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Authors:Lena Leitenmaier, Olof Runborg Pages: 1 - 35 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 1-35, January 2022. In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau--Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau--Lifshitz equation with a highly oscillatory coefficient is chosen as the micro problem for all models. We then show that the averaging errors only depend on $\varepsilon$ and the size of the microscopic oscillations, as well as the size of the averaging domain in time and space and the choice of averaging kernels. Citation: Multiscale Modeling & Simulation PubDate: 2022-01-11T08:00:00Z DOI: 10.1137/21M1409408 Issue No:Vol. 20, No. 1 (2022)

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Authors:Quentin Ayoul-Guilmard, Anthony Nouy, Christophe Binetruy Pages: 36 - 71 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 36-71, January 2022. This paper addresses the complexity reduction of stochastic homogenization of a class of random materials for a stationary diffusion equation. A cost-efficient approximation of the correctors is obtained using a method designed to exploit quasi-periodicity. Accuracy and cost reduction are investigated for local perturbations or small transformations of periodic materials as well as for materials with no periodicity but a mesoscopic structure, for which the limitations of the method are shown. Finally, for materials outside the scope of this method, we propose to use the approximation of homogenized quantities as control variates for variance reduction of a more accurate and costly Monte Carlo estimator (using a multifidelity Monte Carlo method). The resulting cost reduction is illustrated in a numerical experiment and compared with a control variate method from weakly stochastic homogenization. The limits of this variance reduction technique are tested on materials without periodicity or mesoscopic structure. Citation: Multiscale Modeling & Simulation PubDate: 2022-01-13T08:00:00Z DOI: 10.1137/18M1191221 Issue No:Vol. 20, No. 1 (2022)

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Authors:Sylvain Wolf Pages: 72 - 106 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 72-106, January 2022. In our recent work [X. Blanc and S. Wolf, Asymptot. Anal., 126 (2021), pp. 129--155], we have studied the homogenization of the Poisson equation in a class of non-periodically perforated domains. In this paper, we examine the case of the Stokes system. We consider a porous medium in which the characteristic distance between two holes, denoted by $\varepsilon$, is proportional to the characteristic size of the holes. It is well known (see [G. Allaire, Asymptot. Anal., 2 (1989), pp. 203--222; E. Sanchez-Palencia, in Non-Homogeneous Media and Vibration Theory, Springer, New York, 1980, pp. 129--157; L. Tartar, in Non-Homogeneous Media and Vibration Theory, Springer, New York, 1980, Appendix]) that, when the holes are periodically distributed in space, the velocity converges to a limit given by Darcy's law when the size of the holes tends to zero. We generalize these results to the setting of [X. Blanc and S. Wolf, Asymptot. Anal., 126 (2021), pp. 129--155]. The nonperiodic domains are defined as a local perturbation of a periodic distribution of holes. We obtain classical results of the homogenization theory in perforated domains (existence of correctors and regularity estimates uniform in $\varepsilon$) and we prove $H^2$-convergence estimates for particular force fields. Citation: Multiscale Modeling & Simulation PubDate: 2022-01-20T08:00:00Z DOI: 10.1137/21M1390815 Issue No:Vol. 20, No. 1 (2022)

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Authors:Chou Kao, Yu-Yu Liu, Jack Xin Pages: 107 - 117 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 107-117, January 2022. The Arnold--Beltrami--Childress (ABC) flow and the Kolmogorov flow are three-dimensional periodic divergence-free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. First, we give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely, those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but noncoercive Hamilton--Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute front speeds of G-equation based on a semi-Lagrangian scheme with Strang splitting and weighted essentially nonoscillatory interpolation. The Semi-Lagrangian scheme is stable when the ratio of time step and spatial grid size is smaller than a positive constant independent of the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits. Citation: Multiscale Modeling & Simulation PubDate: 2022-01-31T08:00:00Z DOI: 10.1137/20M1387699 Issue No:Vol. 20, No. 1 (2022)

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Authors:Charles-Edouard Bréhier, Shmuel Rakotonirina-Ricquebourg Pages: 118 - 163 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 118-163, February 2022. We introduce and study a notion of asymptotic preserving schemes, related to convergence in distribution, for a class of slow-fast stochastic differential equations. In some examples, crude schemes fail to capture the correct limiting equation resulting from averaging and diffusion approximation procedures. We propose examples of asymptotic preserving schemes: when the time-scale separation vanishes, one obtains a limiting scheme, which is shown to be consistent in distribution with the limiting stochastic differential equation. Numerical experiments illustrate the importance of the proposed asymptotic preserving schemes for several examples. In addition, in the averaging regime, error estimates are obtained, and the proposed scheme is proved to be uniformly accurate. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-07T08:00:00Z DOI: 10.1137/20M1379836 Issue No:Vol. 20, No. 1 (2022)

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Authors:Yongyong Cai, Yan Wang Pages: 164 - 187 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 164-187, March 2022. We propose a class of efficient and uniformly accurate nested Picard iterative integrators (NPI) for solving the nonlinear Dirac equation (NLDE) in the nonrelativistic regime, and apply it to study the convergence rates of the NLDE to its limiting models, the dynamics of traveling waves, and the two-dimensional dynamics. The NLDE involves a dimensionless parameter $\varepsilon\in (0, 1]$, and its solution is highly oscillatory in time with wavelength $O(\varepsilon^2)$ in the nonrelativistic regime. To gain uniform accuracies in time, the NPI method employs an operator decomposition technique for explicitly separating the highly oscillatory phases and utilizes exponential wave integrators for the time integrals. Moreover, with the help of nested Picard iterations, the NPI method could easily achieve uniform first- and second-order accuracies. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-22T08:00:00Z DOI: 10.1137/20M133573X Issue No:Vol. 20, No. 1 (2022)

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Authors:Yifan Chen, Thomas Y. Hou Pages: 188 - 219 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 188-219, March 2022. There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-24T08:00:00Z DOI: 10.1137/20M1372214 Issue No:Vol. 20, No. 1 (2022)

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Authors:Paul Dupuis, Guo-Jhen Wu Pages: 220 - 249 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 220-249, March 2022. Metastability is a formidable challenge to Markov chain Monte Carlo methods. In this paper we present methods for algorithm design to meet this challenge. The design problem we consider is temperature selection for the infinite swapping scheme, which is the limit of the widely used parallel tempering scheme obtained when the swap rate tends to infinity. We use a recently developed tool for the analysis of the empirical measure of a small noise diffusion to transform the variance reduction problem into an explicit optimization problem. Our first analysis of the optimization problem is in the setting of a double-well model, and it shows that the optimal selection of temperature ratios is a geometric sequence except possibly the highest temperature. In the same setting we identify two different sources of variance reduction and show how their competition determines the optimal highest temperature. In the general multiwell setting we prove that the same geometric sequence of temperature ratios as in the two-well case is always nearly optimal, with a performance gap that decays geometrically in the number of temperatures. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-24T08:00:00Z DOI: 10.1137/21M1402029 Issue No:Vol. 20, No. 1 (2022)

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Authors:Samuel Heroy, Dane Taylor, Feng Shi, M. Gregory Forest, Peter J. Mucha Pages: 250 - 281 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 250-281, March 2022. In composite materials composed of a soft polymer matrix and rigid, high aspect-ratio particles, the composite undergoes a transition in mechanical strength when the incorporated particle phase surpasses a critical density. This phenomenon, termed rheological percolation, is well known to occur in many polymer-rod and polymer-platelet composites at a critical density that exceeds the conductivity percolation threshold (which occurs when the conducting particles form a large connected component that spans the composite). Contact percolation in rod-like composites has been routinely exploited to engineer thermal or electrical conductivity in otherwise nonconducting polymers, and the characterization of contact percolation is well established. Mechanical or rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation is due to a rigidity phenomenon, whereby a large rigid component of inclusions spans the composite. Here we build an algorithm to detect the rigidity percolation threshold in rod-polymer composites. We model the composites as systems of randomly distributed, soft-core (intersecting at contact) rods and study the emergence of a giant (i.e., spanning) rigid component. Building on our previous results for two-dimensional composites, we develop an approximate algorithm that identifies spanning rigid components through hierarchically identifying and compressing provably rigid motifs---equivalently, decomposing a giant rigid component into rigid assemblies of a hierarchy of successively smaller rigid components. We apply this algorithm to random monodisperse systems that are generated in Monte Carlo simulations to estimate a rigidity percolation threshold (critical density) and explore its dependence on rod aspect ratio. We show that this transition point---like its contact percolation analogue---scales inversely with the excluded volume of a rod. Moreover, this implies that the critical contact number (i.e., the number of contacts per rod at the rigidity percolation threshold) is constant for aspect ratios above some relatively low value and is lower than the prediction from Maxwell's isostatic condition. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-24T08:00:00Z DOI: 10.1137/21M1401206 Issue No:Vol. 20, No. 1 (2022)

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Authors:Jiachuan Cao, Liqun Cao Pages: 282 - 322 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 282-322, March 2022. This paper discusses the consistency and algorithm for semiconductor Boltzmann transport equations (BTEs) with multivalley arising from the modeling of the carrier transport in semiconductor materials and devices. A direct simulation Monte Carlo (DSMC) method for the BTEs with the space nonhomogeneous and the nonlinear collision terms is presented. The new results obtained in this paper are as follows. First, some theoretical results such as the consistency between the solution of the BTEs and a Markov process, the invariant measure, and the well-posedness for the solution of the BTEs are proved. Second, a DSMC algorithm and the convergence results are proposed. Third, some numerical results are carried out to validate the theoretical results of this paper. Finally, the typical applications for some semiconductor materials and devices such as GaAs and graphene are advanced. Citation: Multiscale Modeling & Simulation PubDate: 2022-02-28T08:00:00Z DOI: 10.1137/19M128750X Issue No:Vol. 20, No. 1 (2022)

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Authors:Xiaoxue Qin, Yejun Gu, Luchan Zhang, Yang Xiang Pages: 323 - 348 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 323-348, March 2022. We present a continuum model to determine the dislocation structure and energy of low angle grain boundaries in three dimensions. The equilibrium dislocation structure is obtained by minimizing the grain boundary energy that is associated with the constituent dislocations subject to the constraint of Frank's formula. The orientation-dependent continuous distributions of dislocation lines on grain boundaries are described conveniently using the dislocation density potential functions, whose contour lines on the grain boundaries represent the dislocations. The energy of a grain boundary is the total energy of the constituent dislocations derived from a discrete dislocation dynamics model, incorporating both the dislocation line energy and reactions of dislocations. The constrained energy minimization problem is solved by the augmented Lagrangian method and projection method. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of the energy and dislocation densities of both planar and curved low angle grain boundaries. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-17T07:00:00Z DOI: 10.1137/20M1366782 Issue No:Vol. 20, No. 1 (2022)

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Authors:Philippe Guyenne, Adilbek Kairzhan, Catherine Sulem Pages: 349 - 378 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 349-378, March 2022. This article concerns the water wave problem in a three-dimensional domain of infinite depth and examines the modulational regime for weakly nonlinear wavetrains. We use the method of normal form transformations near the equilibrium state to provide a new derivation of the Hamiltonian Dysthe equation describing the slow evolution of the wave envelope. A precise calculation of the third-order normal form allows for a refined reconstruction of the free surface. We test our approximation against direct numerical simulations of the three-dimensional Euler system and against predictions from the classical Dysthe equation, and find very good agreement. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-17T07:00:00Z DOI: 10.1137/21M1432788 Issue No:Vol. 20, No. 1 (2022)

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Authors:Konstantin Fackeldey, Mathias Oster, Leon Sallandt, Reinhold Schneider Pages: 379 - 403 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 379-403, March 2022. We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space by a sequence of linear equations. As high degree multipolynomials are needed, the corresponding equations suffer from the curse of dimensionality even in moderate dimensions. We employ Tensor-Train methods to account for this problem. The approximation process within the Policy Iteration is done via a Least-Squares ansatz and the integration is done via Monte-Carlo methods. Numerical evidences are given for the (multidimensional) double-well potential, a three-hole potential, and a 40-dimensional stochastic Van der Pol oscillator. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-21T07:00:00Z DOI: 10.1137/20M1372500 Issue No:Vol. 20, No. 1 (2022)

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Authors:Daria Stepanova, Helen M. Byrne, Philip K. Maini, Tomás Alarcón Pages: 404 - 432 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 404-432, March 2022. Hybrid multiscale modeling has emerged as a useful framework for modeling complex biological phenomena. However, when accounting for stochasticity in the internal dynamics of agents, these models frequently become computationally expensive. Traditional techniques to reduce the computational intensity of such models can lead to a reduction in the richness of the dynamics observed, compared to the original system. Here we use large deviation theory to decrease the computational cost of a spatially extended multiagent stochastic system with a region of multistability by coarse-graining it to a continuous time Markov chain on the state space of stable steady states of the original system. Our technique preserves the original description of the stable steady states of the system and accounts for noise-induced transitions between them. We apply the method to a bistable system modeling phenotype specification of cells driven by a lateral inhibition mechanism. For this system, we demonstrate how the method may be used to explore different pattern configurations and unveil robust patterns emerging on longer timescales. We then compare the full stochastic, coarse-grained, and mean-field descriptions via pattern quantification metrics and in terms of the numerical cost of each method. Our results show that the coarse-grained system exhibits the lowest computational cost while preserving the rich dynamics of the stochastic system. The method has the potential to reduce the computational complexity of hybrid multiscale models, making them more tractable for analysis, simulation, and hypothesis testing. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-22T07:00:00Z DOI: 10.1137/21M1418575 Issue No:Vol. 20, No. 1 (2022)

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Authors:Stephan Gärttner, Peter Frolkovič, Peter Knabner, Nadja Ray Pages: 433 - 461 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 433-461, March 2022. In this paper, we introduce a pore-scale model for reactive flow and transport in evolving porous media exhibiting two competing mineral phases. By formal two-scale asymptotic expansion in a level-set framework an effective micro-macro model is derived. As such, our approach comprises flow and transport equations on the macroscopic scale including effective hydrodynamic parameters calculated from representative unit cells. Conversely, the macroscopic solutes' concentrations alter the unit cells' geometrical structure by triggering dissolution or precipitation processes. The numerical implementation of such micro-macro models poses several challenges, especially in terms of geometry representation and computational complexity. In this research, the Voronoi implicit interface method is applied to characterize and evolve the two-mineral structure and first-order convergence is obtained in a test case where analytical solutions are available. Furthermore, we present a sophisticated overall solution strategy for the introduced fully coupled nonlinear micro-macro problem and conduct numerical simulations. In doing so, the significant performance enhancements arising from machine learning techniques are evaluated. To this end, a convolutional neural network is trained on (realistic) unit cell geometries for permeability prediction and deployed in a micro-macro simulation. The outcome is compared to the respective results obtained by classical methods in terms of predictive power and computational effort. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-22T07:00:00Z DOI: 10.1137/20M1380648 Issue No:Vol. 20, No. 1 (2022)

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Authors:Qingqing Feng, Gregoire Allaire, Pascal Omnes Pages: 462 - 492 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 462-492, March 2022. This paper addresses an enriched nonconforming multiscale finite element method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, et al., Multiscale Model. Simul., 13 (2015), pp. 1146--1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389, 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows one to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-22T07:00:00Z DOI: 10.1137/21M141926X Issue No:Vol. 20, No. 1 (2022)

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Authors:Guillaume Bal, Daniel Massatt Pages: 493 - 523 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 493-523, March 2022. This paper analyzes Floquet topological insulators resulting from the time-harmonic irradiation of electromagnetic waves on two-dimensional materials such as graphene. We analyze the bulk and edge topologies of approximations to the evolution of the light-matter interaction. Topologically protected interface states are created by spatial modulations of the drive polarization across an interface. In the high-frequency modulation regime, we obtain a sequence of topologies that apply to different time scales. Bulk-difference invariants are computed in detail and a bulk-interface correspondence is shown to apply. We also analyze a high-frequency high-amplitude modulation resulting in a large-gap effective topology that remains valid only for moderately long times. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-31T07:00:00Z DOI: 10.1137/21M1392826 Issue No:Vol. 20, No. 1 (2022)

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Authors:Beilei Liu, Huajie Chen, Geneviève Dusson, Jun Fang, Xingyu Gao Pages: 524 - 550 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 524-550, March 2022. We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with the Kohn--Sham density functional theory and the projector augmented wave method, illustrating the efficiency and potential of the algorithm. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-31T07:00:00Z DOI: 10.1137/21M1396241 Issue No:Vol. 20, No. 1 (2022)

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Authors:Julia Schleuß, Kathrin Smetana Pages: 551 - 582 Abstract: Multiscale Modeling & Simulation, Volume 20, Issue 1, Page 551-582, March 2022. We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To prove compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin--Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the $L^2(H^1)$-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time. Citation: Multiscale Modeling & Simulation PubDate: 2022-03-31T07:00:00Z DOI: 10.1137/20M1384294 Issue No:Vol. 20, No. 1 (2022)