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Multiscale Modeling & Simulation

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2005

Volume 4, Issue 4, pp. 1041-1372


Well-Posedness of a Multiscale Model for Concentrated Suspensions

Eric Cancès, Isabelle Catto, Yousra Gati, and Claude Le Bris

Multiscale Model. Simul. 4, pp. 1041-1058 (18 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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In a previous work [E. Cancès, I. Catto, and Y. Gati, SIAM J. Math. Anal., 37 (2005), pp. 60--82], three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles which are subjected to a given time-dependent shear rate. In the present work we extend the model to a more physically relevant situation where the shear rate actually depends on the macroscopic velocity of the fluid. As a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker--Planck-type equation with nonhomogeneous, nonlocal, and possibly degenerate coefficients. We show the existence and the uniqueness of the global-in-time weak solutionto such a system.

Numerical Modeling of Laser Pulse Behavior in Nonlinear Crystal and Application to the Second Harmonic Generation

A. Bourgeade and B. Nkonga

Multiscale Model. Simul. 4, pp. 1059-1090 (32 pages)

Online Publication Date: July 26, 2006

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We propose an efficient numerical strategy for the full wave integration of the nonlinear Maxwell equations by a finite difference approach (finite-difference time-domain). We compare it to the asymptotic limit described by an extended nonlinear Schrödinger equation. This is achieved by considering the nonlinear behavior of high intensity ($\geq 5GW/cm^2$) ultrashort ($ \leq 100 fs$ FWMH) laser beams in a KDP crystal (type I). We assume that this crystal can be characterized by two linear dispersion Lorentz resonances, as well as instantaneous quadratic and cubic nonlinearities. Investigations are performed, for different optical regimes, in the context of the second harmonic generation for one-dimensional and two-dimensional models.

Deblurring and Denoising of Images by Nonlocal Functionals

Stefan Kindermann, Stanley Osher, and Peter W. Jones

Multiscale Model. Simul. 4, pp. 1091-1115 (25 pages) | Cited 37 times

Online Publication Date: July 26, 2006

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This paper investigates the use of regularization functionals with nonlocal correlation terms for the problem of image denoising and image deblurring. These functionals are expressed as integrals over the Cartesian product of the pixel space. We show that the class of neighborhood filters can be described in this framework. Using these functionals we can consider the functional analytic properties of some of these neighborhood filters and show how they can be seen as regularization terms using a smoothed version of the Prokhorov metric. Moreover, we define a nonlocal variant of the well-known bounded variation regularization, which does not suffer from the staircase effect. We show existence of a minimizer of the corresponding regularization functional for the denoising and deblurring problem, and we present some numerical examples comparing the nonlocal version to the bounded variation regularization and the nonlocal mean filter.

Direct Algorithms for Thermal Imaging of Small Inclusions

Habib Ammari, Ekaterina Iakovleva, Hyeonbae Kang, and Kyoungsun Kim

Multiscale Model. Simul. 4, pp. 1116-1136 (21 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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The goal of this paper is to reconstruct a collection of small inclusions inside a homogeneous object by applying a heat flux and measuring the induced temperature on its boundary. Taking advantage of the smallness of the inclusions, we design efficient noniterative algorithms for locating the inclusions from boundary measurements of the temperature. We illustrate the feasibility and the viability of our algorithms by numerical examples.

Multiscale Angiogenesis Modeling Using Mixed Finite Element Methods

Shuyu Sun, Mary F. Wheeler, Mandri Obeyesekere, and Charles Patrick, Jr.

Multiscale Model. Simul. 4, pp. 1137-1167 (31 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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In this paper, we present a deterministic two-scale tissue-cellular approach for modeling growth factor-induced angiogenesis. The bioreaction-diffusion of chemotactic growth factors (CGF) is modeled at a tissue scale, whereas cell proliferation, capillary extension, branching, and anastomosis are modeled at a cellular scale. The capillary indicator function is used to bridge these two scales. The complete system of equations consists of parabolic PDEs coupled nonlinearly with a varying number of ODEs and algebraic equations. Our proposed schemes involve applying mixed finite element methods to approximate concentrations of CGF and a point-to-point tracking method to simulate sprout branching and anastomosis. Capillary extensions are computed by a system of ODEs. Here, both the continuous and discrete-in-time algorithms are analyzed using some new techniques for treating the nonlinear coupling terms. Error bounds for each of the processes---CGF reaction-diffusion, capillary extension, sprout branching, and anastomosis---and overall error bounds for their coupled nonlinear interactions are established. Optimal order estimates in the mesh size are obtained for the continuous-in-time schemes, and optimal order estimates in both the mesh and the time step sizes are derived for the fully discretized schemes. In addition, we address several implementation issues, including an equivalent cell-centered finite difference formulation, time splitting techniques, and object-oriented programming strategies for efficient scientific computing. Finally, a representative simulation example is provided.

Signal Recovery by Proximal Forward-Backward Splitting

Patrick L. Combettes and Valérie R. Wajs

Multiscale Model. Simul. 4, pp. 1168-1200 (33 pages) | Cited 163 times

Online Publication Date: July 26, 2006

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We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

Multiscale Representations for Manifold-Valued Data

Inam Ur Rahman, Iddo Drori, Victoria C. Stodden, David L. Donoho, and Peter Schröder

Multiscale Model. Simul. 4, pp. 1201-1232 (32 pages) | Cited 10 times

Online Publication Date: July 26, 2006

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We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.

Scaling Laws and Fokker--Planck Equations for 3-Dimensional Porous Media with Fractal Mesoscale

Moongyu Park, Natalie Kleinfelter, and John H. Cushman

Multiscale Model. Simul. 4, pp. 1233-1244 (12 pages) | Cited 6 times

Online Publication Date: July 26, 2006

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Transport is studied in three-scale porous media with fractal mesoscale. On the microscale the dispersive mixing is governed by a stochastic ordinary differential equation with stationary, ergodic, Markovian drift velocity and an $\alpha$-stable Lévy diffusion. The inclusion of Lévy diffusion allows one to study self-motile particles such as flagellated microbes. On the mesoscale it is assumed that a fractal Eulerian flow field with spatially homogeneous increments gives rise to a fractal drift velocity with temporally stationary increments. The drift velocity is assumed to be Lévy. These processes have stationary increments and fractal graphs. The diffusive structure on the mesoscale is determined by the asymptotic microscale process. On the macroscale there is no additional drift; i.e., the physics is controlled solely by the asymptotic behavior of the mesoscale process. Scaling laws from the micro- to meso- and meso- to macroscales are obtained as well as the Fokker--Planck equations for the transition densities on the meso- and macroscales. These latter equations are fractional (pseudodifferential).

A Coupled Model for Radiative Transfer: Doppler Effects, Equilibrium, and Nonequilibrium Diffusion Asymptotics

Pauline Godillon-Lafitte and Thierry Goudon

Multiscale Model. Simul. 4, pp. 1245-1279 (35 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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This paper is devoted to the asymptotic analysis of a coupled model arising in radiative transfer. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a diffusion equation satisfied by the material temperature. The interaction terms take into account both scattering and absorption/emission phenomena, as well as Doppler corrections. Two asymptotic regimes are identified, depending on the scaling assumptions about the physical parameters and observation scales. In the equilibrium regime, the system is driven only by the material temperature which satisfies a nonlinear drift-diffusion equation. In the nonequilibrium regime, the radiation temperature and the material temperature will be coupled by a system of nonlinear drift-diffusion equations.

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-Plane Structure Transitions

M. Gregory Forest, Ruhai Zhou, and Qi Wang

Multiscale Model. Simul. 4, pp. 1280-1304 (25 pages) | Cited 11 times

Online Publication Date: July 26, 2006

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Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270--6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829--843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184--3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17--37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80--93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17--31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853--870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141--8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B}, to appear].

Operator Upscaling for the Acoustic Wave Equation

Tetyana Vdovina, Susan E. Minkoff, and Oksana Korostyshevskaya

Multiscale Model. Simul. 4, pp. 1305-1338 (34 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarser domain without solving the full fine-scale problem. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, we solve small independent problems for approximate fine-scale information internal to each coarse block. Then we use these subgrid solutions to define an upscaled operator on the coarse grid. The fine-grid velocity field is used throughout the process (i.e.,no averaging of input fields is required). An equivalence between the variational form of the problem and a staggered finite-difference scheme allows us to use finite differences to solve the subgrid wave propagation problems. Due to the homogeneous Neumann boundary conditions imposed on each coarse block, the subgrid problems decouple, which leads to the natural parallelization of the first stage of the method. The algorithm requires none of the ghost cell (edge) communication required by standard data parallelism. Timing studies indicate that the parallel algorithm has near optimal speedup. Three variable velocity numerical experiments illustrate that operator-based upscaling captures the essential fine-scale information (even details contained within a single coarse grid block) and models wave propagation quite accurately at considerably less expense than full finite differences. The resulting coarse-grid solution has the advantage of being less prohibitive to store and manipulate.

Projection Multilevel Methods for Quasi-linear PDEs: V-cycle Theory

Stephen F. McCormick

Multiscale Model. Simul. 4, pp. 1339-1348 (10 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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The projection multilevel method can be an efficient solver for systems of nonlinear partial differential equations that, for certain classes of nonlinearities (including least-squares formulations of the Navier--Stokes equations), requires no linearization anywhere in the algorithm. This paper provides an abstract framework and establishes optimal $V$-cycle convergence theory for this method.

Combined Adaptive Multiscale and Level-Set Parameter Estimation

Martha Lien, Inga Berre, and Trond Mannseth

Multiscale Model. Simul. 4, pp. 1349-1372 (24 pages) | Cited 9 times

Online Publication Date: July 26, 2006

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We propose a solution strategy for parameter estimation, where we combine adaptive multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the nonlinear inverse problem of recovering a coefficient function in a system of differential equations from spatially sparsely distributed measurement data. The specific equations considered in this paper describe two-phase porous-media flow where a coefficient function defining absolute permeability (fluid conductivity) is estimated based on fluid pressure observations in wells. This inverse problem is known to be ill-posed. The spatial variability of the sought coefficient function is unknown and will typically vary within the porous medium. Due to limited information in the available data, mainly coarse-scale features of the existing variability in the coefficient function will be attainable. In AME, one starts out with a single parameter representation of the sought function, whereafter the domain is successively divided into finer rectangular zones, each representing a constant parameter value of the coefficient function. The strong restrictions on the zone geometry may lead to overparameterization. LSE is a method for moving curves and enables adjustments of the zone structure into more general geometries. In order to perform well, LSE requires a reasonable starting point. In this paper, we have developed a methodology to combine AME and LSE, where at each step either refinements or deformation of the zone structure may be conducted, depending on which method promises the better result. The combined approach seems promising with respect to recovering coarse-scale features of the sought coefficient functions with a low number of parameters.
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