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Multiscale Modeling & Simulation

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2010

Volume 8, Issue 5, pp. 1581-2097


A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems

Max Gunzburger and R. B. Lehoucq

Multiscale Model. Simul. 8, pp. 1581-1598 (18 pages) | Cited 2 times

Online Publication Date: September 07, 2010

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We develop a calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical vector calculus. The operators we define do not involve derivatives. We then apply the nonlocal calculus to define weak formulations of nonlocal “boundary-value” problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal problems, we derive a fundamental solution and Green's functions, demonstrate that weak formulations of the nonlocal “boundary-value” problems are well posed, and show how, under appropriate limits, the nonlocal problems reduce to their local analogues.

A Fast Multigrid Algorithm for Energy Minimization under Planar Density Constraints

Dorit Ron, Ilya Safro, and Achi Brandt

Multiscale Model. Simul. 8, pp. 1599-1620 (22 pages)

Online Publication Date: September 07, 2010

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The two-dimensional layout optimization problem reinforced by the efficient space utilization demand has a wide spectrum of practical applications. Formulating the problem as a nonlinear minimization problem under planar equality and/or inequality density constraints, we present a linear time multigrid algorithm for solving a correction to this problem. The method is demonstrated in various graph drawing (visualization) instances.

Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces

Juan Galvis and Yalchin Efendiev

Multiscale Model. Simul. 8, pp. 1621-1644 (24 pages) | Cited 1 time

Online Publication Date: September 16, 2010

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In this paper, robust preconditioners for multiscale flow problems are investigated. We consider elliptic equations with highly varying coefficients. We design and analyze two-level domain decomposition preconditioners that converge independent of the contrast in the media properties. The coarse spaces are constructed using selected eigenvectors of a local spectral problem. Our new construction enriches any given initial coarse space to make it suitable for high-contrast problems. Using the initial coarse space we construct local mass matrices for the local eigenvalue problems. We show that there is a gap in the spectrum of the eigenvalue problem when high-conductivity regions are disconnected. The eigenvectors corresponding to small, asymptotically vanishing eigenvalues are chosen to construct an enrichment of the initial coarse space. Only via a judicious choice of the initial space do we reduce the dimension of the resulting coarse space. Classical coarse basis functions such as multiscale or energy minimizing basis functions can be taken as the basis for the initial coarse space. In particular, if we start with classical multiscale basis, the selected eigenvectors represent only high-conductivity features that cannot be localized within a coarse-grid block, e.g., high-conductivity channels that connect the boundaries of a coarse-grid block. Numerical experiments are presented. The new construction presented here can handle tensor coefficients. The results of this paper substantially extend those presented in [J. Galvis and Y. Efendiev, Multiscale Model. Simul., 8 (2010), pp. 1461–1483], where only scalar coefficients are considered and the coarse space dimension can be very large because the coarse space includes all isolated high-conductivity features that are within a coarse block.

Fractional Brownian Vector Fields

Pouya Dehghani Tafti and Michael Unser

Multiscale Model. Simul. 8, pp. 1645-1670 (26 pages) | Cited 1 time

Online Publication Date: September 16, 2010

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This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel'fand and Vilenkin's stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the general case. These random vector fields result from the transformation of white noise by a special operator whose invariance properties the random field inherits. The said operator combines an inverse fractional Laplacian with a Helmholtz-like decomposition and weighted recombination. Classical fBm's can be obtained by balancing the weights of the Helmholtz components. The introduced random fields exhibit several important properties that are discussed in this paper. In addition, the proposed scheme yields a natural extension of the definition to Hurst exponents greater than one.

Efficiency of Multiscale Hybrid Grid-Particle Vortex Methods

M. El Ossmani and P. Poncet

Multiscale Model. Simul. 8, pp. 1671-1690 (20 pages)

Online Publication Date: September 23, 2010

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This article presents a study of computational cost for the vortex method, a Lagrangian numerical scheme using particles of fluids for simulation of three-dimensional flows. Its main features are to compute accurately transport effects and to be very robust, that is to say, often without prohibitive stability condition. Special attention is given to hybrid grid-particle vortex methods. They are shown to scale as $\mathcal{O}(n\log n)$, even when used in a multiscale context. Furthermore, a discussion is provided on the best strategies for simulations in complex geometry. Computational cost is shown to have the same efficiency when performing multiscale simulations of three-dimensional flows in complex geometry.

Incorporating Active Transport of Cellular Cargo in Stochastic Mesoscopic Models of Living Cells

Andreas Hellander and Per Lötstedt

Multiscale Model. Simul. 8, pp. 1691-1714 (24 pages)

Online Publication Date: September 23, 2010

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We propose a new multiscale method to incorporate active transport of cargo particles in biological cells in stochastic, mesoscopic models of reaction-transport processes. Given a discretization of the computational domain, we find stochastic, convective mesoscopic molecular fluxes over the edges or facets of the subvolumes and relate the process to a corresponding first order finite volume discretization of the linear convection equation. We give an example of how this can be used to model active transport of cargo particles on a microtubule network by the motor proteins kinesin and dynein. In this way we extend mesoscopic reaction-diffusion models of biochemical reaction networks to more general models of molecular transport within the living cell.

A Variational Model for Infinite Perimeter Segmentations Based on Lipschitz Level Set Functions: Denoising while Keeping Finely Oscillatory Boundaries

M. Barchiesi, S. H. Kang, T. M. Le, M. Morini, and M. Ponsiglione

Multiscale Model. Simul. 8, pp. 1715-1741 (27 pages)

Online Publication Date: September 29, 2010

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We propose a new model for segmenting piecewise constant images with irregular object boundaries: a variant of the Chan–Vese model [T. F. Chan and L. A. Vese, IEEE Trans. Image Process., 10 (2000), pp. 266–277], where the length penalization of the boundaries is replaced by the area of their neighborhood of thickness $\varepsilon$. Our aim is to keep fine details and irregularities of the boundaries while denoising additive Gaussian noise. For the numerical computation we revisit the classical $BV$ level set formulation [S. Osher and J. A. Sethian, J. Comput. Phys., 79 (1988), pp. 12–49] considering suitable Lipschitz level set functions instead of $BV$ ones.

Singular Limits of Klein–Gordon–Schrödinger Equations to Schrödinger–Yukawa Equations

Weizhu Bao, Xuanchun Dong, and Shu Wang

Multiscale Model. Simul. 8, pp. 1742-1769 (28 pages)

Online Publication Date: October 07, 2010

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In this paper, we study analytically and numerically the singular limits of the nonlinear Klein–Gordon–Schrödinger (KGS) equations in $\mathbb{R}^d$ ($d=1,2,3$) both with and without a damping term to the nonlinear Schrödinger–Yukawa (SY) equations. By using the two-scale matched asymptotic expansion, formal limits of the solution of the KGS equations to the solution of the SY equations are derived with an additional correction in the initial layer. Then for general initial data, weak and strong convergence results are established for the formal limits to provide rigorous mathematical justification for the matched asymptotic approximation by using the weak compactness argument and the (modulated) energy method, respectively. In addition, for well-prepared initial data, optimal quadratic and linear convergence rates are obtained for the KGS equations both with and without the damping term, respectively, and for ill-prepared initial data, the optimal linear convergence rate is obtained. Finally, numerical results for the KGS equations are presented to confirm the asymptotic and analytic results.

Convergence of the Heterogeneous Multiscale Finite Element Method for Elliptic Problems with Nonsmooth Microstructures

Rui Du and Pingbing Ming

Multiscale Model. Simul. 8, pp. 1770-1783 (14 pages) | Cited 2 times

Online Publication Date: October 07, 2010

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We propose a condition under which the heterogeneous multiscale finite element method converges for elliptic problems with nonsmooth coefficients, and we obtain the optimal convergence rate for elliptic problems with nonsymmetric periodic coefficients that allow for nonsmooth microstructures.

Clustering and Classification through Normalizing Flows in Feature Space

J. P. Agnelli, M. Cadeiras, E. G. Tabak, C. V. Turner, and E. Vanden-Eijnden

Multiscale Model. Simul. 8, pp. 1784-1802 (19 pages)

Online Publication Date: October 14, 2010

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A unified variational methodology is developed for classification and clustering problems and is tested in the classification of tumors from gene expression data. It is based on fluid-like flows in feature space that cluster a set of observations by transforming them into likely samples from $p$ isotropic Gaussians, where $p$ is the number of classes sought. The methodology blurs the distinction between training and testing populations through the soft assignment of both to classes. The observations act as Lagrangian markers for the flows, comparatively active or passive depending on the current strength of the assignment to the corresponding class.

Fast Multiscale Gaussian Wavepacket Transforms and Multiscale Gaussian Beams for the Wave Equation

Jianliang Qian and Lexing Ying

Multiscale Model. Simul. 8, pp. 1803-1837 (35 pages) | Cited 1 time

Online Publication Date: October 14, 2010

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We introduce a new multiscale Gaussian beam method for the numerical solution of the wave equation with smooth variable coefficients. The first computational question addressed in this paper is how to generate a Gaussian beam representation from general initial conditions for the wave equation. We propose fast multiscale Gaussian wavepacket transforms and introduce a highly efficient algorithm for generating the multiscale beam representation for a general initial condition. Starting from this multiscale decomposition of initial data, we propose the multiscale Gaussian beam method for solving the wave equation. The second question is how to perform long time propagation. Based on this new initialization algorithm, we utilize a simple reinitialization procedure that regenerates the beam representation when the beams become too wide. Numerical results in one, two, and three dimensions illustrate the properties of the proposed algorithm. The methodology can be readily generalized to treat other wave propagation problems.

Probability Distance Based Compression of Hidden Markov Models

Hao Wu and Frank Noé

Multiscale Model. Simul. 8, pp. 1838-1861 (24 pages)

Online Publication Date: November 09, 2010

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Large-scale stochastic models are relevant in many different fields such as computational biology, finance, social sciences, communication, and traffic networks. In order to both efficiently simulate and analyze such models and to understand the essential properties of the system, it is desirable to have model reduction techniques that much reduce the dimensionality of the model while at the same time preserving the system's essential dynamical properties. In this paper, a general model reduction technique for the class of discrete space and time hidden Markov models is presented, thereby also including the more special class of discrete Markov chains. The method is illustrated on some model applications.

Homogenization of Dielectric Photonic Quasi Crystals

Guy Bouchitté, Sébastien Guenneau, and Frédéric Zolla

Multiscale Model. Simul. 8, pp. 1862-1881 (20 pages)

Online Publication Date: November 09, 2010

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We adapt two-scale convergence to the homogenization of photonic quasi-periodic structures such as Penrose tilings. This convergence relies upon the irrational nature of a parameter characterizing a quasi-crystalline phase through its associated cut-and-projection matrix of permittivity: We generate quasi crystals by considering a periodic structure in an upper-dimensional space. We apply this tool to the homogenization of the vector Maxwell system.

Simplified Modelling of a Thermal Bath, with Application to a Fluid Vortex System

Svetlana Dubinkina, Jason Frank, and Ben Leimkuhler

Multiscale Model. Simul. 8, pp. 1882-1901 (20 pages)

Online Publication Date: November 16, 2010

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Based on the thermodynamic concept of a reservoir, we investigate a computational model for interaction with unresolved degrees of freedom (a thermal bath). We assume that a finite restricted system can be modelled by a generalized canonical ensemble, described by a density which is a smooth function of the energy of the restricted system. A thermostat is constructed to continuously perturb the resolved dynamics, while leaving the desired equilibrium distribution invariant. We build on a thermostatting framework developed and tested in the setting of molecular dynamics, using stochastic perturbations to control (and stabilize) the invariant measure. We also apply these techniques in the setting of a simplified point vortex flow on a disc, in which a modified Gibbs distribution (modelling a finite, rather than infinite, bath of weak vortices) provides a regularizing formulation for restricted system dynamics. Numerical experiments, effectively replacing many vortices by a few artificial degrees of freedom, are in excellent agreement with the two-scale simulations of Bühler [Phys. Fluids, 14 (2002), pp. 2139–2149].

Justification of the Cavity Model in the Numerical Simulation of Patch Antennas by the Method of Matched Asymptotic Expansions

Abderrahmane Bendali, Abdelkader Makhlouf, and Sébastien Tordeux

Multiscale Model. Simul. 8, pp. 1902-1922 (21 pages)

Online Publication Date: December 02, 2010

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The cavity model is a widespread empirical approach for the numerical simulation of patch antennas. An attempt to give a rigorous mathematical background for this way to proceed is presented. The justification is carried out in the framework of a two-dimensional representation of the underlying radiation problem. It is obtained by a suitable application of the method of matched asymptotic expansions. Furthermore, it is shown how to improve the cavity model by pushing the asymptotic expansion to the next order. A remarkable outcome of the asymptotic expansions is that they clearly confirm that the way to feed the antenna only determines the level of its radiation pattern but not its shape. Some numerical experiments are given to illustrate the theoretical developments.

Gyrokinetic Vlasov Equation in Three Dimensional Setting. Second Order Approximation

Mihai Bostan

Multiscale Model. Simul. 8, pp. 1923-1957 (35 pages) | Cited 1 time

Online Publication Date: December 02, 2010

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One of the main applications in plasma physics concerns the energy production through thermonuclear fusion. The controlled fusion requires the confinement of the plasma into a bounded domain, and for this we appeal to the magnetic confinement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large stationary inhomogeneous magnetic fields. The first order corrections are computed as well: electric cross field drift, magnetic gradient drift, magnetic curvature drift, etc. The mathematical analysis relies on average techniques and ergodicity.

Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates

Lei Zhang, Leonid Berlyand, Maxim V. Fedorov, and Houman Owhadi

Multiscale Model. Simul. 8, pp. 1958-1980 (23 pages)

Online Publication Date: December 07, 2010

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This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. Most atomistic-to-continuum methods are constrained by a crystalline order or a periodic setting and therefore cannot be directly applied to modeling of soft matter. By contrast, the energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. In this paper we use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations.

Source Localization in Random Acoustic Waveguides

Liliana Borcea, Leila Issa, and Chrysoula Tsogka

Multiscale Model. Simul. 8, pp. 1981-2022 (42 pages)

Online Publication Date: December 07, 2010

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Mode coupling due to scattering by weak random inhomogeneities in waveguides leads to loss of coherence of wave fields at long distances of propagation. This in turn leads to serious deterioration of coherent source localization methods, such as matched field. We study with analysis and numerical simulations how such deterioration occurs and introduce a novel incoherent approach for long range source localization in random waveguides. It is based on a special form of transport theory for the incoherent fluctuations of the wave field. We study theoretically the statistical stability of the method and illustrate its performance with numerical simulations. We also show how it can be used to estimate the correlation function of the random fluctuations of the wave speed.

Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

B. Amaziane, S. Antontsev, L. Pankratov, and A. Piatnitski

Multiscale Model. Simul. 8, pp. 2023-2047 (25 pages)

Online Publication Date: December 16, 2010

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This paper is devoted to the homogenization of a coupled system of diffusion-convection equations in a domain with periodic microstructure, modeling the flow and transport of immiscible compressible, such as water-gas, fluids through porous media. The problem is formulated in terms of a nonlinear parabolic equation for the nonwetting phase pressure and a nonlinear degenerate parabolic diffusion-convection equation for the wetting saturation phase with rapidly oscillating porosity function and absolute permeability tensor. We obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using two-scale convergence. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

A Narrow Band Method for the Convex Formulation of Discrete Multilabel Problems

Antonio Baeza, Vicent Caselles, Pau Gargallo, and Nicolas Papadakis

Multiscale Model. Simul. 8, pp. 2048-2078 (31 pages)

Online Publication Date: December 16, 2010

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We study a narrow band type algorithm to solve a discrete formulation of the convex relaxation of energy functionals with total variation regularization and nonconvex data terms. We prove that this algorithm converges to a local minimum of the original nonlinear optimization problem. We illustrate the algorithm with experiments for disparity computation in stereo and a multilabel segmentation problem, and we check experimentally that the energy of the local minimum is very near to the energy of the global minimum obtained without the narrow band type method.

Markov Chain Stochastic Parametrizations of Essential Variables

K. Nimsaila and I. Timofeyev

Multiscale Model. Simul. 8, pp. 2079-2096 (18 pages)

Online Publication Date: December 21, 2010

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We analyze the performance of the novel Markov chain stochastic modeling technique for derivation of effective equations for a set of essential variables. This technique is an empirical approach where the right-hand side of the essential variables is modeled by a Markov chain. We demonstrate that the Markov chain modeling approach performs well in a prototype model without scale separation between the essential and the nonessential variables. Moreover, we utilize the truncated Burgers–Hopf model to show that the Markov chain should be properly conditioned on the essential variables to reproduce the structure of two-point statistical quantities. On the other hand, the conditioning can be rather straightforward and unsophisticated.

Erratum: P-Splines Using Derivative Information

Christopher P. Calderon, Josue G. Martinez, Raymond J. Carroll, and Danny C. Sorensen

Multiscale Model. Simul. 8, pp. 2097-2097 (1 page)

Online Publication Date: December 21, 2010

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This is a correction to the author's article [Multiscale Model. Simul., 8 (2010), pp. 1562–1580].
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