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Multiscale Modeling & Simulation

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2011

Volume 9, Issue 4, pp. 1301-1865


Heterogeneous Multiscale Simulations of Suspension Flow

Eric Lorenz and Alfons G. Hoekstra

Multiscale Model. Simul. 9, pp. 1301-1326 (26 pages)

Online Publication Date: October 20, 2011

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The macroscopically emergent rheology of suspensions is dictated by details of fluid-particle and particle-particle interactions. For systems where the typical spatial scale on the particle level is much smaller than that of macroscopic properties, the scales can be split. We present a heterogeneous multiscale method (HMM) approach to modeling suspension flow in which at the macroscale the suspension is treated as a non-Newtonian fluid. The local shear-rate and particle volume fraction are input to a simulation of fully resolved suspension microdynamics. With the help of these simulations, the apparent viscosity and shear-induced diffusivities can be computed for a given shear-rate and volume fraction, and are then used to complete the information needed in the constitutional relations on the macroscopic level. On both levels, the lattice-Boltzmann method (LBM) is applied to model the fluid phase and coupled to a Lagrangian model for the advection-diffusion of the solid phase. Down and upward mapping of viscosity and diffusivity related quantities will be discussed, as well as information exchanged between the phases on both scales. Temporal scale splitting between viscous and diffusive dynamics has also been exploited to accelerate the macroscopic equilibration dynamics. Additionionally, Galileian and rotational symmetries allow us to make very efficient use of a database where the results of previous simulations can be stored, again reducing the computational effort by factors of several orders of magnitude. The HMM suspension model is applied to the simulation of a 2-dimensional flow through a straight channel of macroscopic width. The equilibration dynamics of flow and volume fraction profiles and equilibrium profiles of volume fraction, diffusivity, velocity, shear-rate, and viscosity are discussed. We show that the proposed HMM model not only reproduces experimental findings for low Reynolds numbers but also predicts additional dependencies introduced by shear-thickening effects not covered by existing macroscopic suspension flow models.

Weight Function in a Bimaterial Strip Containing an Interfacial Crack and an Imperfect Interface. Application to Bloch–Floquet Analysis in a Thin Inhomogeneous Structure with Cracks

A. Vellender, G. S. Mishuris, and A. B. Movchan

Multiscale Model. Simul. 9, pp. 1327-1349 (23 pages)

Online Publication Date: October 20, 2011

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We define a weight function in a bimaterial strip containing a semi-infinite crack and an imperfect interface and analyze a problem of antiplane shear. We then present an asymptotic algorithm that uses the weight function to evaluate the coefficients in asymptotics of solutions to problems of wave propagation in a thin bimaterial strip containing a periodic array of cracks situated at the interface between two materials.

Variational Multiscale Finite Element Method for Flows in Highly Porous Media

O. Iliev, R. Lazarov, and J. Willems

Multiscale Model. Simul. 9, pp. 1350-1372 (23 pages)

Online Publication Date: November 01, 2011

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We present a two-scale finite element method (FEM) for solving Brinkman’s and Darcy’s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes’ equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy’s equations. In order to reduce the “resonance error” and to ensure convergence to the global fine solution, the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.

Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast

Houman Owhadi and Lei Zhang

Multiscale Model. Simul. 9, pp. 1373-1398 (26 pages) | Cited 1 time

Online Publication Date: November 01, 2011

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We construct finite-dimensional approximations of solution spaces of divergence-form operators with L∞-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in H1 if source terms are in the unit ball of L2 instead of the unit ball of H−1. Approximation spaces are generated by solving elliptic PDEs on localized subdomains with source terms corresponding to approximation bases for H2. The H1-error estimates show that O(h−d)-dimensional spaces with basis elements localized to subdomains of diameter O(hα ln math) (with α ∊ [½,1)) result in an O(h2−2α) accuracy for elliptic, parabolic, and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved, provided that localized subdomains contain buffer zones of width O(hα ln math), where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

Joint-MAP Tomographic Reconstruction with Patch Similarity Based Mixture Prior Model

Yang Chen, Yinsheng Li, Weimin Yu, Limin Luo, Wufan Chen, and Christine Toumoulin

Multiscale Model. Simul. 9, pp. 1399-1419 (21 pages)

Online Publication Date: November 17, 2011

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Tomographic reconstruction from noisy projections do not yield adequate results. Mathematically, this tomographic reconstruction represents an ill-posed problem due to information missing caused by the presence of noise. Maximum a posteriori (MAP) or Bayesian reconstruction methods offer possibilities to improve the image quality as compared with analytical methods in particular by introducing a prior to guide the reconstruction and regularize the noise. With an aim to achieve robust utilization of continuity/connectivity information and overcome the heuristic weight update for other nonlocal prior methods, this paper proposes a novel patch similarity based mixture (PSM) prior model for tomographic reconstruction. This prior is defined by a weighted Gaussian distance between neighborhood intensities. The weight quantifies the similarity between local neighborhoods and is computed using a maximization entropy constraint. This prior is then introduced within a joint image/weight MAP computed tomography reconstruction algorithm. Several acceleration trials including Compute Unified Device Architecture (CUDA) parallelization is applied to alleviate the intensive patch distance computation involved in the joint algorithm. The method was tested with both synthetic phantoms and clinical computed tomography data and compared in accuracy with five other reconstruction algorithms which are filtered back-projection and Bayesian-based. Reconstruction results show that the proposed reconstructions are able to produce high-quality images with ensured iteration convergence.

A Computational Model of Cell Polarization and Motility Coupling Mechanics and Biochemistry

Ben Vanderlei, James J. Feng, and Leah Edelstein-Keshet

Multiscale Model. Simul. 9, pp. 1420-1443 (24 pages)

Online Publication Date: November 17, 2011

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The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in two dimensions, resides in the fact that the problem is inherently one of deforming, nonstationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed-boundary method to address this problem. We adopt a simple reaction-diffusion (RD) system that represents an internal regulatory mechanism controlling the polarization of a cell and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the RD system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.

Asymptotic Approximation for the Weight Function in a Solid with a Surface-Breaking Crack and Small Voids

M. J. Nieves, A. B. Movchan, and I. S. Jones

Multiscale Model. Simul. 9, pp. 1444-1458 (15 pages)

Online Publication Date: November 17, 2011

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We derive an asymptotic approximation for the weight function in an edge-cracked half-plane containing multiple holes. Simplified asymptotic formulas are also presented for the case when the weight function is evaluated along the face of the crack, and we show how this formula can be used to obtain asymptotics of Mode I stress intensity factors. Numerical experiments are given, which illustrate the effect of the perturbation of the weight function and stress intensity factor due to the presence of voids.

Coupling Molecular Dynamics and Continua with Weak Constraints

Konstantin Fackeldey, Dorian Krause, Rolf Krause, and Christoph Lenzen

Multiscale Model. Simul. 9, pp. 1459-1494 (36 pages)

Online Publication Date: November 17, 2011

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One of the most challenging problems in dynamic concurrent multiscale simulations is the reflectionless transfer of physical quantities between the different scales. In particular, when coupling molecular dynamics and finite element discretizations in solid body mechanics, often spurious wave reflections are introduced by the applied coupling technique. The reflected waves are typically of high frequency and are arguably of little importance in the domain where the finite element discretization drives the simulation. In this work, we provide an analysis of this phenomenon. Based on the gained insight, we derive a new coupling approach, which neatly separates high and low frequency waves. Whereas low frequency waves are permitted to bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of wavelengths is achieved by employing a discrete L2-projection, which acts as a low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization, this approach has the advantage of a small number of constraints, which is computationally efficient. Numerical results in one and two dimensions confirm our theoretical findings and illustrate the performance of our new weak coupling approach.

Analysis of the Quasi-Nonlocal Approximation of Linear and Circular Chains in the Plane

Pavel Bělík and Mitchell Luskin

Multiscale Model. Simul. 9, pp. 1495-1527 (33 pages)

Online Publication Date: November 22, 2011

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We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy–Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones-type nearest neighbor and next nearest neighbor pair-potential interactions. Previous analyses for linear chains have shown that the Cauchy–Born and quasi-nonlocal approximations reproduce (up to the order of the lattice spacing) the atomistic lattice stability for perturbations that are constrained to the line of the chain. However, we show that the Cauchy–Born gives a finite increase for the lattice stability of a linear or circular chain under compression when general perturbations in the plane are allowed. We also analyze the increase of the lattice stability under compression when pair-potential energies are augmented by bond-angle energies. Our estimates of the largest strain for lattice stability (the critical strain) are sharp (exact up to the order of the lattice scale). We then use these stability estimates and modeling error estimates for the linearized Cauchy–Born and quasi-nonlocal energies to give an optimal order (in the lattice scale) a priori error analysis for the approximation of the atomistic strain in ℓε2 due to an external force.

Morozov’s Principle for the Augmented Lagrangian Method Applied to Linear Inverse Problems

Klaus Frick, Dirk A. Lorenz, and Elena Resmerita

Multiscale Model. Simul. 9, pp. 1528-1548 (21 pages)

Online Publication Date: December 01, 2011

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The augmented Lagragian method is an algorithm to compute saddle points for linearly constraint convex minimization problems. Recently it has received much attention, also under the name Bregman iteration, as an approach for regularizing inverse problems by applying the iteration to noisy data and stopping appropriately. Convergence and convergence rates have been shown for a priori stopping rules. This work shows convergence and convergence rates for this method when a special a posteriori rule, namely, Morozov’s discrepancy principle, is chosen as a stopping criterion. Particularly, we treat the case in which this rule degenerates in the sense that the stopping indices do not tend to infinity as the noise level vanishes. Moreover, error estimates for the involved sequence of subgradients are pointed out. As potential fields of application we study implications of these results for particular examples in imaging. These are total-variation regularization as well as ℓq penalties with q ∊ [1,2]. It is shown that Morozov’s principle implies convergence and convergence rates for the iterates with respect to the metric of strict convergence and the ℓq-norm, respectively.

Corrector Theory for MsFEM and HMM in Random Media

Guillaume Bal and Wenjia Jing

Multiscale Model. Simul. 9, pp. 1549-1587 (39 pages)

Online Publication Date: December 06, 2011

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We analyze the random fluctuations of several multiscale algorithms, such as the multiscale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multiscale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one-dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multiscale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size h, we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length ε→0. We next derive the limit of such Gaussian processes as h→0 and compare this limit with the distribution of the random fluctuations of the continuous model. When such limits agree, we conclude that the multiscale algorithm captures the random fluctuations accurately and passes the corrector test. This property serves as an interesting benchmark to assess the behavior of the multiscale algorithm in practical situations where the assumptions necessary for the theory of homogenization are not met. What we find is that the computationally more expensive methods MsFEM, and HMM with a choice of parameter δ = h, correctly capture the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM with δ<h correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.

Diffusion Estimation from Multiscale Data by Operator Eigenpairs

Daan Crommelin and Eric Vanden-Eijnden

Multiscale Model. Simul. 9, pp. 1588-1623 (36 pages)

Online Publication Date: December 06, 2011

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In this paper we present a new procedure for the estimation of diffusion processes from discretely sampled data. It is based on the close relation between eigenpairs of the diffusion operator L and those of the conditional expectation operator Pt, a relation stemming from the semigroup structure Pt = exp(tL) for t ≥ 0. It allows for estimation without making time discretization errors, an aspect that is particularly advantageous in the case of data with low sampling frequency. After estimating eigenpairs of L via eigenpairs of Pt, we infer the drift and diffusion functions that determine L by fitting L to the estimated eigenpairs using a convex optimization procedure. We present numerical examples in which we apply the procedure to one- and two-dimensional diffusions, reversible as well as nonreversible. In the second part of the paper, we consider estimation of coarse-grained (homogenized) diffusion processes from multiscale data. We show that eigenpairs of the homogenized diffusion operator are asymptotically close to eigenpairs of the underlying multiscale diffusion operator. This implies that we can infer the correct homogenized process from data of the multiscale process, using the estimation procedure discussed in the first part of the paper. This is illustrated with numerical examples.

Diffusion of Polarized Light

Arnold D. Kim and Miguel Moscoso

Multiscale Model. Simul. 9, pp. 1624-1645 (22 pages)

Online Publication Date: December 06, 2011

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We study partially polarized light propagation in random media governed by the theory of radiative transport. In particular, we derive a diffusion approximation when scattering is strong and absorption is weak. This diffusion approximation is substantially easier to solve than the vector radiative transport equation because it requires only the solution of a scalar problem. Included in this analysis is the derivation of a boundary layer solution that corrects the diffusion approximation near the boundaries and provides the means to derive boundary conditions for the diffusion approximation. We give an explicit solution for this boundary layer solution in terms of plane wave solutions that we calculate numerically using the discrete ordinate method. We evaluate the effectiveness of this diffusion approximation through comparison with the numerical solution of the full problem.

On Reduced Models for the Chemical Master Equation

Tobias Jahnke

Multiscale Model. Simul. 9, pp. 1646-1676 (31 pages)

Online Publication Date: December 06, 2011

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The chemical master equation plays a fundamental role for the understanding of gene regulatory networks and other discrete stochastic reaction systems. Solving this equation numerically, however, is usually extremely expensive or even impossible due to the huge size of the state space. Thus, the chemical master equation must often be replaced by a reduced model which operates with a considerably smaller number of degrees of freedom but hopefully still provides the essential information about the dynamics of the full system. We prove error bounds for two reduced models which have previously been proposed in the literature. Based on the error analysis, an alternative model reduction approach for the chemical master equation is introduced and discussed, and its advantage is illustrated by numerical examples.

Weak Approximation Properties of Elliptic Projections with Functional Constraints

Robert Scheichl, Panayot S. Vassilevski, and Ludmil T. Zikatanov

Multiscale Model. Simul. 9, pp. 1677-1699 (23 pages)

Online Publication Date: December 06, 2011

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This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble–Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted L2 norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided.

On Analysis of Nonstationary Categorical Data Time Series: Dynamical Dimension Reduction, Model Selection, and Applications To Computational Sociology

Illia Horenko

Multiscale Model. Simul. 9, pp. 1700-1726 (27 pages)

Online Publication Date: December 13, 2011

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Many real-life processes can be described as simplified categorical (or discrete) data models, switching between a finite number of states or regimes. Two main computational problems can be considered in the context of their analysis: (a) dimension reduction (i.e., identification of the essential (discrete) degrees of freedom or categories), and (b) parameterization of reduced dynamical models (e.g., Markov chains or Bernoulli processes) that can be used for analysis and prediction. Existing analysis methods, addressing the above issues (a) and (b), are limited with respect to possibilities for analyzing the influences of exogenous factors and have an implicit assumption of the stationarity built in. A general framework for analysis and online prediction of categorical data dynamics is presented. In contrast to standard approaches of categorical data analysis (like generalized linear discrete models, e.g., logit and probit regression methods) that are based on the transformation of the discrete categorical data into continuous representation, the presented methods allow us to build the (auto) regressive models of the data and to find the reduced data representation directly in the discrete setting. This general framework is based on an extension of the principal component analysis method to structure-preserving dimension reduction of (nonstationary) discrete jump processes, combined with a data-based estimation of nonhomogeneous Markov chain models under the influence of external factors in the reduced representation. Efficiently parallelizable numerical methods for the solution of the resulting mixed discrete-continuous optimization problems are described, applicable to the analysis of systems with a large number of discrete categories. Described methods have a favorable scaling with the dimension (expressed via a number of discrete states). General applicability is illustrated on the generic toy model example and applied to two problems from computational sociology: (i) estimation of the time-dependent transition graphs describing the dynamics of political preferences of the mean German voter, and (ii) parameterization of the discrete jump process describing the transition between the employed and unemployed states of an average German individual. The resulting online predictions are compared to the ones obtained by standard methods of the time series analysis, and the influence of implicit statistical assumptions is discussed.

Particle Systems and Kinetic Equations Modeling Interacting Agents in High Dimension

M. Fornasier, J. Haškovec, and J. Vybíral

Multiscale Model. Simul. 9, pp. 1727-1764 (38 pages)

Online Publication Date: December 13, 2011

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In this paper we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with today’s powerful computers, numerical simulations of such systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high dimension, by random projections via Johnson–Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations by addressing the phenomenon of the delayed curse of dimension, known in information-based complexity for optimal numerical integration problems and measure quantization in high dimensions.

Capillary Displacement in Totally Wetting And Infinitely Long Right Prisms

Roland Glantz and Markus Hilpert

Multiscale Model. Simul. 9, pp. 1765-1800 (36 pages)

Online Publication Date: December 20, 2011

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We study capillary displacement in infinitely long right prisms filled with one wetting fluid and one nonwetting fluid, while restricting ourselves to zero gravity and totally wetting prism walls. At equilibrium, the intersection of the fluid-fluid interface with the cross section of the prism is composed of nonintersecting circular arcs with a uniform radius of curvature inversely proportional to the capillary pressure. We trace the circular arcs during capillary displacement using the so-called connected chordal axis (CCA) of the cross section. We identify the points on the CCA which are associated with the following events in the arcs’ “lives”: emergence, disappearance, end points (i.e., three phase contact points), getting pinned or unpinned, merging with another arc, or splitting into two arcs. The CCA comes with a decomposition of the cross section into triangular and trapezoidal regions marking the stages of the arcs’ lives. We also characterize the (polygonal) cross-sectional shapes for which the fluid-fluid distribution responds continuously and reversibly to a change in capillary pressure, i.e., those cross-sectional shapes for which capillary displacement can be quasi-static. For these shapes the intersections of the nonwetting fluid with the cross section are precisely the morphological openings of the cross section, where the structuring elements are the disks whose radii coincide with those of the circular arcs. For any other (polygonal) shape, this correspondence does not hold, and, equivalently, capillary displacement cannot be quasi-static. We also formulate an assumption on the fluid-fluid distributions in such a cross section, which allows us to describe discontinuous capillary displacement.

On Donoho’s Log-Exp Subdivision Scheme: Choice of Retraction and Time-Symmetry

Esfandiar Nava-Yazdani and Thomas P. Y. Yu

Multiscale Model. Simul. 9, pp. 1801-1828 (28 pages)

Online Publication Date: December 22, 2011

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In recent years a number of different approaches for adapting linear subdivision schemes to manifold-valued data were proposed. In this article, we study the following family:
math
here f is a smooth retraction on M, g is the corresponding local inverse, and (a2ℓ+σ) is the mask of a linear subdivision scheme. This particular way of adapting linear subdivision schemes to manifold, in which the same base point xi is used for both the odd and even rules, is a fundamental building block of the wavelet transform proposed in Ur Rahman et al. [Multiscale Model. Simul., 4 (2005), pp. 1201–1232]. This feature is not shared by the other ways proposed in the recent literature. In this article, we expose the rather subtle smoothness equivalence properties of the above S; here “smoothness equivalence property” refers to how much smoothness the nonlinear S inherits from the underlying linear scheme. We first prove that one always gets C2 equivalence between S and the linear scheme regardless of the choice of f. In contrast, if one wants just one more order of smoothness equivalence, then the choice of f matters. We show that C3 equivalence is guaranteed by a condition on the third order Taylor expansions of f. This condition is further proved to be genuinely geometric in the sense that it is invariant under change of coordinates. Our second main result shows that the most natural choice f = exp in a symmetric space setting always satisfies the condition. Consequently, any third order accurate approximation of the exponential map would satisfy the same condition. This provides the ground for replacing the exponential map by a more computationally efficient approximant. The difficulty is that such an approximant must also be chosen such that it is by itself also a retraction of the underlying symmetric space or Lie group. Fortunately, it is a well-studied problem in the area of numerical geometric integration; many computationally efficient approximations to the exponential map are available for different symmetric spaces. Finally, we discuss the effect of time-symmetry on smoothness.

Multiscale NeighborhoodWise Decision Fusion for Redundancy Detection in Image Pairs

Charles Kervrann, Jérôme Boulanger, Thierry Pécot, Patrick Pérez, and Jean Salamero

Multiscale Model. Simul. 9, pp. 1829-1865 (37 pages)

Online Publication Date: December 22, 2011

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To develop better image change detection algorithms, new models able to capture spatio-temporal regularities and geometries present in an image pair are needed. In this paper, we propose a multiscale formulation for modeling semilocal interimage interactions and detecting local or regional changes in an image pair. By introducing dissimilarity measures to compare patches and binary local decisions, we design collaborative decision rules that use the total number of detections obtained from the neighboring pixels for different patch sizes. We study the statistical properties of the nonparametric detection approach that guarantees small probabilities of false alarms. Experimental results on several applications demonstrate that the detection algorithm (with no optical flow computation) performs well at detecting occlusions and meaningful changes for a variety of illumination conditions and signal-to-noise ratios. The number of control parameters of the algorithm is small, and the adjustment is intuitive in most cases.
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