Wing Tat Leung

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is appropriately chosen. We would like to show a mesh-dependent convergence with a minimal number of basis functions. Our construction starts with an auxiliary multiscale space by solving local spectral problems. In auxiliary multiscale space, we select the basis functions that correspond to small (contrast-dependent) eigenvalues. These basis functions represent the channels (high-contrast features that connect the boundaries of the coarse block). Using the auxiliary space, we propose a constraint energy minimization to construct multiscale spaces. The minimization is performed in the oversampling domain, which is larger than the target coarse block. The constraints allow handling non-decaying components of the local minimizers. If the auxiliary space is correctly chosen, we show that the convergence rate is independent of the contrast (because the basis representing the channels are included in the auxiliary space) and is proportional to the coarse-mesh size (because the constrains handle non-decaying components of the local minimizers). The oversampling size weakly depends on the contrast as our analysis shows. The convergence theorem requires that channels are not aligned with the coarse edges, which hold in many applications, where the channels are oblique with respect to the coarse-mesh geometry. The numerical results confirm our theoretical results. In particular, we show that if the oversampling domain size is not sufficiently large, the errors are large. To remove the contrast-dependence of the oversampling size, we propose a modified construction for basis functions and present numerical results and the analysis.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung et al.

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. The main contributions of this paper are (1) the rigorous analysis of the offline approach (2) the development of the online procedures and their analysis (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.

Eric Chung, Yalchin Efendiev, Wing Tat Leung

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that can not be localized, in general. This is built on our previous work on the Generalized Multiscale Finite Element Method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast-dependence in the convergence. The method's convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.

Mohammed Al Kobaisi, Dmitry Ammosov, Yalchin Efendiev et al.

We develop a multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM) for constructing coarse-scale models in heterogeneous media that simultaneously provide accurate numerical approximations and physically consistent macroscopic equations. Classical multiscale methods efficiently approximate fine-scale solutions on coarse grids using localized basis functions, but they do not offer a systematic pathway for deriving macroscopic governing equations. To overcome this limitation, we introduce a unified framework that integrates multiscale finite element constructions with multicontinuum representations. The proposed method builds on the structure of GMsFEM and exploits a representation of multiscale basis functions that separates coarse variables and their gradients. We construct continuum-dependent basis functions using auxiliary fields defined through local problems with integral constraints, ensuring that each basis function is associated with a specific continuum. This leads to a decomposition of the coarse-scale solution into continuum variables and their gradients, establishing a direct connection between multiscale discretizations and multicontinuum homogenization. Compared to existing multicontinuum approaches, the proposed framework provides greater flexibility in handling heterogeneous media with spatially varying numbers of continua and is naturally embedded within a standard finite element setting. This enables both systematic derivation of macroscopic equations and straightforward numerical implementation. We apply the proposed method to the upscaling of two-phase immiscible flow in heterogeneous porous media, where multiple interacting continua, including mobile and trapped phases, arise. With the proposed approaches, we derive new macroscopic models and show that if classical models are used, the errors can be large.

Siu Wun Cheung, Eric T. Chung, Yalchin Efendiev et al.

The dual continuum model serves as a powerful tool in the modeling of subsurface applications. It allows a systematic coupling of various components of the solutions. The system is of multiscale nature as it involves high heterogeneous and high contrast coefficients. To numerically compute the solutions, some types of reduced order methods are necessary. We will develop and analyze a novel multiscale method based on the recent advances in multiscale finite element methods. Our method will compute multiple local multiscale basis functions per coarse region. The idea is based on some local spectral problems, which are important to identify high contrast channels, and an energy minimization principle. Using these concepts, we show that the basis functions are localized, even in the presence of high contrast long channels and fractures. In addition, we show that the convergence of the method depends only on the coarse mesh size. Finally, we present several numerical tests to show the performance.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung et al.

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space-time coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space-time coarse cells. We construct space-time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary conditions and oversampling. We construct the local spectral decomposition based on our analysis, as presented in the paper. We present numerical results to confirm our theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces. We remark that the proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space-time cells, and (3) local boundary conditions in space-time cells.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [4]). The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.

Min Wang, Siu Wun Cheung, Eric T. Chung et al.

In this paper, we propose a deep-learning-based approach to a class of multiscale problems. THe Generalized Multiscale Finite Element Method (GMsFEM) has been proven successful as a model reduction technique of flow problems in heterogeneous and high-contrast porous media. The key ingredients of GMsFEM include mutlsicale basis functions and coarse-scale parameters, which are obtained from solving local problems in each coarse neighborhood. Given a fixed medium, these quantities are precomputed by solving local problems in an offline stage, and result in a reduced-order model. However, these quantities have to be re-computed in case of varying media. The objective of our work is to make use of deep learning techniques to mimic the nonlinear relation between the permeability field and the GMsFEM discretizations, and use neural networks to perform fast computation of GMsFEM ingredients repeatedly for a class of media. We provide numerical experiments to investigate the predictive power of neural networks and the usefulness of the resultant multiscale model in solving channelized porous media flow problems.

Maria Vasilyeva, Eric T. Chung, Wing Tat Leung et al.

In this work, we present an upscaled model for mixed dimensional coupled flow problem in fractured porous media. We consider both embedded and discrete fracture models (EFM and DFM) as fine scale models which contain coupled system of equations. For fine grid discretization, we use a conservative finite-volume approximation. We construct an upscaled model using the non-local multicontinuum (NLMC) method for the coupled system. The proposed upscaled model is based on a set of simplified multiscale basis functions for the auxiliary space and a constraint energy minimization principle for the construction of multiscale basis functions. Using the constructed NLMC-multiscale basis functions, we obtain an accurate coarse grid upscaled model. We present numerical results for both fine-grid models and upscaled coarse-grid models using our NLMC method. We consider model problems with (1) discrete fracture fine grid model with low and high permeable fractures; (2) embedded fine grid model for two types of geometries with differnet fracture networks and (3) embedded fracture fine grid model with heterogeneous permeability. The simulations using the upscaled model provide very accurate solutions with significant reduction in the dimension of the problem.

Maria Vasilyeva, Eric T. Chung, Wing Tat Leung et al.

In this paper, we present an upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations. Our methodology is based on the recently developed Non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. We will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations, with the aim of handling non-homogeneous boundary conditions on perforations. We start with describing our method for the Laplace equation, and then extending the framework for the elasticity problem and parabolic equations. The resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid. We will present numerical results (1) for steady and unsteady problems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robin non-homogeneous boundary conditions on perforations. Numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedom.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung et al.

We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In addition, we coarsen the corresponding random space through a clustering algorithm. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale SPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. The new method offers considerable savings in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov-Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions, and are related to the trial space. The resulting test functions have a localization property, and can therefore be computed locally. We will prove the stability, and present several numerical results. Our numerical results confirm that our test space gives a good stability, in the sense that the solution error is close to the best approximation error.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung et al.

Numerical homogenization and multiscale finite element methods construct effective properties on a coarse grid by solving local problems and extracting the average effective properties from these local solutions. In some cases, the solutions of local problems can be expensive to compute due to scale disparity. In this setting, one can basically apply a homogenization or multiscale method re-iteratively to solve for the local problems. This process is known as re-iterated homogenization and has many variations in the numerical context. Though the process seems to be a straightforward extension of two-level process, it requires some careful implementation and the concept development for problems without scale separation and high contrast. In this paper, we consider the Generalized Multiscale Finite Element Method (GMsFEM) and apply it iteratively to construct its multiscale basis functions. The main idea of the GMsFEM is to construct snapshot functions and then extract multiscale basis functions (called offline space) using local spectral decompositions in the snapshot spaces. The extension of this construction to several levels uses snapshots and offline spaces interchangebly to achieve this goal. At each coarse-grid scale, we assume that the offline space is a good approximation of the solution and use all possible offline functions or randomization as boundary conditions and solve the local problems in the offline space at the previous (finer) level, to construct snapshot space. We present an adaptivity strategy and show numerical results for flows in heterogeneous media and in perforated domains.

Yating Wang, Wing Tat Leung, Guang Lin

In this work, we propose an adaptive sparse learning algorithm that can be applied to learn the physical processes and obtain a sparse representation of the solution given a large snapshot space. Assume that there is a rich class of precomputed basis functions that can be used to approximate the quantity of interest. We then design a neural network architecture to learn the coefficients of solutions in the spaces which are spanned by these basis functions. The information of the basis functions are incorporated in the loss function, which minimizes the differences between the downscaled reduced order solutions and reference solutions at multiple time steps. The network contains multiple submodules and the solutions at different time steps can be learned simultaneously. We propose some strategies in the learning framework to identify important degrees of freedom. To find a sparse solution representation, a soft thresholding operator is applied to enforce the sparsity of the output coefficient vectors of the neural network. To avoid over-simplification and enrich the approximation space, some degrees of freedom can be added back to the system through a greedy algorithm. In both scenarios, that is, removing and adding degrees of freedom, the corresponding network connections are pruned or reactivated guided by the magnitude of the solution coefficients obtained from the network outputs. The proposed adaptive learning process is applied to some toy case examples to demonstrate that it can achieve a good basis selection and accurate approximation. More numerical tests are performed on two-phase multiscale flow problems to show the capability and interpretability of the proposed method on complicated applications.

Yueqi Wang, Wing Tat Leung, Guanglian Li

We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in homogeneous media with high wavenumber are particularly challenging due to the so-called pollution effect: the mesh size must be significantly smaller than the reciprocal of the wavenumber to achieve a desired accuracy. This challenge is amplified in heterogeneous media, which frequently occur in practical applications such as metamaterial simulations, since resolving the heterogeneity is necessary for obtaining reliable solutions. Our approach overcomes this difficulty by avoiding explicit resolution of the heterogeneity, while employing a mesh size that depends almost linearly on the reciprocal of the wavenumber. The approximation properties and stability of the method rely critically on the development and rigorous analysis of a novel, nonstandard variational formulation, which constitutes the main innovation of this work. Extensive numerical experiments are provided to validate our theoretical findings.

Fan Wang, Yating Wang, Wing Tat Leung et al.

Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation. However, this approach usually assumes scale separation and that the heterogeneity of the solution can be approximated by the solution average in each coarse block. For complex multiscale problems, the computed single effective properties/continuum might be inadequate. In this paper, we propose a novel learning-based multi-continuum model to enrich the homogenized equation and improve the accuracy of the single continuum model for multiscale problems with some given data. Without loss of generalization, we consider a two-continuum case. The first flow equation keeps the information of the original homogenized equation with an additional interaction term. The second continuum is newly introduced, and the effective permeability in the second flow equation is determined by a neural network. The interaction term between the two continua aligns with that used in the Dual-porosity model but with a learnable coefficient determined by another neural network. The new model with neural network terms is then optimized using trusted data. We discuss both direct back-propagation and the adjoint method for the PDE-constraint optimization problem. Our proposed learning-based multi-continuum model can resolve multiple interacted media within each coarse grid block and describe the mass transfer among them, and it has been demonstrated to significantly improve the simulation results through numerical experiments involving both linear and nonlinear flow equations.

Eric Chung, Yalchin Efendiev, Wing Tat Leung et al.

In this work, we propose a multi-agent actor-critic reinforcement learning (RL) algorithm to accelerate the multi-level Monte Carlo Markov Chain (MCMC) sampling algorithms. The policies (actors) of the agents are used to generate the proposal in the MCMC steps; and the critic, which is centralized, is in charge of estimating the long term reward. We verify our proposed algorithm by solving an inverse problem with multiple scales. There are several difficulties in the implementation of this problem by using traditional MCMC sampling. Firstly, the computation of the posterior distribution involves evaluating the forward solver, which is very time consuming for a problem with heterogeneous. We hence propose to use the multi-level algorithm. More precisely, we use the generalized multiscale finite element method (GMsFEM) as the forward solver in evaluating a posterior distribution in the multi-level rejection procedure. Secondly, it is hard to find a function which can generate samplings which are meaningful. To solve this issue, we learn an RL policy as the proposal generator. Our experiments show that the proposed method significantly improves the sampling process

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung et al.

In this paper, we propose a rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods [10]. Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) in [10] and derive appropriate local problems in oversampled regions once we identify important modes representing each continua. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. We note that the use of non-local upscaled model for porous media flows is not new, e.g., in [14], the authors derive non-local approach. Our main contribution is identifying appropriate local problems together with local spectral modes to represent each continua. The model problem for fractures assumes that one can identify fracture networks. The resulting non-local equation (restricted to the oversampling region, which is several times larger compared to the target coarse block) has the same form as \cite{Hamdi_Nonlocal} with much smaller local regions. We present numerical results, which show that the proposed approach can provide good accuracy.

Eric T. Chung, Yalchin Efendiev, Bangti Jin et al.

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid media properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the coarse-grid sizes. In these cases, grid refinement can lead to many degrees of freedom, and thus unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the approach by numerical results for fractured media.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

Local multiscale methods often construct multiscale basis functions in the offline stage without taking into account input parameters, such as source terms, boundary conditions, and so on. These basis functions are then used in the online stage with a specific input parameter to solve the global problem at a reduced computational cost. Recently, online approaches have been introduced, where multiscale basis functions are adaptively constructed in some regions to reduce the error significantly. In multiscale methods, it is desired to have only 1-2 iterations to reduce the error to a desired threshold. Using Generalized Multiscale Finite Element Framework, it was shown that by choosing sufficient number of offline basis functions, the error reduction can be made independent of physical parameters, such as scales and contrast. In this paper, our goal is to improve this. Using our recently proposed approach and special online basis construction in oversampled regions, we show that the error reduction can be made sufficiently large by appropriately selecting oversampling regions. Our numerical results show that one can achieve a three order of magnitude error reduction, which is better than our previous methods. We also develop an adaptive algorithm and enrich in selected regions with large residuals. In our adaptive method, we show that the convergence rate can be determined by a user-defined parameter and we confirm this by numerical simulations. The analysis of the method is presented.

Juan Galvis, Eric Chung, Yalchin Efendiev et al.

We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems and propose a domain decomposition method better performance in terms of the number of iterations. The main novelty of our approaches is the construction of coarse spaces, which are computed using spectral information of local bilinear forms. We present several approaches to incorporate the spectral information into the coarse problem in order to obtain minimal coarse space dimension. We show that using these coarse spaces, we can obtain a domain decomposition preconditioner with the condition number independent of contrast and small scales. To minimize further the number of iterations until convergence, we use this minimal dimensional coarse spaces in a construction combining them with large overlap local problems that take advantage of the possibility of localizing global fields orthogonal to the coarse space. We obtain a condition number close to 1 for the new method. We discuss possible drawbacks and further extensions.

Victor M. Calo, Eric T. Chung, Yalchin Efendiev et al.

We develop a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems and thus, develop a systematic approach for finding reduced-order approximations of the solution. We start from a Petrov-Galerkin framework using optimal weighting functions. We introduce an auxiliary variable to a mixed formulation of the problem. The auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a reduced dimensional space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full solution on the reduced dimensional space that approximates the solution. To find the test space, we reformulate some recent mixed Generalized Multiscale Finite Element Methods. We introduce snapshots and local spectral problems that appropriately define local weight and trial spaces. In particular, we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable. The resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system. We discuss the stability and its relation to the approximation property of the test space. We design online basis functions, which accelerate convergence in the test space, and consequently, improve stability. We present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the Peclet number.

Eric T. Chung, Wing Tat Leung, Sara Pollock

In this paper we develop two goal-oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework. In this methodology, one seeks to determine the number of multiscale basis functions adaptively for each coarse region to efficiently reduce the error in the goal functional. Our first error estimator uses a residual based strategy where local indicators on each coarse neighborhood are the product of local indicators for the primal and dual problems, respectively. In the second approach, viewed as the multiscale extension of the dual weighted residual method (DWR), the error indicators are computed as the pairing of the local H^{-1} residual of the primal problem weighed by a projection into the primal space of the H_0^1 dual solution from an enriched space, over each coarse neighborhood. In both of these strategies, the goal-oriented indicators are then used in place of a standard residual-based indicator to mark coarse neighborhoods of the mesh for further enrichment in the form of additional multiscale basis functions. The method is demonstrated on high-contrast problems with heterogeneous multiscale coefficients, and is seen to outperform the standard residual based strategy with respect to efficient reduction of error in the goal function.

Eric T. Chung, Wing Tat Leung

Numerical simulations of waves in highly heterogeneous media have important applications, but direct computations are prohibitively expensive. In this paper, we develop a new generalized multiscale finite element method with the aim of simulating waves at a much lower cost. Our method is based on a mixed Galerkin type method with carefully designed basis functions that can capture various scales in the solution. The basis functions are constructed based on some local snapshot spaces and local spectral problems defined on them. The spectral problems give a natural ordering of the basis functions in the snapshot space and allow systematically enrichment of basis functions. In addition, by using a staggered coarse mesh, our method is energy conserving and has block diagonal mass matrix, which are desirable properties for wave propagation. We will prove that our method has spectral convergence, and present numerical results to show the performance of the method.

Eric Chung, Yalchin Efendiev, Wing Tat Leung et al.

In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use $l_1$ minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply $l_1$ minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we can not identify a local minimization principle, such as local generalized SVD.

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.