Authors:Rebecca K. Borchering, Scott A. McKinley Pages: 551 - 582 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 551-582, January 2018. In the last decade there has been growing criticism of the use of stochastic differential equations to approximate discrete-state-space, continuous-time Markov chain population models. In particular, several authors have demonstrated the failure of Diffusion Approximation, as it is often called, to approximate expected extinction times for populations that start in a quasi-stationary state. In this work we investigate a related, but distinct, population dynamics property for which Diffusion Approximation is unreliable: invasion probabilities. We consider the situation in which a few individuals are introduced into a population and ask whether their collective lineage can successfully invade. Because the population count is so small during the critical period of success or failure, the process is intrinsically stochastic and discrete. In addition to demonstrating how and why the Diffusion Approximation fails in the large population limit, we contrast this analysis with that of a sometimes more successful alternative WKB-like approach. Through numerical investigations, we also study how these approximations perform in an important intermediate regime. Surprisingly, we find that there are times when the Diffusion Approximation performs well, particularly when parameters are near-critical and the population size is small to intermediate. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-03T07:00:00Z DOI: 10.1137/17M1155259

Authors:Erwan Deriaz, Sébastien Peirani Pages: 583 - 614 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 583-614, January 2018. We present an original adaptive scheme using a dynamically refined grid for the simulation of the six-dimensional Vlasov--Poisson equations. The distribution function is represented in a hierarchical basis that retains only the most significant coefficients. This allows considerable savings in terms of computational time and memory usage. The proposed scheme involves the mathematical formalism of multiresolution analysis and computer implementation of adaptive mesh refinement. We apply a finite difference method to approximate the Vlasov--Poisson equations, although other numerical methods could be considered. Numerical experiments are presented for the $d$-dimensional Vlasov--Poisson equations in the full $2d$-dimensional phase space for $d=1,2$, or 3. The six-dimensional case is compared to a Gadget N-body simulation. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-03T07:00:00Z DOI: 10.1137/16M1108649

Authors:Thomas Y. Hou, De Huang, Ka Chun Lam, PengChuan Zhang Pages: 615 - 678 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 615-678, January 2018. In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix $A$ using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this energy framework, we propose a systematic operator compression scheme for the inverse operator $A^{-1}$. In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely, the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the patch pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-05T07:00:00Z DOI: 10.1137/17M1140686

Authors:Hao Wang, Siyao Yang Pages: 679 - 709 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 679-709, January 2018. We consider the a posteriori error estimation for an atomistic-to-continuum coupling scheme for a generic one-dimensional many-body next-nearest-neighbor interaction model in one dimension. We derive and rigorously prove the efficiency of the residual type estimator. We prove the equivalence between the residual type and the gradient recovery-type estimator in the continuum region and propose a (novel) hybrid a posteriori error estimator by combining the two types of estimators. Our numerical experiments illustrate the optimal convergence rate of the adaptive algorithms using these estimators, whose effectivity indices are also presented. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-17T07:00:00Z DOI: 10.1137/17M1118579

Authors:Rongjie Lai, Jianfeng Lu Pages: 710 - 726 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 710-726, January 2018. The committor functions provide useful information to the understanding of transitions of a stochastic system between disjoint regions in phase space. In this work, we develop a point cloud discretization for Fokker--Planck operators to numerically calculate the committor function, with the assumption that the transition occurs on an intrinsically low dimensional manifold in the ambient potentially high dimensional configurational space of the stochastic system. Numerical examples on model systems validate the effectiveness of the proposed method. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-26T07:00:00Z DOI: 10.1137/17M1123018

Authors:Wangtao Lu, Jianliang Qian, Robert Burridge Pages: 727 - 751 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 727-751, January 2018. Starting from Hadamard's method, we extend Babich's ansatz to the frequency-domain point-source (FDPS) Maxwell's equations in an inhomogeneous medium in the high-frequency regime. First, we develop a novel asymptotic series, dubbed Hadamard's ansatz, to form the fundamental solution of the Cauchy problem for the time-domain point-source (TDPS) Maxwell's equations in the region close to the source. Governing equations for the unknowns in Hadamard's ansatz are then derived. In order to derive the initial data for the unknowns in the ansatz, we further propose a condition for matching Hadamard's ansatz with the homogeneous-medium fundamental solution at the source. Directly taking the Fourier transform of Hadamard's ansatz in time, we obtain a new ansatz, dubbed the Hadamard--Babich ansatz, for the FDPS Maxwell's equations. Next, we elucidate the relation between the Hadamard--Babich ansatz and a recently proposed Babich-like ansatz for solving the same FDPS Maxwell's equations. Finally, incorporating the first two terms of the Hadamard--Babich ansatz into a planar-based Huygens sweeping algorithm, we solve the FDPS Maxwell's equations at high frequencies in the region where caustics occur. Numerical experiments demonstrate the accuracy of our method. Citation: Multiscale Modeling & Simulation PubDate: 2018-04-26T07:00:00Z DOI: 10.1137/17M1130381

Authors:Sambasiva Rao Chinnamsetty, Michael Griebel, Jan Hamaekers Pages: 752 - 776 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 752-776, January 2018. In this paper, we introduce a new scheme for the efficient numerical treatment of the electronic Schrödinger equation for molecules. It is based on the combination of a many-body expansion, which corresponds to the bond order dissection ANOVA approach introduced in [M. Griebel, J. Hamaekers, and F. Heber, Extraction of Quantifiable Information from Complex Systems, Springer, New York, pp. 211--235; F. Heber, Ph.D. thesis, Intitut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014], with a hierarchy of basis sets of increasing order. Here, the energy is represented as a finite sum of contributions associated to subsets of nuclei and basis sets in a telescoping sum like fashion. Under the assumption of data locality of the electronic density (nearsightedness of electronic matter), the terms of this expansion decay rapidly and higher terms may be neglected. We further extend the approach in a dimension-adaptive fashion to generate quasi-optimal approximations, i.e., a specific truncation of the hierarchical series such that the total benefit is maximized for a fixed amount of costs. This way, we are able to achieve substantial speed up factors compared to conventional first principles methods depending on the molecular system under consideration. In particular, the method can deal efficiently with molecular systems which include only a small active part that needs to be described by accurate but expensive models. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-01T07:00:00Z DOI: 10.1137/16M1088119

Authors:Gabriel Stoltz, Zofia Trstanova Pages: 777 - 806 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 777-806, January 2018. We study Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of Boltzmann--Gibbs distributions. In particular, this kinetic energy can be nonglobally Lipschitz, which raises issues for the stability of discretizations of the associated Langevin dynamics. We first prove the exponential convergence of the law of the continuous process to the Boltzmann--Gibbs measure by a hypocoercive approach and characterize the asymptotic variance of empirical averages over trajectories. We next develop numerical schemes which are stable and of weak order two by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We use the newly developed schemes for two applications: optimizing the shape of the kinetic energy for the so-called adaptively restrained Langevin dynamics (which considers perturbations of standard quadratic kinetic energies vanishing around the origin) and reducing the metastability of some toy models using nonglobally Lipschitz kinetic energies. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-08T07:00:00Z DOI: 10.1137/16M110575X

Authors:Adrian Muntean, Sina Reichelt Pages: 807 - 832 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 807-832, January 2018. The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermodiffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The term “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter $\varepsilon$ of the heat conduction-diffusion interaction terms, while the “high-contrast” is considered particularly in terms of the heat conduction properties of the composite material. As a main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling leads to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with $\varepsilon$-independent estimates for the thermal and concentration fields and for their coupled fluxes. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-15T07:00:00Z DOI: 10.1137/16M109538X

Authors:Pavel Gurevich, Sina Reichelt Pages: 833 - 856 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 833-856, January 2018. This paper is devoted to pulse solutions in FitzHugh--Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh--Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-15T07:00:00Z DOI: 10.1137/17M1143708

Authors:Stefan Neukamm, Mario Varga Pages: 857 - 899 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 857-899, January 2018. The aim of our work is to provide a simple homogenization and discrete-to-continuum procedure for energy driven problems involving stochastic rapidly oscillating coefficients. Our intention is to extend the periodic unfolding method to the stochastic setting. Specifically, first we recast the notion of stochastic two-scale convergence in the mean by introducing an appropriate stochastic unfolding operator. This operator admits similar properties as the periodic unfolding operator, leading to an uncomplicated method for stochastic homogenization. Second, we analyze the discrete-to-continuum (resp., stochastic homogenization) limit for a rate-independent system describing a network of linear elasto-plastic springs with random coefficients. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-15T07:00:00Z DOI: 10.1137/17M1141230

Authors:Di Fang, Shi Jin, Christof Sparber Pages: 900 - 921 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 900-921, January 2018. The Ehrenfest dynamics, representing a quantum-classical mean-field type coupling, is a widely used approximation in quantum molecular dynamics. In this paper, we propose a time-splitting method for an Ehrenfest dynamics, in the form of a nonlinearly coupled Schrödinger--Liouville system. We prove that our splitting scheme is stable uniformly with respect to the semiclassical parameter and, moreover, that it preserves a discrete semiclassical limit. Thus one can accurately compute physical observables using time steps induced only by the classical Liouville equation, i.e., independent of the small semiclassical parameter---in addition to classical mesh sizes for the Liouville equation. Numerical examples illustrate the validity of our meshing strategy. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-15T07:00:00Z DOI: 10.1137/17M1112789

Authors:Junshan Lin, Hai Zhang Pages: 922 - 953 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 922-953, January 2018. This is the first part in a series of two papers that are concerned with the quantitative analysis of the electromagnetic field enhancement and anomalous diffraction by a periodic array of perfect conducting (PEC) subwavelength slits. The scattering problem in the diffraction regime is investigated in this part, for which the size of the period is comparable to the incident wavelength. We distinguish scattering resonances and real eigenvalues, and derive their asymptotic expansions when they are away from the Rayleigh cutoff frequencies. Furthermore, we present quantitative analysis of the field enhancement at resonant frequencies, by quantifying both the enhancement order and the associated resonant modes. The field enhancement near the Rayleigh cutoff frequencies is also investigated. It is demonstrated that the field enhancement at resonant frequencies becomes weaker if those frequencies are close to one of the Rayleigh cutoff frequencies. Finally, we also characterize the embedded eigenvalues for the underlying periodic structure, and point out that transmission anomalies such as the Fano resonant phenomenon do not occur for the PEC narrow slit array. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-17T07:00:00Z DOI: 10.1137/17M1133774

Authors:Junshan Lin, Hai Zhang Pages: 954 - 990 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 2, Page 954-990, January 2018. This is the second part in a series of two papers that are concerned with the quantitative analysis of the electromagnetic field enhancement and anomalous diffraction by a periodic array of subwavelength slits in a perfect conducting slab. In this part, we explore the scattering problem in the homogenization regimes, where the period of the structure is much smaller than the incident wavelength. In particular, two homogenization regimes are investigated: in the first regime, the width of the slits is comparable to the period, while in the second regime, the width of the slits is much smaller than the period. By presenting rigorous asymptotic analysis, we demonstrate that a surface plasmonic effect mimicking that of plasmonic metals occurs in the first regime. In addition, for incident plane waves, we discover and justify a novel phenomenon of total transmission which occurs either at certain frequencies for all incident angles or at a special incident angle but for all frequencies. For the second regime, the nonresonant field enhancement is investigated. It is shown that the fast transition of the magnetic field in the slits induces strong electric field enhancement. Moreover, the enhancement becomes stronger when the coupling between the slits is weaker. Citation: Multiscale Modeling & Simulation PubDate: 2018-05-17T07:00:00Z DOI: 10.1137/17M1133786

Authors:Ting Wang, Petr Plecháč, David Aristoff Pages: 1 - 27 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 1-27, January 2018. We propose two algorithms for simulating continuous time Markov chains in the presence of metastability. We show that the algorithms correctly estimate, under the ergodicity assumption, stationary averages of the process. Both algorithms, based on the idea of the parallel replica method, use parallel computing in order to explore metastable sets more efficiently. The algorithms require no assumptions on the Markov chains beyond ergodicity and the presence of identifiable metastability. In particular, there is no assumption on reversibility. For simpler illustration of the algorithms, we assume that a synchronous architecture is used throughout the paper. We present error analyses, as well as numerical simulations on multiscale stochastic reaction network models in order to demonstrate consistency of the method and its efficiency. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-02T08:00:00Z DOI: 10.1137/16M1108716

Authors:Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno, Ariane Trescases Pages: 28 - 77 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 28-77, January 2018. We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-09T08:00:00Z DOI: 10.1137/17M1135207

Authors:M. Ganesh, T. Thompson Pages: 78 - 105 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 78-105, January 2018. Multi-scale/level computer models are efficient for simulating solutions of partial differential equations on nontrivial geometries using a large number of modes. This is because fine-scale structures in the quantities of interest cannot, in practice, be resolved using a single-level finite element method (FEM) applied directly to the full model, or standard spectral basis functions are not appropriate for the nontrivial geometry and related constraints. The multi-scale/level FEM (MFEM) is based on first applying a FEM for a reduced model problem, obtained by restriction to either (i) several local subdomains or (ii) a dominant linear differential operator in the model. The resulting structure-aware precomputed solutions are then used as basis functions to simulate the full model in the MFEM framework. If these are eigenfunctions of a reduced model eigenvalue problem, such basis functions provide spectrally accurate approximations to the solution of the full model and the resulting method is called the spectral MFEM (SMFEM). In this article we develop, analyze, and implement time-space fully discrete implicit SMFEM (ISMFEM) algorithms for efficiently simulating a Ginzburg--Landau (GL) system modeling superconductivity on a class of superconducting surfaces $S$. The spatial linear differential operator in the time-dependent nonlinear GL model is the Schrödinger operator $( i \nabla + {A}_0)^2$ on $S$, with magnetic vector potential ${A}_0$. For our ISMFEM, the spectrally accurate basis functions are eigenfunctions of a reduced stationary linear model governed by $(i \nabla + {A}_0)^2$. In a recent work we developed, analyzed, and implemented an algorithm to compute these functions using an analytic-numeric hybrid method, in conjunction with a high-order FEM. Using these precomputed solutions as basis functions, the spectrally accurate algorithms developed and analyzed for our new GL computer models on nontrivial geometries are validated using both nonlinear and linearized implicit time discretizations. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-11T08:00:00Z DOI: 10.1137/16M1096487

Authors:Kyle R. Steffen, Yekaterina Epshteyn, Jingyi Zhu, Megan J. Bowler, Jody W. Deming, Kenneth M. Golden Pages: 106 - 124 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 106-124, January 2018. Sea ice hosts a rich ecosystem of flora and fauna, from microscale to macroscale. Algae living in its porous brine microstructure, such as the diatom Melosira arctica, secrete gelatinous exopolymeric substances (EPS) which are thought to protect these communities from their cold and highly saline environment. Recent experimental work has shown significant changes in the structure and properties of young sea ice with entrained Melosira EPS, such as increased brine volume fraction, salt retention, pore tortuosity, and decreased fluid permeability. In particular, we find that the cross-sectional areas of the brine inclusions are described by a bimodal--lognormal distribution, which generalizes the classic lognormal distribution of Perovich and Gow. We propose a model for the effective fluid permeability of young, EPS-laden sea ice, consisting of a random network of pipes with cross-sectional areas chosen from this bimodal distribution. We consider an equilibrium model posed on a square lattice, incorporating only the most basic features of the geometry and connectivity of the brine microstructure, and find good agreement between our model and the observed drop in fluid permeability. Our model formulation suggests future directions for experimental work, focused on measuring the inclusion size distribution and fluid permeability of sea ice with entrained EPS as functions of brine volume fraction. The drop in fluid permeability observed in experimental work and predicted by the model is significant, and should be taken into account, for example, in physical or ecological process models involving fluid or nutrient transport. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-11T08:00:00Z DOI: 10.1137/17M1117513

Authors:Weihua Deng, Buyang Li, Wenyi Tian, Pingwen Zhang Pages: 125 - 149 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 125-149, January 2018. To characterize the Brownian motion in a bounded domain $\Omega$, it is well known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the Lévy flights or tempered Lévy flights in a bounded domain, the boundary conditions involve the information of a solution in the complementary set of $\Omega$, i.e., $\Bbb{R}^n\backslash \Omega$, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying “boundary” conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-18T08:00:00Z DOI: 10.1137/17M1116222

Authors:Eduard Feireisl, Mária Lukáčová-Medviďová, Šárka Nečasová, Antonín Novotný, Bangwei She Pages: 150 - 183 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 150-183, January 2018. We study the convergence of numerical solutions of the compressible Navier--Stokes system to its incompressible limit. The numerical solution is obtained by a combined finite element--finite volume method based on the linear Crouzeix--Raviart finite element for the velocity and piecewise constant approximation for the density. The convective terms are approximated using upwinding. The distance between a numerical solution of the compressible problem and the strong solution of the incompressible Navier--Stokes equations is measured by means of a relative energy functional. For barotropic pressure exponent $\gamma \geq 3/2$ and for well-prepared initial data we obtain uniform convergence of order ${\cal O}(\sqrt{\Delta t}, h^a, \varepsilon)$, $a = \min \{ \frac{2 \gamma - 3 }{ \gamma}, 1\}$. Extensive numerical simulations confirm that the numerical solution of the compressible problem converges to the solution of the incompressible Navier--Stokes equations as the discretization parameters $\Delta t$, $h$ and the Mach number $\varepsilon$ tend to zero. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-24T08:00:00Z DOI: 10.1137/16M1094233

Authors:Jessica K. Hargreaves, Marina I. Knight, Jon W. Pitchford, Rachael J. Oakenfull, Seth J. Davis Pages: 184 - 214 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 184-214, January 2018. Rhythmic processes are found at all biological and ecological scales, and are fundamental to the efficient functioning of living systems in changing environments. The biochemical mechanisms underpinning these rhythms are therefore of importance, especially in the context of anthropogenic challenges such as pollution or changes in climate and land use. Here we develop and test a new method for clustering rhythmic biological data with a focus on circadian oscillations. The method combines locally stationary wavelet time series modelling with functional principal components analysis and thus extracts the time-scale patterns arising in a range of rhythmic data. We demonstrate the advantages of our methodology over alternative approaches, by means of a simulation study and real data applications, using both a published circadian dataset and a newly generated one. The new dataset records plant response to various levels of stress induced by a soil pollutant, a biological system where existing methods which assume stationarity are shown to be inappropriate. Our method successfully clusters the circadian data in an interesting way, thereby facilitating wider ranging analyses of the response of biological rhythms to environmental changes. Citation: Multiscale Modeling & Simulation PubDate: 2018-01-24T08:00:00Z DOI: 10.1137/16M1108078

Authors:Zuoqiang Shi, Jian Sun, Minghao Tian Pages: 215 - 247 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 215-247, January 2018. In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We formulate the harmonic extension as solving a Laplace--Beltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) proposed in [Z. Li, Z. Shi, and J. Sun, Commun. Comput. Phys., 22 (2017), pp. 228--258; Z. Shi and J. Sun, Res. Math. Sci., to appear; Z. Li and Z. Shi, Multiscale Model. Simul., 14 (2016), pp. 874--905] to solve the Laplace--Beltrami equation. The basic idea of the PIM method is to approximate the Laplace equation using an integral equation, which is easy to discretize from points. Based on the integral equation, we found that the traditional graph Laplacian method (GLM) fails to approximate the harmonic functions near the boundary. One important application of the harmonic extension in machine learning is semisupervised learning. We run a popular semisupervised learning algorithm by Zhu, Ghahramani, and Lafferty [Machine Learning, Proceedings of the Twentieth International Conference (ICML 2003), 2003, Washington, DC, 2003, ARAI Press, Menlo Park, CA pp. 912--919] over a couple of well-known datasets and compare the performance of the aforementioned approaches. Our experiments show the PIM performs the best. We also apply PIM to an image recovery problem and show it outperforms GLM. Finally, on the model problem of the Laplace--Beltrami equation with Dirichlet boundary, we prove the convergence of the point integral method. Citation: Multiscale Modeling & Simulation PubDate: 2018-02-06T08:00:00Z DOI: 10.1137/16M1098747

Authors:Jakob Witzig, Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber Pages: 248 - 265 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 248-265, January 2018. In this paper, we present a new, optimization-based method to exhibit cyclic behavior in nonreversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing nonreversible biological processes, in particular, molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring oscillator with three genes, we show that this approach can detect cycles that have a productivity one magnitude larger than classical spectral analysis methods. Our method applies to general nonequilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst. Citation: Multiscale Modeling & Simulation PubDate: 2018-02-15T08:00:00Z DOI: 10.1137/16M1091162

Authors:Joshua P. Schneider, Paul N. Patrone, Dionisios Margetis Pages: 266 - 299 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 266-299, January 2018. By linking atomistic and mesoscopic scales, we formally show how a local steric effect can hinder crystal growth and lead to a buildup of adsorbed atoms (adatoms) on a supersaturated, (1+1)-dimensional surface. Starting from a many-adatom master equation of a kinetic restricted solid-on-solid (KRSOS) model with external material deposition, we heuristically extract a coarse-grained, mesoscale description that defines the motion of a line defect (i.e., a step) in terms of statistical averages over KRSOS microstates. Near thermodynamic equilibrium, we use error estimates to show that this mesoscale picture can deviate from the standard Burton--Cabrera--Frank (BCF) step flow model in which the adatom flux at step edges is linear in the adatom supersaturation. This deviation is caused by the accumulation of adatoms near the step, which block one another from being incorporated into the crystal lattice. In the mesoscale picture, this deviation manifests as a significant contribution from many-adatom microstates to the corresponding statistical averages. We carry out kinetic Monte Carlo simulations to numerically demonstrate how certain parameters control the aforementioned deviation. From these results, we discuss empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step. We also discuss how this work could be used to understand the kinetic interplay between accumulation of adatoms and step motion in recent experiments of ice surfaces. Citation: Multiscale Modeling & Simulation PubDate: 2018-02-15T08:00:00Z DOI: 10.1137/16M1110017

Authors:I. Niyonzima, R. V. Sabariego, P. Dular, K. Jacques, C. Geuzaine Pages: 300 - 326 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 300-326, January 2018. In this paper we develop magnetic induction conforming multiscale formulations for magnetoquasistatic problems involving periodic materials. The formulations are derived using the periodic homogenization theory and applied within a heterogeneous multiscale approach. Therefore the fine-scale problem is replaced by a macroscale problem defined on a coarse mesh that covers the entire domain and many mesoscale problems defined on finely-meshed small areas around some points of interest of the macroscale mesh (e.g., numerical quadrature points). The exchange of information between these macro and meso problems is thoroughly explained in this paper. For the sake of validation, we consider a two-dimensional geometry of an idealized periodic soft magnetic composite. (An erratum is attached.) Citation: Multiscale Modeling & Simulation PubDate: 2018-02-22T08:00:00Z DOI: 10.1137/16M1081609

Authors:Lijian Jiang, Na Ou Pages: 327 - 355 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 327-355, January 2018. In the paper, we present a strategy for accelerating posterior inference for unknown inputs in time fractional diffusion models. In many inference problems, the posterior may be concentrated in a small portion of the entire prior support. It will be much more efficient if we build and simulate a surrogate only over the significant region of the posterior. To this end, we construct a coarse model using the generalized multiscale finite element method (GMsFEM) and solve a least-squares problem for the coarse model with a regularizing Levenberg--Marquart algorithm. An intermediate distribution is built based on the approximate sampling distribution. For Bayesian inference, we use GMsFEM and the least-squares stochastic collocation method to obtain a reduced coarse model based on the intermediate distribution. To increase the sampling speed of Markov chain Monte Carlo, the DREAM$_{ZS}$ algorithm is used to explore the surrogate posterior density, which is based on the surrogate likelihood and the intermediate distribution. The proposed method with lower generalized polynomial chaos order gives the approximate posterior as accurately as the surrogate model directly based on the original prior. A few numerical examples for time fractional diffusion equations are carried out to demonstrate the performance of the proposed method with applications of the Bayesian inversion. Citation: Multiscale Modeling & Simulation PubDate: 2018-02-22T08:00:00Z DOI: 10.1137/17M1110535

Authors:Habib Ammari, Francisco Romero, Matias Ruiz Pages: 356 - 384 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 356-384, January 2018. In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and the cases of both a single and multiple particles. We clarify the strong dependency of the heat generation on the geometry of the particles as it depends on the eigenvalues of the associated Neumann--Poincaré operator. For close-to-touching nanoparticles, we show that the temperature field deviates significantly from the one generated by two single nanoparticles. The results of this paper formally explain experimental results reported in the nanomedical literature. They open a door for solving the challenging problems of detecting plasmonic nanoparticles in biological media and monitoring temperature elevation in tissue generated by nanoparticle heating. Citation: Multiscale Modeling & Simulation PubDate: 2018-02-22T08:00:00Z DOI: 10.1137/17M1125893

Authors:Mario Ohlberger, Barbara Verfurth Pages: 385 - 411 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 385-411, January 2018. In this paper, we suggest a new heterogeneous multiscale method (HMM) for the Helmholtz equation with high contrast. The method is constructed for a setting as in Bouchitte and Felbacq [C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 377-382], where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We revisit existing homogenization approaches for this special setting and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to the Helmholtz equation with discontinuous diffusion matrix. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and an a priori error estimate under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. Numerical experiments confirm our theoretical convergence results and examine the resolution condition. Moreover, the numerical simulation gives a good insight and explanation of the physical phenomenon of frequency band gaps. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-01T08:00:00Z DOI: 10.1137/16M1108820

Authors:Matthias Maier, Rolf Rannacher Pages: 412 - 428 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 412-428, January 2018. This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators that are derived in the general context of the dual weighted residual (DWR) method. Based on the optimization approach a postprocessing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, “goal-oriented” tuning of effective models} with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with a strong microscopic, random advection field. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-01T08:00:00Z DOI: 10.1137/16M1105670

Authors:Daniel Massatt, Stephen Carr, Mitchell Luskin, Christoph Ortner Pages: 429 - 451 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 429-451, January 2018. To make the investigation of electronic structure of incommensurate heterostructures computationally tractable, effective alternatives to Bloch theory must be developed. In [Multiscale Model. Simul., 15(2017), pp. 476--499] we developed and analyzed a real space scheme that exploits spatial ergodicity and near-sightedness. In the present work, we present an analogous scheme formulated in momentum space, which we prove has significant computational advantages in specific incommensurate systems of physical interest, e.g., bilayers of a specified class of materials with small rotation angles. We use our theoretical analysis to obtain estimates for improved rates of convergence with respect to total CPU time for our momentum space method that are confirmed in computational experiments. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-06T08:00:00Z DOI: 10.1137/17M1141035

Authors:Hui Ji, Zuowei Shen, Yufei Zhao Pages: 452 - 476 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 452-476, January 2018. Digital Gabor filters are indispensable tools of local time-frequency analysis in signal processing. With strong orientation selectivity, discrete (tight) Gabor frames generated by two-dimensional Gabor filters also see their wide applications in image processing and volume data processing. However, owing to the lack of multiscale structures, discrete Gabor frames are less effective than multiresolution analysis (MRA) based wavelet (tight) frames when being used for modeling data composed of local structures with varying sizes. Recently, it was shown that digital Gabor filters do generate MRA-based wavelet tight frames via the unitary extension principle. However, the corresponding window function has to be a constant window, which has poor joint time-frequency resolution. In this paper, we showed that digital Gabor filters with smooth window function can generate MRA-based wavelet biframes. The MRA-based wavelet biframes generated by digital Gabor filters have both the advantages of Gabor systems on local time-frequency analysis and the advantages of wavelet systems on multiscale analysis. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-15T07:00:00Z DOI: 10.1137/17M1138789

Authors:Guanglian Li Pages: 477 - 502 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 477-502, January 2018. In this work, we investigate the low-rank approximation of elliptic problems in heterogeneous media by means of Kolmogrov $n$-width and asymptotic expansion. This class of problems arises in many practical applications involving high-contrast media, and their efficient numerical approximation often relies crucially on certain low-rank structure of the solutions. We provide conditions on the permeability coefficient $\kappa$ that ensure a favorable low-rank approximation. These conditions are expressed in terms of the distribution of the inclusions in the coefficient $\kappa$, e.g., the values, locations, and sizes of the heterogeneous regions. Further, we provide a new asymptotic analysis for high-contrast elliptic problems based on the perfect conductivity problem and layer potential techniques, which allows deriving new estimates on the spectral gap for such high-contrast problems. These results provide theoretical underpinnings for several multiscale model reduction algorithms. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-20T07:00:00Z DOI: 10.1137/17M1120737

Authors:Timothy B. Costa, Stephen D. Bond, David J. Littlewood Pages: 503 - 527 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 503-527, January 2018. In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. In this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. We conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-27T07:00:00Z DOI: 10.1137/16M1090351

Authors:Emiliano Cristiani, Andrea Tosin Pages: 528 - 549 Abstract: Multiscale Modeling & Simulation, Volume 16, Issue 1, Page 528-549, January 2018. In this paper we investigate the possibility of reducing the complexity of a system composed of a large number of interacting agents, whose dynamics feature a symmetry breaking. We consider first order stochastic differential equations describing the behavior of the system at the particle (i.e., Lagrangian) level and we get its continuous (i.e., Eulerian) counterpart via a kinetic description. However, the resulting continuous model alone fails to describe adequately the evolution of the system, due to the loss of granularity which prevents it from reproducing the symmetry breaking of the particle system. By suitably coupling the two models we are able to reduce considerably the necessary number of particles while still keeping the symmetry breaking and some of its large-scale statistical properties. We describe such a multiscale technique in the context of opinion dynamics, where the symmetry breaking is induced by the results of some opinion polls reported by the media. Citation: Multiscale Modeling & Simulation PubDate: 2018-03-27T07:00:00Z DOI: 10.1137/17M113397X