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2012

Volume 10, Issue 1, pp. 1-283


Importance Sampling for Multiscale Diffusions

Paul Dupuis, Konstantinos Spiliopoulos, and Hui Wang

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.

A Simple Linear Response Closure Approximation for Slow Dynamics of a Multiscale System with Linear Coupling

Rafail V. Abramov

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.

Numerical Computation of Solutions of the Critical Nonlinear Schrödinger Equation after the Singularity

Panos Stinis

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.

Estimating the Eigenvalue Error of Markov State Models

Natasa Djurdjevac, Marco Sarich, and Christof Schütte

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.

Buckling Instability of Viral Capsids—A Continuum Approach

Sebastian Aland, Andreas Rätz, Matthias Röger, and Axel Voigt

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.

Multiscale Discrete Approximation of Fourier Integral Operators

Fredrik Andersson, Maarten V. de Hoop, and Herwig Wendt

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.

Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics

David F. Anderson and Desmond J. Higham

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.

Optimal Distribution of the Nonoverlapping Conducting Disks

Vladimir Mityushev and Natalia Rylko

Multiscale Model. Simul. 10, pp. 180-190 (11 pages)

Online Publication Date: March 08, 2012

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Conducting nonoverlapping identical disks are embedded in a two-dimensional background. The set of disks is infinite. The disks are distributed in such a way that the obtained composite is macroscopically isotropic. Let the conductivity of inclusions be higher than the conductivity of the matrix. It is proved that the hexagonal (triangular) lattice of disks possess the minimal effective conductivity when the concentration is not high.

Nonequilibrium Shear Viscosity Computations with Langevin Dynamics

Rémi Joubaud and Gabriel Stoltz

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.

Coupled Wideangle Wave Approximations

Josselin Garnier and Knut Sølna

Multiscale Model. Simul. 10, pp. 217-244 (28 pages)

Online Publication Date: March 13, 2012

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In this paper we analyze wave propagation in three-dimensional random media. We consider a source with limited spatial and temporal support that generates spherically diverging waves. The waves propagate in a random medium whose fluctuations have small amplitude and correlation radius larger than the typical wavelength but smaller than the propagation distance. In a regime of separation of scales we prove that the wave is modified in two ways by the interaction with the random medium: first, its time profile is affected by a deterministic diffusive and dispersive convolution; second, the wave fronts are affected by random perturbations that can be described in terms of a Gaussian process whose amplitude is of the order of the wavelength and whose correlation radius is of the order of the correlation radius of the medium. Both effects depend on the two-point statistics of the random medium.

The AL Basis for the Solution of Elliptic Problems in Heterogeneous Media

L. Grasedyck, I. Greff, and S. Sauter

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.

An Efficient High Order Heterogeneous Multiscale Method for Elliptic Problems

Ruo Li, Pingbing Ming, and Fengyang Tang

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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