear part of the response is sufficiently small. In theory, one can add an extra forcing to the physical perturbation of the system, called synthetic forcing, as long as this extra forcing preserves the invariant measure of the reference system. The aim is to find synthetic perturbations allowing one to reduce the nonlinear part of the response as much as possible. We present a mathematical framework for quantifying the quality of synthetic forcings, in the context of linear response theory, and discuss various possible choices for them. Our findings are illustrated with numerical results in low-dimensional systems.
Extending the Regime of Linear Response with Synthetic Forcings
10.1137/23M1557611
Multiscale Modeling & Simulation
2023-11-14T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Renato Spacek
Gabriel Stoltz
Extending the Regime of Linear Response with Synthetic Forcings
21
4
1602
1643
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1557611
https://epubs.siam.org/doi/abs/10.1137/23M1557611?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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High-order Contrast Bounds for Piezoelectric Constants of Two-phase Fibrous Composites
https://epubs.siam.org/doi/abs/10.1137/23M1559907?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1644-1666, December 2023. <br/> Abstract. The constructive theory of analytical higher-order contrast bounds for the effective constants of dispersed conducting and piezoelectric fibrous composites is developed. The lower-order bounds, e.g., Wiener and Hashin–Shtrikman bounds, are universal for composites but do not take into account interactions among inclusions corresponding to their location. To study the variety of dispersed random composites, we use computationally effective structural sums directly relating the location of inclusions to the effective constants. The present paper is the first report where the structural sums are applied to higher-order contrast bounds instead of the virtually impossible in computation multipoint correlation functions. We concentrate our attention on two-phase conducting fibrous composites. Rylko’s matrix decomposition is used for the higher-order contrast bounds to extend the obtained analytical bounds to piezoelectric fibrous composites. The supplementary materials contain the results of numerical-symbolic computations, the long analytical formulas for the effective constants and bounds up to [math], where [math] stands for concentration.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1644-1666, December 2023. <br/> Abstract. The constructive theory of analytical higher-order contrast bounds for the effective constants of dispersed conducting and piezoelectric fibrous composites is developed. The lower-order bounds, e.g., Wiener and Hashin–Shtrikman bounds, are universal for composites but do not take into account interactions among inclusions corresponding to their location. To study the variety of dispersed random composites, we use computationally effective structural sums directly relating the location of inclusions to the effective constants. The present paper is the first report where the structural sums are applied to higher-order contrast bounds instead of the virtually impossible in computation multipoint correlation functions. We concentrate our attention on two-phase conducting fibrous composites. Rylko’s matrix decomposition is used for the higher-order contrast bounds to extend the obtained analytical bounds to piezoelectric fibrous composites. The supplementary materials contain the results of numerical-symbolic computations, the long analytical formulas for the effective constants and bounds up to [math], where [math] stands for concentration.
High-order Contrast Bounds for Piezoelectric Constants of Two-phase Fibrous Composites
10.1137/23M1559907
Multiscale Modeling & Simulation
2023-11-27T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Vladimir Mityushev
High-order Contrast Bounds for Piezoelectric Constants of Two-phase Fibrous Composites
21
4
1644
1666
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1559907
https://epubs.siam.org/doi/abs/10.1137/23M1559907?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Preconditioned Algorithm for Difference of Convex Functions with Applications to Graph Ginzburg–Landau Model
https://epubs.siam.org/doi/abs/10.1137/23M1561270?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1667-1689, December 2023. <br/> Abstract. In this work, we propose and study a preconditioned framework with a graphic Ginzburg–Landau functional for image segmentation and data clustering by parallel computing. Solving nonlocal models is usually challenging due to the huge computation burden. For the nonconvex and nonlocal variational functional, we propose several damped Jacobi and generalized Richardson preconditioners for the large-scale linear systems within a difference of convex functions algorithm framework. These preconditioners are efficient for parallel computing with GPU and can leverage the computational cost. Our framework also provides flexible step sizes with a global convergence guarantee. Numerical experiments show the proposed algorithms are very competitive compared to the singular value decomposition based spectral method.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1667-1689, December 2023. <br/> Abstract. In this work, we propose and study a preconditioned framework with a graphic Ginzburg–Landau functional for image segmentation and data clustering by parallel computing. Solving nonlocal models is usually challenging due to the huge computation burden. For the nonconvex and nonlocal variational functional, we propose several damped Jacobi and generalized Richardson preconditioners for the large-scale linear systems within a difference of convex functions algorithm framework. These preconditioners are efficient for parallel computing with GPU and can leverage the computational cost. Our framework also provides flexible step sizes with a global convergence guarantee. Numerical experiments show the proposed algorithms are very competitive compared to the singular value decomposition based spectral method.
Preconditioned Algorithm for Difference of Convex Functions with Applications to Graph Ginzburg–Landau Model
10.1137/23M1561270
Multiscale Modeling & Simulation
2023-12-04T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Xinhua Shen
Hongpeng Sun
Xuecheng Tai
Preconditioned Algorithm for Difference of Convex Functions with Applications to Graph Ginzburg–Landau Model
21
4
1667
1689
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1561270
https://epubs.siam.org/doi/abs/10.1137/23M1561270?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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Analysis and Simulation of Optimal Control for a Two-Time-Scale Fractional Advection-Diffusion-Reaction Equation with Space-Time-Dependent Order and Coefficients
https://epubs.siam.org/doi/abs/10.1137/23M1573537?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1690-1716, December 2023. <br/> Abstract. We investigate an optimal control model with pointwise constraints governed by a two-time-scale time-fractional advection-diffusion-reaction equation with space-time-dependent fractional order and coefficients, which describes, e.g., the contaminant in groundwater under various transport scales or miscible displacement of hydrocarbon by injected fluid through heterogeneous porous media. To accommodate for the effects of complex fractional order and coefficients, an auxiliary equation method is proposed, along with the Fredholm alternative for compact operators, to analyze the well-posedness of the state equation. Additionally, a bootstrapping argument is utilized to progressively improve the solution regularity through a carefully designed pathway, leading to the maximal regularity estimates. Subsequently, we analyze the adjoint equation derived from the first-order optimality condition, which requires more subtle treatments due to the presence of hidden-memory variable-order fractional operators. Based on these findings, we ultimately analyze the well-posedness, first-order optimality conditions and maximal regularity estimates for the optimal control problem, and we conduct numerical experiments to investigate its behavior in potential applications.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1690-1716, December 2023. <br/> Abstract. We investigate an optimal control model with pointwise constraints governed by a two-time-scale time-fractional advection-diffusion-reaction equation with space-time-dependent fractional order and coefficients, which describes, e.g., the contaminant in groundwater under various transport scales or miscible displacement of hydrocarbon by injected fluid through heterogeneous porous media. To accommodate for the effects of complex fractional order and coefficients, an auxiliary equation method is proposed, along with the Fredholm alternative for compact operators, to analyze the well-posedness of the state equation. Additionally, a bootstrapping argument is utilized to progressively improve the solution regularity through a carefully designed pathway, leading to the maximal regularity estimates. Subsequently, we analyze the adjoint equation derived from the first-order optimality condition, which requires more subtle treatments due to the presence of hidden-memory variable-order fractional operators. Based on these findings, we ultimately analyze the well-posedness, first-order optimality conditions and maximal regularity estimates for the optimal control problem, and we conduct numerical experiments to investigate its behavior in potential applications.
Analysis and Simulation of Optimal Control for a Two-Time-Scale Fractional Advection-Diffusion-Reaction Equation with Space-Time-Dependent Order and Coefficients
10.1137/23M1573537
Multiscale Modeling & Simulation
2023-12-06T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Yiqun Li
Hong Wang
Xiangcheng Zheng
Analysis and Simulation of Optimal Control for a Two-Time-Scale Fractional Advection-Diffusion-Reaction Equation with Space-Time-Dependent Order and Coefficients
21
4
1690
1716
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1573537
https://epubs.siam.org/doi/abs/10.1137/23M1573537?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
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A Bending-Torsion Theory for Thin and Ultrathin Rods as a [math]-Limit of Atomistic Models
https://epubs.siam.org/doi/abs/10.1137/22M1517640?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1717-1745, December 2023. <br/> Abstract. The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as [math]-limits of three-dimensional atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness [math] and interatomic distance [math]. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibers ([math]), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where [math] and recover a nonlinear rod model—the modern version of Kirchhoff’s rod theory.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1717-1745, December 2023. <br/> Abstract. The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as [math]-limits of three-dimensional atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness [math] and interatomic distance [math]. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibers ([math]), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where [math] and recover a nonlinear rod model—the modern version of Kirchhoff’s rod theory.
A Bending-Torsion Theory for Thin and Ultrathin Rods as a [math]-Limit of Atomistic Models
10.1137/22M1517640
Multiscale Modeling & Simulation
2023-12-06T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Bernd Schmidt
Jiří Zeman
A Bending-Torsion Theory for Thin and Ultrathin Rods as a [math]-Limit of Atomistic Models
21
4
1717
1745
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/22M1517640
https://epubs.siam.org/doi/abs/10.1137/22M1517640?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics
-
Bottom-Up Transient Time Models in Coarse-Graining Molecular Systems
https://epubs.siam.org/doi/abs/10.1137/23M1548451?ai=s9&mi=3csssi&af=R
Multiscale Modeling &Simulation, <a href="https://epubs.siam.org/toc/mmsubt/21/4">Volume 21, Issue 4</a>, Page 1746-1774, December 2023. <br/> Abstract. This work presents a systematic methodology for describing the transient dynamics of coarse-grained molecular systems inferred from all-atom simulated data. We suggest Langevin-type dynamics where the coarse-grained interaction potential depends explicitly on time to efficiently approximate the transient coarse-grained dynamics. We apply the path-space force matching approach at the transient dynamics regime to learn the proposed model parameters. In particular, we parameterize the coarse-grained potential both with respect to the pair distance of the coarse-grained particles and the time, and we obtain an evolution model that is explicitly time dependent. Moreover, we follow a data-driven approach to estimate the friction kernel, given by appropriate correlation functions directly from the underlying all-atom molecular dynamics simulations. To explore and validate the proposed methodology we study a benchmark system of a moving particle in a box. We examine the suggested model’s effectiveness in terms of the system’s correlation time and find that the model can well approximate the transient time regime of the system, depending on the correlation time of the system. As a result, in the less correlated case, it can represent the dynamics for a longer time interval. We present an extensive study of our approach to a realistic high-dimensional water molecular system. Posing the water system initially out of thermal equilibrium we collect trajectories of all-atom data for the, empirically estimated, transient time regime. Then, we infer the suggested model and strengthen the model’s validity by comparing it with simplified Markovian models.
Multiscale Modeling & Simulation, Volume 21, Issue 4, Page 1746-1774, December 2023. <br/> Abstract. This work presents a systematic methodology for describing the transient dynamics of coarse-grained molecular systems inferred from all-atom simulated data. We suggest Langevin-type dynamics where the coarse-grained interaction potential depends explicitly on time to efficiently approximate the transient coarse-grained dynamics. We apply the path-space force matching approach at the transient dynamics regime to learn the proposed model parameters. In particular, we parameterize the coarse-grained potential both with respect to the pair distance of the coarse-grained particles and the time, and we obtain an evolution model that is explicitly time dependent. Moreover, we follow a data-driven approach to estimate the friction kernel, given by appropriate correlation functions directly from the underlying all-atom molecular dynamics simulations. To explore and validate the proposed methodology we study a benchmark system of a moving particle in a box. We examine the suggested model’s effectiveness in terms of the system’s correlation time and find that the model can well approximate the transient time regime of the system, depending on the correlation time of the system. As a result, in the less correlated case, it can represent the dynamics for a longer time interval. We present an extensive study of our approach to a realistic high-dimensional water molecular system. Posing the water system initially out of thermal equilibrium we collect trajectories of all-atom data for the, empirically estimated, transient time regime. Then, we infer the suggested model and strengthen the model’s validity by comparing it with simplified Markovian models.
Bottom-Up Transient Time Models in Coarse-Graining Molecular Systems
10.1137/23M1548451
Multiscale Modeling & Simulation
2023-12-06T08:00:00Z
© 2023 Society for Industrial and Applied Mathematics
Georgia Baxevani
Vagelis Harmandaris
Evangelia Kalligiannaki
Ivi Tsantili
Bottom-Up Transient Time Models in Coarse-Graining Molecular Systems
21
4
1746
1774
2023-12-31T08:00:00Z
2023-12-31T08:00:00Z
10.1137/23M1548451
https://epubs.siam.org/doi/abs/10.1137/23M1548451?ai=s9&mi=3csssi&af=R
© 2023 Society for Industrial and Applied Mathematics