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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics Multiscale Model. Simul. 10, pp. 146-179 (34 pages) Online Publication Date: March 08, 2012
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We show how to extend a recently proposed multilevel Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the system to a specified accuracy. The extension is nontrivial, exploiting a coupling of the requisite processes that is easy to simulate while providing a small variance for the estimator. Further, and in a stark departure from other implementations of multilevel Monte Carlo, we show how to produce an unbiased estimator that is significantly less computationally expensive than the usual unbiased estimator arising from exact algorithms in conjunction with crude Monte Carlo. We thereby dramatically improve, in a quantifiable manner, the basic computational complexity of current approaches that have many names and variants across the scientific literature, including the Bortz–Kalos–Lebowitz algorithm, discrete event simulation, dynamic Monte Carlo, kinetic Monte Carlo, the n-fold way, the next reaction method, the residence-time algorithm, the stochastic simulation algorithm, Gillespie's algorithm, and tau-leaping. The new algorithm applies generically, but we also give an example where the coupling idea alone, even without a multilevel discretization, can be used to improve efficiency by exploiting system structure. Stochastically modeled chemical reaction networks provide a very important application for this work. Hence, we use this context for our notation, terminology, natural scalings, and computational examples. |
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Multiscale Model. Simul. 10, pp. 285-305 (21 pages) Online Publication Date: April 03, 2012
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We present modeling of the incompressible viscous flows in the domain containing unconfined fluid and a porous medium in the case when the flow in the unconfined domain dominates. For such a setting a rigorous derivation of the Beavers–Joseph–Saffman interface condition was undertaken by Jäger and Mikelić [SIAM J. Appl. Math., 60 (2000), pp. 1111–1127] using the homogenization method. So far the interface law for the pressure was conceived and confirmed only numerically. In this article we derive the Beavers and Joseph law for a general body force by estimating the pressure field approximation. Different from the Poiseuille flow case, the velocity approximation is not divergence-free and the precise pressure estimation is essential. This new estimate allows us to rigorously justify the pressure jump condition using the Navier boundary layer, already used to calculate the constant in the law by Beavers and Joseph. Finally, our results confirm that the position of the interface influences the solution only at the order of physical permeability and therefore the choice of this position does not pose problems. |
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Multiscale Developments of the Cellular Potts Model Multiscale Model. Simul. 10, pp. 342-382 (41 pages) Online Publication Date: April 12, 2012
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Multiscale problems are ubiquitous and fundamental in all biological phenomena that emerge naturally from the complex interaction of processes which occur at various levels. A number of both discrete and continuous mathematical models and methods have been developed to address such an intricate network of organization. One of the most suitable individual cell-based model for this purpose is the well-known cellular Potts model (CPM). The CPM is a discrete, lattice-based, flexible technique that is able to accurately identify and describe the phenomenological mechanisms which are responsible for innumerable biological (and nonbiological) phenomena. In this work, we first give a brief overview of its biophysical basis and discuss its main limitations. We then propose some innovative extensions, focusing on ways of integrating the basic mesoscopic CPM with accurate continuous models of microscopic dynamics of individuals. The aim is to create a multiscale hybrid framework that is able to deal with the typical multilevel organization of biological development, where the behavior of the simulated individuals is realistically driven by their internal state. Our CPM extensions are then tested with sample applications that show a qualitative and quantitative agreement with experimental data. Finally, we conclude by discussing further possible developments of the method. |
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An Efficient High Order Heterogeneous Multiscale Method for Elliptic Problems Multiscale Model. Simul. 10, pp. 259-283 (25 pages) Online Publication Date: March 29, 2012
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We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost as the linear macroscopic solver. Optimal error bounds are proved for the elliptic problem. Numerical results demonstrate that the new method significantly reduces the cost without loss of accuracy. |
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Importance Sampling for Multiscale Diffusions Multiscale Model. Simul. 10, pp. 1-27 (27 pages) Online Publication Date: January 24, 2012
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We construct importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. Standard Monte Carlo methods perform poorly for these problems in the small noise limit. With multiscale processes there are additional complications, and indeed the straightforward adaptation of importance sampling methods for standard small noise diffusions will not produce efficient schemes. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Examples and simulation results are provided. |
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Multiscale Model. Simul. 10, pp. 28-47 (20 pages) Online Publication Date: February 02, 2012
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Many applications of contemporary science involve multiscale dynamics, which are typically characterized by the time and space scale separation of patterns of motion, with fewer slowly evolving variables and a much larger set of faster evolving variables. This time-space scale separation causes direct numerical simulation of the evolution of the dynamics to be computationally expensive due to both the large number of variables and the necessity to choose a small discretization time step in order to resolve the fast components of dynamics. In this work we propose a simple method of determining the closed model for slow variables alone, which requires only a single computation of appropriate statistics for the fast dynamics with a certain fixed state of the slow variables. The method is based on the first-order Taylor expansion of the averaged coupling term with respect to the slow variables, which can be computed using the linear fluctuation-dissipation theorem. We show that, with simple linear coupling in both slow and fast variables, this method produces quite comparable statistics to what is exhibited by a complete two-scale model. The main advantage of the method is that it applies even when the statistics of the full multiscale model cannot be simulated due to computational complexity, which makes it practical for real-world large scale applications. |
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Multiscale Discrete Approximation of Fourier Integral Operators Multiscale Model. Simul. 10, pp. 111-145 (35 pages) Online Publication Date: March 08, 2012
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We develop a discretization and computational procedures for approximation of the action of Fourier integral operators the canonical relations of which are graphs. Such operators appear, for instance, in the formulation of imaging and inverse scattering of seismic reflection data. Our discretization and algorithms are based on a multiscale low-rank expansion of the action of Fourier integral operators using the dyadic parabolic decomposition of phase space and on explicit constructions of low-rank separated representations using prolate spheroidal wave functions, which directly reflect the geometry of such operators. The discretization and computational procedures connect to the discrete almost symmetric wave packet transform. Numerical wave propagation and imaging examples illustrate our computational procedures. |
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Multiscale Model. Simul. 8, pp. 1621-1644 (24 pages) 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. |
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Nonequilibrium Shear Viscosity Computations with Langevin Dynamics Multiscale Model. Simul. 10, pp. 191-216 (26 pages) Online Publication Date: March 13, 2012
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We study the mathematical properties of a nonequilibrium Langevin dynamics which can be used to estimate the shear viscosity of a system. More precisely, we prove a linear response result which allows us to relate averages over the nonequilibrium stationary state of the system to equilibrium canonical expectations. We then write a local conservation law for the average longitudinal velocity of the fluid and show how, under some closure approximation, the viscosity can be extracted from this profile. We finally characterize the asymptotic behavior of the velocity profile, in the limit where either the transverse or the longitudinal friction goes to infinity. Some numerical illustrations of the theoretical results are also presented. |
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Estimating the Eigenvalue Error of Markov State Models Multiscale Model. Simul. 10, pp. 61-81 (21 pages) Online Publication Date: February 28, 2012
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We consider a continuous-time, ergodic Markov process on a large continuous or discrete state space. The process is assumed to exhibit a number of metastable sets. Markov state models (MSMs) are designed to represent the effective dynamics of such a process by a Markov chain that jumps between the metastable sets with the transition rates of the original process. MSMs have been used for a number of applications, including molecular dynamics (cf. [F. Noé et al., Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 19011–19016]), for more than a decade. The rigorous and fully general (no zero temperature limit or comparable restrictions) analysis of their approximation quality, however, has only recently begun. Our first article on this topics [M. Sarich, F. Noé, and Ch. Schütte, Multiscale Model. Simul., 8 (2010), pp. 1154–1177] introduces an error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales. Herein we provide upper bounds for the error in the eigenvalues between the MSM and the original process, which means that we analyze how well the longest timescales in the original process are approximated by the MSM. Our findings are illustrated by numerical experiments. |
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Buckling Instability of Viral Capsids—A Continuum Approach Multiscale Model. Simul. 10, pp. 82-110 (29 pages) Online Publication Date: March 06, 2012
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The crystallographic structure of spherical viruses is modeled using a multiscale approach combining a macroscopic Helfrich model for morphology evolution with a microscopic approximation of a classical density functional theory for the protein interactions. The derivation of the model is based on energy dissipation and conservation of protein number density. The resulting set of equations is solved within a diffuse domain approach using finite elements and shows buckling transitions of spherical shapes into faceted viral shapes. |
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A Review of Image Denoising Algorithms, with a New One Multiscale Model. Simul. 4, pp. 490-530 (41 pages) Online Publication Date: July 26, 2006
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The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove image fine structures. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the "method noise," defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods are compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of L2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method. |
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Multiscale Model. Simul. 10, pp. 48-60 (13 pages) Online Publication Date: February 28, 2012
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We present numerical results for the solution of the one-dimensional critical nonlinear Schrödinger with periodic boundary conditions and initial data that give rise to a finite time singularity. We construct, through the Mori–Zwanzig formalism, a reduced model which allows us to follow the solution after the formation of the singularity. The computed postsingularity solution exhibits the same characteristics as the postsingularity solutions constructed recently by Terence Tao. |
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A Fast Total Variation Minimization Method for Image Restoration Multiscale Model. Simul. 7, pp. 774-795 (22 pages) Online Publication Date: August 06, 2008
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In this paper, we study a fast total variation minimization method for image restoration. In the proposed method, we use the modified total variation minimization scheme to denoise the deblurred image. An alternating minimization algorithm is employed to solve the proposed total variation minimization problem. Our experimental results show that the quality of restored images by the proposed method is competitive with those restored by the existing total variation restoration methods. We show the convergence of the alternating minimization algorithm and demonstrate that the algorithm is very efficient. |
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Long‐Time Protein Folding Dynamics from Short‐Time Molecular Dynamics Simulations Multiscale Model. Simul. 5, pp. 1214-1226 (13 pages) Online Publication Date: December 28, 2006
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Protein folding involves physical timescales—microseconds to seconds—that are too long to be studied directly by straightforward molecular dynamics simulation, where the fundamental timestep is constrained to femtoseconds. Here we show how the long‐time statistical dynamics of a simple solvated biomolecular system can be well described by a discrete‐state Markov chain model constructed from trajectories that are an order of magnitude shorter than the longest relaxation times of the system. This suggests that such models, appropriately constructed from short molecular dynamics simulations, may have utility in the study of long‐time conformational dynamics. |
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Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities Multiscale Model. Simul. 10, pp. 306-341 (36 pages) Online Publication Date: April 12, 2012
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The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevskiĭ (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the maximal growth rate and associated wavenumber of unstable perturbations and (2) the separatrix between convective and absolute instabilities. The instability properties of the dark soliton are directly related to those of oblique dispersive shock wave (DSW) solutions. Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS/GP equation. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media. |
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The AL Basis for the Solution of Elliptic Problems in Heterogeneous Media Multiscale Model. Simul. 10, pp. 245-258 (14 pages) Online Publication Date: March 13, 2012
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In this paper, we will show that, for elliptic problems in heterogeneous media, there exists a local (generalized) finite element basis (AL basis) consisting of $O\big( \big( \log\frac{1}{H}\big) ^{d+1}\big)$ basis functions per nodal point such that the convergence rates of the classical finite element method for Poisson-type problems are preserved. Here $H$ denotes the mesh width of the finite element mesh and $d$ is the spatial dimension. We provide several numerical examples beyond our theory, where even $O(1)$ basis functions per nodal point are sufficient to preserve the convergence rates. |
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Finite Element Heterogeneous Multiscale Method for the Wave Equation Multiscale Model. Simul. 9, pp. 766-792 (27 pages) Online Publication Date: June 29, 2011
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A finite element heterogeneous multiscale method is proposed for the wave equation with highly oscillatory coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational work involved is independent of the highly heterogeneous nature of the medium at the smallest scale. Optimal error estimates in the energy norm and the L2 norm and convergence to the homogenized solution are proved, when both the macro and the micro scales are refined simultaneously. Numerical experiments corroborate the theoretical convergence rates and illustrate the behavior of the numerical method for periodic and heterogeneous media. |
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A Hybrid Particle-Continuum Method for Hydrodynamics of Complex Fluids Multiscale Model. Simul. 8, pp. 871-911 (41 pages) Online Publication Date: April 02, 2010
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A previously developed hybrid particle-continuum method [J. B. Bell, A. Garcia, and S. A. Williams, Multiscale Model. Simul., 6 (2008), pp. 1256–1280] is generalized to dense fluids and two- and three-dimensional flows. The scheme couples an explicit fluctuating compressible Navier–Stokes solver with the isotropic direct simulation Monte Carlo (DSMC) particle method [A. Donev, A. L. Garcia, and B. J. Alder, J. Stat. Mech. Theory Exp., 2009 (2009), article P11008]. To achieve bidirectional dynamic coupling between the particle (microscale) and continuum (macroscale) regions, the continuum solver provides state-based boundary conditions to the particle subdomain, while the particle solver provides flux-based boundary conditions for the continuum subdomain. This type of coupling ensures both state and flux continuity across the particle-continuum interface analogous to coupling approaches for deterministic parabolic partial differential equations; here, when fluctuations are included, a small ($<1\%$) mismatch is expected and observed in the mean density and temperature across the interface. By calculating the dynamic structure factor for both a “bulk” (periodic) and a finite system, it is verified that the hybrid algorithm accurately captures the propagation of spontaneous thermal fluctuations across the particle-continuum interface. The equilibrium diffusive (Brownian) motion of a large spherical bead suspended in a particle fluid is examined, demonstrating that the hybrid method correctly reproduces the velocity autocorrelation function of the bead but only if thermal fluctuations are included in the continuum solver. Finally, the hybrid is applied to the well-known adiabatic piston problem, and it is found that the hybrid correctly reproduces the slow nonequilibrium relaxation of the piston toward thermodynamic equilibrium but, again, only if the continuum solver includes stochastic (white-noise) flux terms. These examples clearly demonstrate the need to include fluctuations in continuum solvers employed in hybrid multiscale methods. |
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Nonlocal Operators with Applications to Image Processing Multiscale Model. Simul. 7, pp. 1005-1028 (24 pages) Online Publication Date: November 12, 2008
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We propose the use of nonlocal operators to define new types of flows and functionals for image processing and elsewhere. A main advantage over classical PDE-based algorithms is the ability to handle better textures and repetitive structures. This topic can be viewed as an extension of spectral graph theory and the diffusion geometry framework to functional analysis and PDE-like evolutions. Some possible applications and numerical examples are given, as is a general framework for approximating Hamilton–Jacobi equations on arbitrary grids in high demensions, e.g., for control theory. |
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