Authors:Eduardo Abreu, Ciro Díaz, Juan Galvis, John Pérez Pages: 1375 - 1408 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 4, Page 1375-1408, January 2020. We present and discuss a novel approach to deal with conservation properties for the simulation of nonlinear complex porous media flows in the presence of the following: (1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model and (2) multiscale wave structures resulting from interactions of shock waves and rarefaction from the nonlinear hyperbolic-transport model. For the pressure-velocity Darcy flow problem, we revisit a recent high-order and volumetric residual-based Lagrange multipliers saddle point problem to impose local mass conservation on convex polygons. We clarify and improve conservation properties on applications. For the hyperbolic-transport problem we introduce a new locally conservative Lagrangian--Eulerian finite volume method. For the purpose of this work, we recast our method within the Crandall and Majda treatment of the stability and convergence properties of conservation-form, monotone difference, in which the scheme converges to the physical weak solution satisfying the entropy condition. This multiscale coupling approach was applied to several nontrivial examples to show that we are computing qualitatively correct reference solutions. We combine these procedures for the simulation of the fundamental two-phase flow problem with high-contrast multiscale porous medium, but recalling state-of-the-art paradigms on the notion of solution in related multiscale applications. This is a first step to deal with out-of-reach multiscale systems with traditional techniques. We provide robust numerical examples for verifying the theory and illustrating the capabilities of the approach being presented. Citation: Multiscale Modeling & Simulation PubDate: 2020-10-05T07:00:00Z DOI: 10.1137/20M1320250 Issue No:Vol. 18, No. 4 (2020)

Authors:Jingrun Chen, Dingjiong Ma, Zhiwen Zhang Pages: 1409 - 1434 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 4, Page 1409-1434, January 2020. The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wave function develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. To address this problem, in this paper we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial grid size is only proportional to the semiclassical parameter and (under suitable conditions) an almost first-order convergence rate is achieved in the random space with respect to the sample number. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for the Schrödinger equation with correlated random potentials in both 1-dimensional and 2-dimensional space. Citation: Multiscale Modeling & Simulation PubDate: 2020-10-08T07:00:00Z DOI: 10.1137/19M127389X Issue No:Vol. 18, No. 4 (2020)

Authors:Hayden Schaeffer, Giang Tran, Rachel Ward, Linan Zhang Pages: 1435 - 1461 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 4, Page 1435-1461, January 2020. Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from undersampled and possibly noisy state-space measurements. The learning problem takes the form of a sparse least-squares fitting over a large set of candidate functions. Based on a Bernstein-like inequality for partly dependent random variables, we provide theoretical guarantees on the recovery rate of the sparse coefficients and the identification of the candidate functions for the corresponding problem. Computational results are demonstrated on datasets generated by the Lorenz 96 equation, the viscous Burgers' equation, and the two-component reaction-diffusion equations. Our formulation includes theoretical guarantees of success and is shown to be efficient with respect to the ambient dimension and the number of candidate functions. Citation: Multiscale Modeling & Simulation PubDate: 2020-10-19T07:00:00Z DOI: 10.1137/18M1194730 Issue No:Vol. 18, No. 4 (2020)

Authors:Eric Cancès, Virginie Ehrlacher, Frédéric Legoll, Benjamin Stamm, Shuyang Xiang Pages: 1179 - 1209 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1179-1209, January 2020. This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Cancès et al., C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 801--806]), where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. The motivation for considering such embedded corrector problems is made clear in the companion article [Cancès et al., J. Comput. Phys., 407 (2020), 109254], where a very efficient algorithm is presented for the resolution of such problems for particular heterogeneous materials. In the present article, we prove that the three different approximations we introduce converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. Citation: Multiscale Modeling & Simulation PubDate: 2020-07-13T07:00:00Z DOI: 10.1137/18M120035X Issue No:Vol. 18, No. 3 (2020)

Authors:Hui Ji, Zuowei Shen, Yufei Zhao Pages: 1210 - 1241 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1210-1241, January 2020. Graph-structured signal enables rich description of data defined in the domain with irregular structure, which has seen its rapid growth in many applications including social, energy, transportation, sensor, neuronal networks, and many others. This paper aims at generalizing discrete framelet transform defined for regular grids in Euclidean space to finite undirected weighted graphs. By leveraging the intuition from classic framelet transform for signals on regular grids, we proposed an approach for constructing multiscale undecimal framelet transform for signals defined on finite graphs with a perfect reconstruction property. The proposed method is based on the definition of basic blocks involved in framelet transform, including graph shift operator, convolution, and band-limited down/up-sampling. These blocks enable a painless construction of a class of multilevel undecimal framelet transforms in vertex domain by directly calling wavelet filter banks of existing wavelet biframes and tight frames. The proposed discrete framelet transform on graphs keeps most desired properties of its counterpart on regular grids, and one can see its usage in applications. Citation: Multiscale Modeling & Simulation PubDate: 2020-07-22T07:00:00Z DOI: 10.1137/19M1259201 Issue No:Vol. 18, No. 3 (2020)

Authors:Sijing Li, Zhiwen Zhang, Hongkai Zhao Pages: 1242 - 1271 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1242-1271, January 2020. We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic approximate low-dimensional structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low-dimensional space and its basis are extracted from solution samples to achieve significant dimension reduction in the solution space. At the online stage, the extracted data-driven basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of approximate low-dimensional structure is established in two scenarios based on (1) high separability of the underlying Green's functions, and (2) smooth dependence of the parameters in the random coefficients. Various online construction methods are proposed for different problem setups. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present extensive numerical examples to demonstrate the accuracy and efficiency of the proposed method. Citation: Multiscale Modeling & Simulation PubDate: 2020-07-27T07:00:00Z DOI: 10.1137/19M1277485 Issue No:Vol. 18, No. 3 (2020)

Authors:Hannes Vandecasteele, Przemysław Zieliński, Giovanni Samaey Pages: 1272 - 1298 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1272-1298, January 2020. We discuss through multiple numerical examples the accuracy and efficiency of a micro-macro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of some slow macroscopic state variables. The algorithm interleaves a short simulation of the stiff SDE with extrapolation of the macroscopic state variables over a longer time interval. After extrapolation, we obtain the reconstructed microscopic state via a matching procedure: we compute the probability distribution that is consistent with the extrapolated state variables, while minimally altering the microscopic distribution that was available just before the extrapolation. In this work, we numerically study the accuracy and efficiency of micro-macro acceleration as a function of the extrapolation time step and as a function of the chosen macroscopic state variables. Additionally, we compare the effect of different hierarchies of macroscopic state variables. We illustrate that the method can take significantly larger time steps than the inner microscopic integrator, while simultaneously being more accurate than approximate macroscopic models. Citation: Multiscale Modeling & Simulation PubDate: 2020-07-28T07:00:00Z DOI: 10.1137/19M1246158 Issue No:Vol. 18, No. 3 (2020)

Authors:Habib Ammari, Bryn Davies, Sanghyeon Yu Pages: 1299 - 1317 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1299-1317, January 2020. In this paper, we study the behavior of the coupled subwavelength resonant modes when two high-contrast acoustic resonators are brought close together. We consider the case of spherical resonators and use bispherical coordinates to derive explicit representations for the capacitance coefficients which, we show, capture the system's resonant behavior at leading order. We prove that the pair of resonators has two subwavelength resonant modes whose frequencies have different leading-order asymptotic behavior. We also derive estimates for the rate at which the gradient of the scattered pressure wave blows up as the resonators are brought together. Citation: Multiscale Modeling & Simulation PubDate: 2020-07-30T07:00:00Z DOI: 10.1137/20M1313350 Issue No:Vol. 18, No. 3 (2020)

Authors:Jean-Pierre Fouque, Ruimeng Hu Pages: 1318 - 1342 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1318-1342, January 2020. Empirical studies indicate the presence of multiscales in the volatility of underlying assets: A fast-scale on the order of days and a slow-scale on the order of months. In our previous works, we have studied the portfolio optimization problem in a Markovian setting under each single scale, the slow one in [J.-P. Fouque and R. Hu, SIAM J. Control Optim., 55 (2017), pp. 1990--2023], and the fast one in [R. Hu, Asymptotic optimal portfolio in fast mean-reverting stochastic environments, in Proceedings of the 2018 IEEE CDC, 2018, pp. 5771--5776]. This paper is dedicated to the analysis when the two scales co-exist in a Markovian setting. We study the terminal wealth utility maximization problem when the volatility is driven by both fast- and slow-scale factors. We first propose a zeroth order strategy, and rigorously establish the first order approximation of the associated problem value. This is done by analyzing the corresponding linear partial differential equation (PDE) via regular and singular perturbation techniques, as in the single-scale cases. Then, we show the asymptotic optimality of our proposed strategy by comparing its performance to admissible strategies of a specific form. Interestingly, we highlight that a pure PDE approach does not work in the multiscale case and, instead, we use the so-called epsilon-martingale decomposition. This completes the analysis of portfolio optimization in both fast mean-reverting and slowly varying Markovian stochastic environments. Citation: Multiscale Modeling & Simulation PubDate: 2020-08-03T07:00:00Z DOI: 10.1137/19M1245967 Issue No:Vol. 18, No. 3 (2020)

Authors:S. N. Gomes, G. A. Pavliotis, U. Vaes Pages: 1343 - 1370 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1343-1370, January 2020. In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure. Citation: Multiscale Modeling & Simulation PubDate: 2020-09-02T07:00:00Z DOI: 10.1137/19M1258116 Issue No:Vol. 18, No. 3 (2020)

Authors:Rachael T. Keller, Jeremy L. Marzuola, Braxton Osting, Michael I. Weinstein Pages: 1371 - 1373 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 3, Page 1371-1373, January 2020. We correct the statement and proof of Corollary 4.2 in Keller et al. [Multiscale Model. Simul., 16 (2018), pp. 1684--1731] corresponding to the case of admissible potentials, which are also reflection invariant ($\rho$-invariant). We also include a short addendum on implications, in this case, for the effective Hamiltonian corresponding to states which are spectrally localized near the ${\bf M}$ point. Citation: Multiscale Modeling & Simulation PubDate: 2020-09-30T07:00:00Z DOI: 10.1137/20M1337727 Issue No:Vol. 18, No. 3 (2020)

Authors:Konrad Simon, Jörn Behrens Pages: 543 - 571 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 543-571, January 2020. Long simulation times in climate science typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and cannot be neglected for reliable long-term simulations. This is typically done via parametrizations, but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel upscaling approach inspired by multiscale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advection-dominated problems arising in climate simulations, the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general, and we demonstrate it as a proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion. Citation: Multiscale Modeling & Simulation PubDate: 2020-04-08T07:00:00Z DOI: 10.1137/18M117248X Issue No:Vol. 18, No. 2 (2020)

Authors:Patrick Murphy, Paul C. Bressloff, Sean D. Lawley Pages: 572 - 588 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 572-588, January 2020. Single-particle tracking experiments have recently found that C. elegans zygotes rely on space-dependent switching diffusivities to form intracellular gradients during cell polarization. The relevant proteins switch between fast-diffusing and slow-diffusing states on timescales that are much shorter than the timescale of diffusion or gradient formation. This manifests in models as a small parameter, allowing an asymptotic analysis of the gradient formation. In this paper we consider how this mechanism of rapidly switching diffusive states interacts with a locally varying periodic microstructure in the cell, incorporated through a second small parameter. We show that an asymptotic analysis based on both small parameters yields different results based on the order of limits taken and suggest an explicit relation between the two parameters for when each type of analysis is appropriate. We further investigate a mean first passage time problem for a diffusing protein to gain insight into the effects of the microstructure on the global environment. Citation: Multiscale Modeling & Simulation PubDate: 2020-04-21T07:00:00Z DOI: 10.1137/19M1271245 Issue No:Vol. 18, No. 2 (2020)

Authors:Kit Newton, Qin Li, Andrew M. Stuart Pages: 589 - 611 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 589-611, January 2020. Many naturally occurring models in the sciences are well approximated by simplified models using multiscale techniques. In such settings it is natural to ask about the relationship between inverse problems defined by the original problem and by the multiscale approximation. We develop an approach to this problem and exemplify it in the context of optical tomographic imaging. Optical tomographic imaging is a technique for inferring the properties of biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a medical imaging methodology. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE from boundary measurements. We study this problem in the Bayesian framework, focussing on the strong scattering regime. In this regime the forward RTE is close to the diffusion equation (DE). We study the RTE in the asymptotic regime where the forward problem approaches the DE and prove convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings. Convergence is proved by studying the distance between the two posterior distributions using the Hellinger metric and using the Kullback--Leibler divergence. Citation: Multiscale Modeling & Simulation PubDate: 2020-04-22T07:00:00Z DOI: 10.1137/19M1247346 Issue No:Vol. 18, No. 2 (2020)

Authors:Dong An, Lin Lin Pages: 612 - 645 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 612-645, January 2020. The dynamics of a closed quantum system is often studied with the direct evolution of the Schrödinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant to physical observables) of the Schrödinger equation can be generally nonoptimal for numerical simulation. This can limit, and in some cases severely limit, the time step size. We find that the optimal gauge choice is given by a parallel transport formulation. This parallel transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The parallel transport dynamics can be derived from a Hamiltonian structure and is thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the parallel transport dynamics in the context of the singularly perturbed linear Schrödinger equation and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrödinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse. Citation: Multiscale Modeling & Simulation PubDate: 2020-04-22T07:00:00Z DOI: 10.1137/18M1179304 Issue No:Vol. 18, No. 2 (2020)

Authors:David Aristoff, Daniel M. Zuckerman Pages: 646 - 673 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 646-673, January 2020. We propose parameter optimization techniques for weighted ensemble sampling of Markov chains in the steady-state regime. Weighted ensemble consists of replicas of a Markov chain, each carrying a weight, that are periodically resampled according to their weights inside of each of a number of bins that partition state space. We derive, from first principles, strategies for optimizing the choices of weighted ensemble parameters, in particular the choice of bins and the number of replicas to maintain in each bin. In a simple numerical example, we compare our new strategies with more traditional ones and with direct Monte Carlo. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-06T07:00:00Z DOI: 10.1137/18M1212100 Issue No:Vol. 18, No. 2 (2020)

Authors:Martin Storath, Andreas Weinmann Pages: 674 - 706 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 674-706, January 2020. In this paper, we consider the sparse regularization of manifold-valued data with respect to an interpolatory wavelet/multiscale transform. We propose and study variational models for this task and provide results on their well-posedness. We present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results to show the potential of the proposed schemes for applications. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-07T07:00:00Z DOI: 10.1137/19M1249801 Issue No:Vol. 18, No. 2 (2020)

Authors:Gao Tang, Haizhao Yang Pages: 707 - 736 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 707-736, January 2020. Multiresolution mode decomposition (MMD) is an adaptive tool to analyze a time series $f(t)=\sum_{k=1}^K f_k(t)$, where $f_k(t)$ is a multiresolution intrinsic mode function (MIMF) of the form $ f_k(t)=\sum\nolimits_{n=-N/2}^{N/2-1} a_{n,k}\cos(2\pi n\phi_k(t))s_{cn,k}(2\pi N_k\phi_k(t))+\sum\nolimits_{n=-N/2}^{N/2-1}b_{n,k} \sin(2\pi n\phi_k(t))s_{sn,k}(2\pi N_k\phi_k(t)) $ with time-dependent amplitudes, frequencies, and waveforms. The multiresolution expansion coefficients $\{a_{n,k}\}$, $\{b_{n,k}\}$ and the shape function series $\{s_{cn,k}(t)\}$ and $\{s_{sn,k}(t)\}$ provide innovative features for adaptive time series analysis. The MMD aims at identifying these MIMFs (including their multiresolution expansion coefficients and shape functions series) from their superposition. However, due to the lack of efficient algorithms to solve the MMD problem, the application of MMD for large-scale data science is prohibitive, especially for real-time data analysis. This paper proposes a fast algorithm for solving the MMD problem based on recursive diffeomorphism-based spectral analysis (RDSA). RDSA admits highly efficient numerical implementation via the nonuniform fast Fourier transform; its convergence and accuracy can be guaranteed theoretically. Numerical examples from synthetic data and natural phenomena are given to demonstrate the efficiency of the proposed method. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-07T07:00:00Z DOI: 10.1137/18M1220649 Issue No:Vol. 18, No. 2 (2020)

Authors:Russel Caflisch, Hung Hsu Chou, Jonathan W. Siegel Pages: 737 - 757 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 737-757, January 2020. Signal fragmentation is the (approximate) representation of a signal as a sum of signal fragments, each of which has compact support. It has been proposed as a method for transmitting a low frequency signal over an array of small antennas. We present a mathematical analysis of signal fragmentation for an idealized model of antenna transmission. In the simplest form of signal fragmentation, each fragment has the same waveform, but the $n$th fragment has an amplitude $a_n$ and is shifted in time by an amount $t_n = n \Delta$. We analyze the spectral leakage (i.e., the error in the Fourier representation) and energy efficiency of signal fragmentation. For a special choice of wavelet the spectral leakage can be eliminated for sinusoidal signals. We also formulate a measure of energy efficiency and perform a scaling analysis of the efficiency with a large number of fragments. Although the efficiency is poor for the original wavelet expansion, an alternative form of fragmentation has efficiency that scales well with the number of fragments. We then find the fragment waveform that optimizes the energy efficiency for a given choice of support size, and we generalize some of these results to AM signals. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-07T07:00:00Z DOI: 10.1137/18M1220595 Issue No:Vol. 18, No. 2 (2020)

Authors:Habib Ammari, Bowen Li, Jun Zou Pages: 758 - 797 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 758-797, January 2020. We study the anomalous electromagnetic scattering in the homogenization regime, by a subwavelength thin layer consisting of periodically distributed plasmonic nanoparticles on a perfectly conducting plane. By using quasi-periodic layer potential techniques, we derive the asymptotic expansion of the electromagnetic field away from the thin layer and quantitatively analyze the field enhancement induced by the excitation of the mixed collective plasmonic resonances, which can be characterized by the spectra of two types of periodic Neumann--Poincaré operators. Based on the asymptotic behavior of the scattered field in the macroscopic scale, characterize the reflection scattering matrix for the thin layer and demonstrate that the optical effect of this metasurface can be effectively approximated by a Leontovich impedance boundary condition, which is uniformly valid no matter whether the incident frequency is near the resonant range. The quantitative approximation clearly shows the blow-up of the field energy and the conversion of the field polarization when the resonance occurs, resulting in a significant change of the reflection property of the conducting plane. These results confirm essential physical changes of electromagnetic metasurface at resonances mathematically, whose occurrence was verified earlier for the acoustic case and the transverse magnetic case. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-11T07:00:00Z DOI: 10.1137/19M1275097 Issue No:Vol. 18, No. 2 (2020)

Authors:Josselin Garnier, Etienne Gay, Eric Savin Pages: 798 - 823 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 798-823, January 2020. We consider the scattering of acoustic waves emitted by an active source above a plane turbulent shear layer. The layer is modeled by a moving random medium with small spatial and temporal fluctuations of its mean velocity, and constant density and speed of sound. We develop a multiscale perturbative analysis for the acoustic pressure field transmitted by the layer and derive its power spectral density when the correlation function of the velocity fluctuations is known. Our aim is to compare the proposed analytical model with some experimental results obtained for jet flows in open wind tunnels. We start with the Euler equations for an ideal fluid flow and linearize them about an ambient, unsteady inhomogeneous flow. We study the transmitted pressure field without fluctuations of the ambient flow velocity to obtain the Green's function of the unperturbed medium with constant characteristics. Then we use a Lippmann--Schwinger equation to derive an analytical expression of the transmitted pressure field, as a function of the velocity fluctuations within the layer. Its power spectral density is subsequently computed invoking a stationary-phase argument, assuming in addition that the source is time-harmonic and the layer is thin. We finally study the influence of the source tone frequency and ambient flow velocity on the power spectral density of the transmitted pressure field and compare our results with other analytical models and experimental data. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-11T07:00:00Z DOI: 10.1137/19M1276492 Issue No:Vol. 18, No. 2 (2020)

Authors:Nikolaos Sfakianakis, Anotida Madzvamuse, Mark A. J. Chaplain Pages: 824 - 850 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 824-850, January 2020. The ability to locally degrade the extracellular matrix (ECM) and interact with the tumor microenvironment is a key process distinguishing cancer cells from normal cells, and is a critical step in the metastatic spread of the tumor. The invasion of the surrounding tissue involves the coordinated action of the cancer cells, the ECM, the matrix degrading enzymes, and the epithelial-to-mesenchymal transition. In this paper, we present a mathematical model which describes the transition from an epithelial invasion strategy of the epithelial-like cells (ECs) to an individual invasion strategy for the mesenchymal-like cells (MCs). We achieve this by formulating a genuinely multiscale and hybrid system consisting of partial and stochastic differential equations that describe the evolution of the ECs and the MCs while accounting for the transitions between them. This approach allows one to reproduce, in a very natural way, fundamental qualitative features of the current biomedical understanding of cancer invasion that are not easily captured by classical modelling approaches, for example, the invasion of the ECM by self-generated gradients, and the formation of EC invasion islands outside of the main body of the tumor. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-13T07:00:00Z DOI: 10.1137/18M1189026 Issue No:Vol. 18, No. 2 (2020)

Authors:Mariya Ptashnyk, Chandrasekhar Venkataraman Pages: 851 - 886 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 851-886, January 2020. We present and analyze a model for cell signaling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces and both inter- and intracellular signaling. Using techniques from the theory of two-scale convergence as well the unfolding method, we show convergence of the solutions to the model to solutions of a two-scale macroscopic problem. We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model. We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-18T07:00:00Z DOI: 10.1137/18M1185661 Issue No:Vol. 18, No. 2 (2020)

Authors:Sean D. Lawley, Varun Shankar Pages: 887 - 915 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 887-915, January 2020. Volume transmission is an important neural communication pathway in which neurons in one brain region influence the neurotransmitter concentration in the extracellular space of a distant brain region. In this paper, we apply asymptotic analysis to a stochastic partial differential equation model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model involves the diffusion equation in a three-dimensional domain with interior holes that randomly switch between being either sources or sinks. These holes model nerve varicosities that alternate between releasing and absorbing neurotransmitter according to when they fire action potentials. In the case that the holes are small, we compute analytically the first two nonzero terms in an asymptotic expansion of the average neurotransmitter concentration. The first term shows that the concentration is spatially constant to leading order and that this constant is independent of many details in the problem. Specifically, this constant first term is independent of the number and location of nerve varicosities, neural firing correlations, and the size and geometry of the extracellular space. The second term shows how these factors affect the concentration at second order. Interestingly, the second term is also spatially constant under some mild assumptions. We verify our asymptotic results by high-order numerical simulation using radial-basis-function--generated finite differences. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-26T07:00:00Z DOI: 10.1137/18M1230773 Issue No:Vol. 18, No. 2 (2020)

Authors:Qingguo Hong, Johannes Kraus, Maria Lymbery, Mary F. Wheeler Pages: 916 - 941 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 916-941, January 2020. We consider flux-based multiple-porosity/multiple-permeability poroelasticity systems describing mulitple-network flow and deformation in a poroelastic medium, also referred to as MPET models. The focus of the paper is on the convergence analysis of the fixed-stress split iteration, a commonly used coupling technique for the flow and mechanics equations defining poromechanical systems. We formulate the fixed-stress split method in this context and prove its linear convergence. The contraction rate of this fixed-point iteration does not depend on any of the physical parameters appearing in the model. This is confirmed by numerical results which further demonstrate the advantage of the fixed-stress split scheme over a preconditioned MinRes solver accelerated by norm-equivalent preconditioning. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-26T07:00:00Z DOI: 10.1137/19M1253988 Issue No:Vol. 18, No. 2 (2020)

Authors:S. Frei, T. Richter Pages: 942 - 969 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 942-969, January 2020. In this article we address flow problems that carry a multiscale character in time. In particular we consider the Navier--Stokes flow in a channel on a fast scale that influences the movement of the boundary which undergoes a deformation on a slow scale in time. We derive an averaging scheme that is of first order with respect to the ratio of time scales $\epsilon$. In order to cope with the problem of unknown initial data for the fast-scale problem, we assume near-periodicity in time. Moreover, we construct a second-order accurate time discretization scheme and derive a complete error analysis for a corresponding simplified ODE system. The resulting multiscale scheme does not ask for the continuous simulation of the fast-scale variable and shows powerful speedups up to 1:10,000 compared to a resolved simulation. Finally, we present some numerical examples for the full Navier--Stokes system to illustrate the convergence and performance of the approach. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-26T07:00:00Z DOI: 10.1137/19M1258396 Issue No:Vol. 18, No. 2 (2020)

Authors:Anaïs Crestetto, Christian Klingenberg, Marlies Pirner Pages: 970 - 998 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 970-998, January 2020. This work is devoted to the numerical simulation of the Bhatnagar--Gross--Krook (BGK) equation for two species in the fluid limit using a particle method. Thus, we are interested in a gas mixture consisting of two species without chemical reactions assuming that the number of particles of each species remains constant. We consider the kinetic two species model proposed by Klingenberg, Pirner, and Puppo in [Kinetic Rel. Models, 10 (2017), pp. 445--465], which separates the intra- and interspecies collisions. We want to study numerically the influence of the two relaxation terms, one corresponding to intraspecies and the other to interspecies collisions. For this, we use the method of micro-macro decomposition. First, we derive an equivalent model based on the micro-macro decomposition (see Bennoune, Lemou, and Mieussens [J. Comput. Phys., 227 (2008), pp. 3781--3803] and Crestetto, Crouseilles, and Lemou [Kinetic Rel. Models, 5 (2012), pp. 787--816]). The kinetic micro part is solved by a particle method, whereas the fluid macro part is discretized by a standard finite volume scheme. The main advantages of this approach are that (i) the noise inherent to the particle method is reduced compared to a standard (without micro-macro decomposition) particle method, and (ii) the computational cost of the method is reduced in the fluid limit since a small number of particles is then sufficient. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-26T07:00:00Z DOI: 10.1137/17M1141023 Issue No:Vol. 18, No. 2 (2020)

Authors:Norbert J. Mauser, Yong Zhang, Xiaofei Zhao Pages: 999 - 1024 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 999-1024, January 2020. We consider numerics/asymptotics for the rotating nonlinear Klein--Gordon (RKG) equation, an important PDE in relativistic quantum physics that can model a rotating galaxy in Minkowski metric and serves also as a model, e.g., for a “cosmic superfluid.” First, we formally show that in the nonrelativistic limit RKG converges to coupled rotating nonlinear Schrödinger equations (RNLS), which are used to describe the particle-antiparticle pair dynamics. Investigations of the vortex state of RNLS are carried out. Second, we propose three different numerical methods to solve RKG from relativistic regimes to nonrelativistic regimes in polar and Cartesian coordinates. In relativistic regimes, a semi-implicit finite difference Fourier spectral method is proposed in polar coordinates where both rotation terms are diagonalized simultaneously. In nonrelativistic regimes, to overcome the fast temporal oscillations, we adopt the rotating Lagrangian coordinates and introduce two efficient multiscale methods with uniform accuracy, i.e., the multirevolution composition method and the exponential integrator. Various numerical results confirm (uniform) accuracy of our methods. Simulations of vortices dynamics are presented. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-27T07:00:00Z DOI: 10.1137/18M1233509 Issue No:Vol. 18, No. 2 (2020)

Authors:T. Zhang, A. Parker, R. P. Carlson, P. S. Stewart, I. Klapper Pages: 1025 - 1052 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1025-1052, January 2020. Models of microbial community dynamics generally rely on a subscale description for microbial metabolisms. In systems such as distributed multispecies communities like biofilms, where it may not be reasonable to simplify to a small number of limiting substrates, tracking the large number of active metabolites likely requires measurement or estimation of large numbers of kinetic and regulatory parameters. Alternatively, a largely kinetics-free framework is proposed combining cellular level constrained, steady state flux analysis of metabolism with macroscale microbial community models. This multiscale setup naturally allows coupling of macroscale information, including measurement data, with cell scale metabolism. Further, flexibility in methodology is stressed: choices at the microscale (e.g., flux balance analysis or elementary flux modes) and at the macroscale (e.g., physical-chemical influences relevant to biofilm or planktonic environments) are available to the user. Illustrative computations in the context of a biofilm, including comparisons of systemic and Nash equilibration as well as an example of coupling experimental data into predictions, are provided. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-27T07:00:00Z DOI: 10.1137/18M1234096 Issue No:Vol. 18, No. 2 (2020)

Authors:Daniela Calvetti, Jamie Prezioso, Rossana Occhipinti, Walter F. Boron, Erkki Somersalo Pages: 1053 - 1075 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1053-1075, January 2020. The mechanism of gas transport across cell membranes remains a topic of considerable interest, particularly regarding the extent to which lipids versus specific membrane proteins provide conduction pathways. Studies of transmembrane carbon dioxide ($CO_2$) transport often rely on data collected under controlled conditions, using pH-sensitive microelectrodes at the extracellular surface to record changes due to extracellular $CO_2$ diffusion and reactions. Although recent detailed computational models can predict a qualitatively correct behavior, a mismatch between the dynamical ranges of the predicted and observed pH curves raises the question of whether the discrepancy may be due to a bias introduced by the pH electrode itself. More specifically, it is reasonable to ask whether bringing the electrode tip near or in contact with the membrane creates a local microenvironment between the electrode tip and the membrane, so that the measured data refer to the microenvironment rather than to the free surface. Here, we introduce a detailed computational model, designed to address this question. We find that, as long as a zone of free diffusion exists between the tip and the membrane, the microenvironment behaves effectively as the free membrane. However, according to our model, when the tip contacts the membrane, partial quenching of extracellular diffusion by the electrode rim leads to a significant increase in the pH dynamics under the electrode, matching values measured in physiological experiments. The computational schemes for the model predictions are based on semidiscretization by a finite element method and on an implicit-explicit time integration scheme to capture the different time scales of the system. Citation: Multiscale Modeling & Simulation PubDate: 2020-05-28T07:00:00Z DOI: 10.1137/19M1262875 Issue No:Vol. 18, No. 2 (2020)

Authors:Carina Bringedal, Lars von Wolff, Iuliu Sorin Pop Pages: 1076 - 1112 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1076-1112, January 2020. We consider a model for precipitation and dissolution in a porous medium, where ions transported by a fluid through the pores can precipitate at the pore walls and form mineral. Also, the mineral can dissolve and become part of the fluid as ions. These processes lead to changes in the flow domain, which are not known a priori but depend on the concentration of the ions dissolved in the fluid. Such a system can be formulated through conservation equations for mass, momentum, and solute in a domain that evolves in time. In this case the fluid and mineral phases are separated by a sharp interface, which also evolves. We consider an alternative approach by introducing a phase field variable, which has a smooth, diffuse transition of nonzero width between the fluid and mineral phases. The evolution of the phase field variable is determined through the Allen--Cahn equation. We show that as the width of the diffuse transition zone approaches zero, the sharp-interface formulation is recovered. When we consider a periodically perforated domain mimicking a porous medium, the phase field formulation is upscaled to Darcy scale by homogenization. Then, the average of the phase field variable represents the porosity. Through cell problems, the effective diffusion and permeability matrices are dependent on the phase field variable. We consider numerical examples to show the behavior of the phase field formulation. We show the effect of flow on the mineral dissolution, and we address the effect of the width of the diffuse interface in the cell problems for both a perforated porous medium and a thin strip. Citation: Multiscale Modeling & Simulation PubDate: 2020-06-11T07:00:00Z DOI: 10.1137/19M1239003 Issue No:Vol. 18, No. 2 (2020)

Authors:Thomas Hudson, Xingjie H. Li Pages: 1113 - 1135 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1113-1135, January 2020. The Mori--Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the importance of memory effects in determining the correct temporal behavior of such systems. Citation: Multiscale Modeling & Simulation PubDate: 2020-06-15T07:00:00Z DOI: 10.1137/18M1222533 Issue No:Vol. 18, No. 2 (2020)

Authors:Marc Dambrine, Helmut Harbrecht Pages: 1136 - 1152 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1136-1152, January 2020. This article combines shape optimization and homogenization techniques by looking for the optimal design of the microstructure in composite materials and of scaffolds. The development of materials with specific properties is of huge practical interest, for example, for medical applications or for the development of lightweight structures in aeronautics. In particular, the optimal design of microstructures leads to fundamental questions for porous media: what is the sensitivity of homogenized coefficients with respect to the shape of the microstructure' We compute Hadamard's shape gradient for the problem of realizing a prescribed effective tensor and demonstrate the applicability and feasibility of our approach through numerical experiments. Citation: Multiscale Modeling & Simulation PubDate: 2020-06-18T07:00:00Z DOI: 10.1137/19M1274638 Issue No:Vol. 18, No. 2 (2020)

Authors:Ke Chen, Qin Li, Jianfeng Lu, Stephen J. Wright Pages: 1153 - 1177 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 2, Page 1153-1177, January 2020. In the framework of generalized finite element methods for elliptic equations with rough coefficients, efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling strategies that construct approximations to the optimal set of basis functions of a given dimension, and proposes a quantitative criterion to analyze and compare these sampling strategies. Numerical evidence shows that the best results are achieved by two strategies, Random Gaussian and Smooth Boundary sampling. Citation: Multiscale Modeling & Simulation PubDate: 2020-06-25T07:00:00Z DOI: 10.1137/18M1166432 Issue No:Vol. 18, No. 2 (2020)

Authors:Leonid Berlyand, Pierre-Emmanuel Jabin, Mykhailo Potomkin, Elżbieta Ratajczyk Pages: 1 - 20 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 1-20, January 2020. The study of active matter consisting of many self-propelled (active) swimmers in an imposed flow is important for many applications. Self-propelled swimmers may represent both living and artificial ones such as bacteria and chemically driven bimetallic nanoparticles. In this work we focus on a kinetic description of active matter represented by self-propelled rods swimming in a viscous fluid confined by a wall. It is well known that walls may significantly affect the trajectories of active rods in contrast to unbounded or periodic containers. Among such effects are accumulation at walls and upstream motion (also known as negative rheotaxis). Our first main result is the rigorous derivation of boundary conditions for the active rods' probability distribution function in the limit of vanishing inertia. Finding such a limit is important because (i) in many examples of active matter inertia is negligible, since swimming occurs in the low Reynolds number regime, and (ii) this limit allows us to reduce the dimension---and so computational complexity---of the kinetic description. For the resulting model, we derive the system in the limit of vanishing translational diffusion which is also typically negligible for active particles. This system allows for tracking separately active particles accumulated at walls and active particles swimming in the bulk of the fluid. Citation: Multiscale Modeling & Simulation PubDate: 2020-01-02T08:00:00Z DOI: 10.1137/19M1263510 Issue No:Vol. 18, No. 1 (2020)

Authors:Ustim Khristenko, Andrei Constantinescu, Patrick Le Tallec, J. Tinsley Oden, Barbara Wohlmuth Pages: 21 - 43 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 21-43, January 2020. Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigue analysis usually employ simple microstructure, consisting of ideal geometries such as ellipsoidal inclusions, which generally does not capture complex geometries. In this paper, we introduce a simple but flexible surrogate microstructure model for two-phase materials through a level-cut of a Gaussian random field with covariance of Matérn class. Such parametrization of the covariance function allows for the representation of a few key design parameters while representing the geometry of inclusions in a more general setting for a large class of random heterogeneous two-phase media. In addition to the traditional morphology descriptors such as porosity, size, and aspect ratio, it provides control of the regularity of the inclusions interface and sphericity. These parameters are estimated from a small number of real material images using Bayesian inversion. An efficient process of evaluating the samples, based on the fast Fourier transform, makes possible the use of Monte Carlo methods to estimate statistical properties for the quantities of interest in a given material class. We demonstrate the overall framework of the use of the surrogate material model in application to the uncertainty quantification in fatigue analysis, its feasibility and efficiency, and its role in the microstructure design. Citation: Multiscale Modeling & Simulation PubDate: 2020-01-08T08:00:00Z DOI: 10.1137/19M1259286 Issue No:Vol. 18, No. 1 (2020)

Authors:Liliana Borcea, Josselin Garnier Pages: 44 - 78 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 44-78, January 2020. We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation, and evanescent modes with amplitudes that are random fields, due to scattering. We obtain a detailed characterization of these amplitudes and thus quantify the transfer of power between the two waveguides in terms of their separation distance. The results show that, no matter how small the fluctuations of the interfaces are, they have a significant effect at a sufficiently large distance of propagation, which manifests in two ways: The first effect is well known and consists of power leakage from the guided modes to the radiation ones. The second effect consists of blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power. Its quantification is the main practical result of the paper. Citation: Multiscale Modeling & Simulation PubDate: 2020-01-16T08:00:00Z DOI: 10.1137/18M1230591 Issue No:Vol. 18, No. 1 (2020)

Authors:Szu-Pei P. Fu, Rolf Ryham, Andreas Klöckner, Matt Wala, Shidong Jiang, Yuan-Nan Young Pages: 79 - 103 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 79-103, January 2020. In this paper, a mathematical model for long-range, hydrophobic attraction between amphiphilic particles is developed to quantify the macroscopic assembly and mechanics of a lipid bilayer membrane in solvents. The nonlocal interactions between amphiphilic particles are obtained from the first domain variation of a hydrophobicity functional, giving rise to forces and torques (between particles) that dictate the motion of both particles and the surrounding solvent. The functional minimizer (that accounts for hydrophobicity at molecular-aqueous interfaces) is a solution to a boundary value problem of the screened Laplace equation. We reformulate the boundary value problem as a second-kind integral equation (SKIE), discretize the SKIE using a Nyström discretization and Quadrature by Expansion (QBX), and solve the resulting linear system iteratively using GMRES. We evaluate the required layer potentials using the GIGAQBX fast algorithm, a variant of the Fast Multipole Method (FMM), yielding the required particle interactions with asymptotically optimal cost. A mobility problem formulation supplies the motion for the rigid particles in a viscous fluid. The simulated fluid-particle systems exhibit a variety of multiscale behaviors over both time and length. Over short time scales, the numerical results show self-assembly for model lipid particles. For large system simulations, the particles form realistic configurations like micelles and bilayers. Over long time scales, the bilayer shapes emerging from the simulation appear to minimize a form of bending energy. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-04T08:00:00Z DOI: 10.1137/18M1219503 Issue No:Vol. 18, No. 1 (2020)

Authors:Jay Chu, John M. Hong, Hsin-Yi Lee Pages: 104 - 130 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 104-130, January 2020. In this paper, we study the approximation and existence of vacuum states in the multiscale gas flows governed by the Cauchy problem of compressible Euler equations containing a small parameter $\eta$ in the initial density. The system of compressible Euler equations is reduced to a hyperbolic resonant system at the vacuum so that the weak solution of the Riemann problem is not suitable as the building block of the Glimm (or Godunov) scheme to establish the existence of vacuum states. We construct a new type of approximate solutions, the weak solutions of the regularized Riemann problem for the leading-order system derived from asymptotic expansions around vacuum states. Such an approximate solution obtained by solving the pressureless Euler equations with generalized Riemann data consists of constant states separated by a composite hyperbolic wave. We show the stability of the regularized Riemann solution, together with the numerical simulations, under the small perturbations of initial data. Adopting the approximate solution as the building block of the generalized Glimm scheme, we prove the existence of the vacuum states by showing the stability and consistency of the scheme as $\eta \rightarrow 0$. The numerical simulation indicates that for any small fixed $t>0$, the approximate solutions converge to the exact solutions of the Cauchy problem in $L^1$ as $\eta \rightarrow 0$. The results of this paper can be applied to some hyperbolic resonant systems of balance laws. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-04T08:00:00Z DOI: 10.1137/19M1290723 Issue No:Vol. 18, No. 1 (2020)

Authors:Gasta͂o A. Braga, Frederico Furtado, Vincenzo Isaia, Long Lee Pages: 131 - 162 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 131-162, January 2020. We systematically study a numerical procedure that reveals the asymptotically self-similar dynamics of solutions of partial differential equations (PDEs). This procedure, based on the renormalization group (RG) theory for PDEs, appeared initially in a conference proceedings [G. A. Braga, F. Furtado, and V. Isaia, in Proceedings of the Fifth International Conference on Dynamical Systems and Differential Equations, Pomona, CA, 2004, pp. 1--13]. A numerical version of the RG method, dubbed nRG, rescales the temporal and spatial variables in each iteration and drives the solutions to a fixed point exponentially fast, which corresponds to the self-similar dynamics of the equations. In this paper, we carefully examine and validate this class of algorithms by comparing the numerical solutions with either exact or asymptotic solutions of model equations found in the literature. The other contribution of the current paper is that we present several examples to demonstrate that this class of nRG algorithms can be applied to a wide range of PDEs to shed light on long time self-similar dynamics of certain physical systems modeled by PDEs. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-04T08:00:00Z DOI: 10.1137/18M120004X Issue No:Vol. 18, No. 1 (2020)

Authors:Pierre Degond, Marina A. Ferreira, Sara Merino-Aceituno, Mickaël Nahon Pages: 163 - 197 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 163-197, January 2020. Swelling media (e.g., gels, tumors) are usually described by mechanical constitutive laws (e.g., Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e., the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets, while in the parallel direction, it is determined by a Laplace--Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-04T08:00:00Z DOI: 10.1137/18M1203158 Issue No:Vol. 18, No. 1 (2020)

Authors:Fei Xu Pages: 198 - 220 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 198-220, January 2020. This paper introduces a cascadic adaptive finite element method for nonlinear eigenvalue equations arising from quantum physics following the multilevel correction strategy. Instead of the classical scheme, which requires solving nonlinear eigenvalue equations on a series of adaptive spaces directly, the new scheme consists of several smoothing processes on an adaptive space sequence and nonlinear eigenvalue equations being solved in a very low dimensional space. The main feature of the proposed scheme is that large-scale nonlinear eigenvalue problem solving is avoided, and the associated smoothing process can be executed efficiently based on the appropriate number of smoothing steps. Thus, efficiency can be enhanced by the proposed cascadic adaptive method. The good performance of the new finite element strategy is examined by various numerical experiments. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-12T08:00:00Z DOI: 10.1137/17M1155569 Issue No:Vol. 18, No. 1 (2020)

Authors:Saumik Dana, Joel Ita, Mary F. Wheeler Pages: 221 - 239 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 221-239, January 2020. We establish a link between the decoupling constraint in a two-grid staggered solution algorithm for consolidation in heterogeneous porous media and the concepts of Voigt and Reuss bounds commonly encountered in the theory of computational homogenization of multiphase composites. Our analysis involves deriving bounds on a tuning parameter in the decoupling constraint for determining the speed and accuracy of the algorithm. An upper bound is obtained from theoretical convergence of the algorithm which leads to the fastest convergence. A lower bound is established by employing the concepts of Voigt and Reuss bounds. From these bounds, we conclude that there is a value for the tuning parameter between the bounds that gives the most accurate solution. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-12T08:00:00Z DOI: 10.1137/18M1187660 Issue No:Vol. 18, No. 1 (2020)

Authors:Habib Ammari, Durga Prasad Challa, Anupam Pal Choudhury, Mourad Sini Pages: 240 - 293 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 240-293, January 2020. We deal with the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. We derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain $\Omega$ of $\mathbb{R}^3$. We handle two situations. (1) In the first one, we distribute the bubbles to occupy a three dimensional domain. For this case, we show that the equivalent speed of propagation changes sign when the medium is excited with frequencies smaller or larger than (but not necessarily close to) the Minnaert resonance. As a consequence, this medium behaves as reflective or absorbing depending on whether the used frequency is smaller or larger than this resonance. In addition, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above a certain threshold, then the medium behaves as a “wall,” i.e., allowing no incident sound to penetrate. (2) In the second one, we distribute the bubbles to occupy a two dimensional (open or closed) surface, not necessarily flat. For this case, we show that the equivalent medium is modeled by a Dirac potential supported on that surface. The sign of the surface potential changes for frequencies smaller or larger than the Minnaert resonance, i.e., it behaves as a smart metasurface reducing or amplifying the transmitted sound across it. As in the three dimensional case, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above an appropriate threshold, then the surface allows no incident sound to be transmitted across the surface, i.e., it behaves as a white screen. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-19T08:00:00Z DOI: 10.1137/19M1237259 Issue No:Vol. 18, No. 1 (2020)

Authors:Martin Heida, Ralf Kornhuber, Joscha Podlesny Pages: 294 - 314 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 294-314, January 2020. Inspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of nonseparating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces, and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into $L^2$ and $H^s$, $s Citation: Multiscale Modeling & Simulation PubDate: 2020-02-19T08:00:00Z DOI: 10.1137/18M1204759 Issue No:Vol. 18, No. 1 (2020)

Authors:Konstantinos Spiliopoulos, Matthew R. Morse Pages: 315 - 350 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 315-350, January 2020. We consider systems of slow-fast diffusions with small noise in the slow component. We construct provably logarithmic asymptotically optimal importance schemes for the estimation of rare events based on the moderate deviations principle. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Moderate deviations--based importance sampling offers a viable alternative to large deviations importance sampling when the events are not too rare. In particular, in many cases of interest one can indeed construct the required change of measure in closed form, a task which is more complicated using the large deviations--based importance sampling, especially when it comes to multiscale dynamically evolving processes. The presence of multiple scales and the fact that we do not make any periodicity assumptions for the coefficients driving the processes complicate the design and the analysis of efficient importance sampling schemes. Simulation studies illustrate the theory. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-19T08:00:00Z DOI: 10.1137/18M1192962 Issue No:Vol. 18, No. 1 (2020)

Authors:Giacomo Dimarco, Lorenzo Pareschi Pages: 351 - 382 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 351-382, January 2020. The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable of considerably accelerating the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates. We show that the additional degrees of freedom can be used to further improve the variance reduction properties of multiscale control variate methods. Citation: Multiscale Modeling & Simulation PubDate: 2020-02-25T08:00:00Z DOI: 10.1137/18M1231985 Issue No:Vol. 18, No. 1 (2020)

Authors:Preston Donovan, Muruhan Rathinam Pages: 383 - 414 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 383-414, January 2020. We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process $\varepsilon^2 Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and that $\varepsilon (Z(t/\varepsilon^2)-\bar{U}t/\varepsilon^2)$ converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our framework allows for the modeling of very general forms of interactions such as attraction, repulsion, and bonding. The case of interest to us is that of null drift $\bar{U}=0$ and we provide some sufficient conditions for null drift that include certain symmetries of the graph. We also provide a formal asymptotic derivation of the effective diffusivity in analogy with homogenization theory for PDEs. For the case of reversible jump rates, we derive an equivalent variational formulation. This derivation involves developing notions of gradient for functions on the graph's nodes, divergence for ${\mathbb R}^d$-valued functions on the graph's edges, and a divergence theorem. Citation: Multiscale Modeling & Simulation PubDate: 2020-03-03T08:00:00Z DOI: 10.1137/18M1213981 Issue No:Vol. 18, No. 1 (2020)

Authors:Weihua Deng, Xudong Wang, Pingwen Zhang Pages: 415 - 443 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 415-443, January 2020. The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit process of isotropic Lévy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing anisotropic normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional Laplacian $\Delta_m^{\beta/2}$ and the tempered one $\Delta_m^{\beta/2,\lambda}$ in $\mathbb{R}^n$. Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new anisotropic diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their well-posedness of the Galerkin weak formulation in $\mathbb{R}^n$. To obtain the coercivity, we claim that the probability density function $Y$ should be nondegenerate. Citation: Multiscale Modeling & Simulation PubDate: 2020-03-18T07:00:00Z DOI: 10.1137/18M1184990 Issue No:Vol. 18, No. 1 (2020)

Authors:Nicolas Crouseilles, Shi Jin, Mohammed Lemou, Florian Méhats Pages: 444 - 474 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 444-474, January 2020. In this paper, we propose a numerical method solving the one space dimensional semiclassical kinetic graphene model introduced in [O. Morandi and F. Schürrer, J. Phys. A, 44 (2012), pp. 265--301] involving fast oscillations in time, space, and momentum. This method can numerically capture the oscillatory space-time quantum solution pointwisely even without numerically resolving the frequency. We prove that the underlying micro-macro equations have smooth (up to a certain order of derivatives) solutions with respect to the frequency, and then we prove the uniform accuracy of the numerical discretization for a scalar model equation exhibiting the same oscillatory behavior. Numerical experiments verify the theory. Citation: Multiscale Modeling & Simulation PubDate: 2020-03-24T07:00:00Z DOI: 10.1137/18M1173770 Issue No:Vol. 18, No. 1 (2020)

Authors:Eric Chung, Yalchin Efendiev, Yanbo Li, Qin Li Pages: 475 - 501 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 475-501, January 2020. The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime and study the numerical behavior of the generalized multiscale finite element method (GMsFEM) in the fluid regime when high oscillation in the media presents. The GMsFEM is a general approach [E. Chung, Y. Efendiev, and T. Y. Hou, J. Comput. Phys., 320 (2016), pp. 69--95] to numerically treat equations with multiscale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the well-posedness of the method on the Boltzmann equation and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number. Citation: Multiscale Modeling & Simulation PubDate: 2020-03-25T07:00:00Z DOI: 10.1137/19M1256282 Issue No:Vol. 18, No. 1 (2020)

Authors:José A. Carrillo, Serafim Kalliadasis, Sergio P. Perez, Chi-Wang Shu Pages: 502 - 541 Abstract: Multiscale Modeling & Simulation, Volume 18, Issue 1, Page 502-541, January 2020. Well-balanced and free energy dissipative first- and second-order accurate finite-volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Lyapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analogous property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories. Citation: Multiscale Modeling & Simulation PubDate: 2020-03-30T07:00:00Z DOI: 10.1137/18M1230050 Issue No:Vol. 18, No. 1 (2020)