Eric T. Chung

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, Guanglian Li et al.

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales (see Figure 1 for the illustration of a perforated domain). Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works [14, 18, 17], where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can accurately approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes equations in perforated domain. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere.

Eric T. Chung, Yalchin Efendiev, Shubin Fu

In this paper, we discuss the application of Generalized Multiscale Finite Element Method (GMsFEM) to elasticity equation in heterogeneous media. Our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom.

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

Simulating complex processes in fractured media requires some type of model reduction. Well-known approaches include multi-continuum techniques, which have been commonly used in approximating subgrid effects for flow and transport in fractured media. Our goal in this paper is to (1) show a relation between multi-continuum approaches and Generalized Multiscale Finite Element Method (GMsFEM) and (2) to discuss coupling these approaches for solving problems in complex multiscale fractured media. The GMsFEM, a systematic approach, constructs multiscale basis functions via local spectral decomposition in pre-computed snapshot spaces. We show that GMsFEM can automatically identify separate fracture networks via local spectral problems. We discuss the relation between these basis functions and continuums in multi-continuum methods. The GMsFEM can automatically detect each continuum and represent the interaction between the continuum and its surrounding (matrix). For problems with simplified fracture networks, we propose a simplified basis construction with the GMsFEM. This simplified approach is effective when the fracture networks are known and have simplified geometries. We show that this approach can achieve a similar result compared to the results using the GMsFEM with spectral basis functions. Further, we discuss the coupling between the GMsFEM and multi-continuum approaches. In this case, many fractures are resolved while for unresolved fractures, we use a multi-continuum approach with local Representative Volume Element (RVE) information. As a result, the method deals with a system of equations on a coarse grid, where each equation represents one of the continua on the fine grid. We present various basis construction mechanisms and numerical results.

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.

Maria Vasilyeva, Eric T. Chung, Siu Wun Cheung et al.

Our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media. We consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures. To construct our upscaled model, we will apply the nonlocal multicontinua (NLMC) upscaling technique. The upscaled coefficients are obtained by using some multiscale basis functions, which are solutions of local problems defined on oversampled regions. For each continuum within a target coarse element, we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers, with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise. The resulting multiscale basis functions have been shown to have good approximation properties. To illustrate the idea of our approach, we will consider a dual continua background model consisting of discrete fractures in two space dimensions, that is, we consider a system with three continua. We will present several numerical examples, and they show that our method is able to capture the interaction between matrix continua and discrete fractures on the coarse grid efficiently.

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.

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

Simulating flow in a highly heterogeneous reservoir with multiscale characteristics could be considerably demanding. To tackle this problem, we propose a numerical scheme coupling the Generalized Multiscale Finite Element Method (GMsFEM) with a triple-continuum model aimed at a faster simulator framework that can explicitly represent the interactions among different continua. To further enrich the descriptive ability of our proposed model, we combine the Discrete Fracture Model (DFM) to model the local effects of discrete fractures. In the proposed model, GMsFEM, as an advanced model reduction technique, enables capturing the multiscale flow dynamics. This is accomplished by systematically generating an approximation space through solving a series of local snapshot and spectral problems. The resulting eigenfunctions can pass the local features to the global level when acting as basis functions in coarse problems. Our goal in this paper is to further improve the accuracy of flow simulation in complicated reservoirs especially for the case when multiple discrete fractures located in single coarse neighborhood and multiscale finite element methods fail. Together with a detailed description of the model, several numerical experiments are conducted to confirm the success of our proposed method. A rigid proof is also given in the aspect of numerical analysis.

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.

Robert Altmann, Eric Chung, Roland Maier et al.

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

Mengnan Li, Eric Chung, Lijian Jiang

In this paper, we present a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in $L^2$ norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is derived and analyzed. A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.

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.

Yanfang Yang, Shubin Fu, Eric T Chung

In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an exten- sion of the online mixed generalized multiscale method [6]. The multiscale online basis functions are computed by solving a Neumann problem in an over-sampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method. Extensive numerical experiments are presented to demonstrate the performance of our methods for both steady-state flow, and two-phase flow and transport problems.

Eric T. Chung, Yanbo Li

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on the generalized multiscale finite element method. We will first construct a suitable snapshot space, and a dimensional reduction procedure to identify important modes of the solution. We next develop and analyze an a posteriori error indicator, and the corresponding adaptive algorithm. In addition, we will construct a coupled offline-online adaptive algorithm, which provides an adaptive strategy to the selection of offline and online basis functions. Our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media. We present several numerical results to illustrate the performance of our method.

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.

Tak Shing Au Yeung, Eric T. Chung, Gunther Uhlmann

In this paper, we consider the inverse problem of determining an unknown function defined in three space dimensions from its geodesic X-ray transform. The standard X-ray transform is defined on the Euclidean metric and is given by the integration of a function along straight lines. The geodesic X-ray transform is the generalization of the standard X-ray transform in Riemannian manifolds and is defined by integration of a function along geodesics. This paper is motivated by Uhlmann and Vasy's theoretical reconstruction algorithm for geodesic X-ray transform and mathematical formulation for traveltime tomography to develop a novel numerical algorithm for the stated goal. Our numerical scheme is based on a Neumann series approximation and a layer stripping approach. In particular, we will first reconstruct the unknown function by using a convergent Neumann series for each small neighborhood near the boundary. Once the solution is constructed on a layer near the boundary, we repeat the same procedure for the next layer, and continue this process until the unknown function is recovered on the whole domain. One main advantage of our approach is that the reconstruction is localized, and is therefore very efficient, compared with other global approaches for which the reconstructions are performed on the whole domain. We illustrate the performance of our method by showing some test cases including the Marmousi model. Finally, we apply this method to a travel time tomography in 3D, in which the inversion of the geodesic X-ray transform is one important step, and present several numerical results to validate the scheme.

Yanfang Yang, Eric T. Chung, Shubin Fu

In this paper, we develop an online basis enrichment method with the mortar mixed finite element method, using the oversampling technique, to solve for flow problems in highly heterogeneous media. We first compute a coarse grid solution with a certain number of offline basis functions per edge, which are chosen as standard polynomials basis functions. We then iteratively enrich the multiscale solution space with online multiscale basis functions computed by using residuals. The iterative solution converges to the fine scale solution rapidly. We also propose an oversampling online method to achieve faster convergence speed. The oversampling refers to using larger local regions in computing the online multiscale basis functions. We present extensive numerical experiments(including both 2D and 3D) to demonstrate the performance of our methods for both steady state flow, and two-phase flow and transport problems. In particular, for the time dependent two-phase flow and transport problems, we apply the online method to the initial model, without updating basis along the time evolution. Our numerical results demonstrate that by using a few number of online basis functions, one can achieve a fast convergence.

Siu Wun Cheung, Eric Chung, Hyea Hyun Kim

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid-structure interaction problems. We will use the immersed boundary method to model the fluid-structure interaction, while the fluid flow is governed by the incompressible Navier-Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid-structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier-Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method.

Eric T. Chung, Patrick Ciarlet, Xingguang Jin et al.

The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the \texttt{T}-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.

Xuehan Zhang, Lijian Jiang, Eric T. Chung

Deep learning-based surrogate models have been extensively developed for efficiently approximating multiscale systems with random input fields. However, most existing approaches require retraining neural networks from scratch when source terms, boundary conditions, or differential operators change, resulting in significant computational costs and limited adaptability. To address this challenge, we integrate our previous CNN-based reduced-order model (ROM) framework with the multiscale finite element method (MsFEM) and propose an MsFEM-inspired transfer learning strategy, termed MITL. The CNN-based ROM consists of two components: Basis CNNs, which learn reduced basis functions, and Coef CNNs, which predict the corresponding linear combination coefficients. To enhance the transferability of learned multiscale representations, global MsFEM basis problems are employed as source tasks during pretraining. For new target problems, MITL requires training only lightweight adaptation networks to construct task-specific reduced bases and coefficients, thereby substantially reducing the computational burden. Numerical experiments demonstrate that MITL achieves accurate and efficient predictions across a range of target tasks, with particularly significant advantages in data-scarce scenarios.

Eric T. Chung, Sai-Mang Pun, Zhiwen Zhang

In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within each coarse grid block for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical results are presented to demonstrate the efficiency of the proposed method in solving time-dependent PDEs with multiscale and random features.

Siu Wun Cheung, Eric T. Chung

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.

Eric T. Chung, Ming Fai Lam, Chi Yeung Lam

In this paper, we present a staggered discontinuous Galerkin (SDG) method for a class of nonlinear elliptic equations in two dimensions. The SDG methods have some distinctive advantages, and have been successfully applied to a wide range of problems including Maxwell equations, acoustic wave equation, elastodynamics and incompressible Navier-Stokes equations. Among many advantages of the SDG methods, one can apply a local post-processing technique to the solution, and obtain superconvergence. We will analyze the stability of the method and derive a priori error estimates. We solve the resulting nonlinear system using the Newton's method, and the numerical results confirm the theoretical rates of convergence and superconvergence.

Eric T. Chung, Yalchin Efendiev, Yanbo Li

In this paper, we propose a space-time GMsFEM for transport equations. Multiscale transport equations occur in many geoscientific applications, which include subsurface transport, atmospheric pollution transport, and so on. Most of existing multiscale approaches use spatial multiscale basis functions or upscaling, and there are very few works that design space-time multiscale functions to solve the transport equation on a coarse grid. For the time dependent problems, the use of space-time multiscale basis functions offers several advantages as the spatial and temporal scales are intrinsically coupled. By using the GMsFEM idea with a space-time framework, one obtains a better dimension reduction taking into account features of the solutions in both space and time. In addition, the time-stepping can be performed using much coarser time step sizes compared to the case when spatial multiscale basis are used. Our scheme is based on space-time snapshot spaces and model reduction using space-time spectral problems derived from the analysis. We give the analysis for the well-posedness and the spectral convergence of our method. We also present some numerical examples to demonstrate the performance of the method. In all examples, we observe a good accuracy with a few basis functions.

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.

Xinshi Chen, Eric Chung

The thesis is about an application of the shape optimization to the morphological evolution of Golgi stack. Golgi stack consists of multiple layers of cisternae. It is an organelle in the biological cells. Inspired by the Helfrich Model \cite{Helfrich}, which is a model for vesicles typically applied to biological cells, a new model specially designed for Golgi stack is developed and then implemented using FEM in this thesis. In the Golgi model, each cisternae of the Golgi stack is viewed as a closed vesicle without topological changes, and our model is adaptable to both single-vesicle case and multiple-vesicle case. The main idea of the math model is to minimize the elastic energy(bending energy) of the vesicles, with some constraints designed regarding the biological properties of Golgi stack. With these constraints attached to the math model, we could extend this model to an obstacle-type problem. Hence, in the thesis, not only the simulations of Golgi stack are shown, but some interesting examples without biological meanings are also demonstrated. Also, as multiple cisternaes are considered as a whole, this is also a model handling multiple objects. A set of numerical examples is shown to compare with the observed shape of Golgi stack, so we can lay down some possible explanations to the morphological performance of trans-Golgi cisternae.

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

Lina Zhao, Eun-Jae Park, Eric Chung

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that $κh$ is sufficiently small, where $κ$ is the wave number and $h$ is the mesh size. Error estimates for both the scalar and vector variables in $L^2$ norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.

Eric Chung, Yalchin Efendiev, Yanbo Li et al.

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 Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach to numerically treat equations with multi-scale 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 wellposedness 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.

Maria Vasilyeva, Eric T. Chung, Yalchin Efendiev et al.

Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method.

Shubin Fu, Eric T. Chung

In this paper, we consider the offline and online Constraint Energy Minimizing Generalized Mul- tiscale Finite Element Method (CEM-GMsFEM) for high-contrast linear elasticity problem. Offline basis construction starts with an auxiliary multiscale space by solving local spectral problems. We select eigenfunctions that correspond to a few small eigenvalues to form the auxiliary space. Using the auxiliary space, we solve a constraint energy minimization problem to construct offline multiscale spaces. The minimization problem is defined in the oversampling domain, which is larger than the target coarse block. To get a good approximation space, the oversampling domain should be large enough. We also propose a relaxed minimization problem to construct multiscale basis functions, which will yield more accurate and robust solution. To take into account the influence of input pa- rameters, such as source terms, we propose the construction of online multiscale basis and an adaptive enrichment algorithm. We provide extensive numerical experiments on 2D and 3D models to show the performance of the proposed method.

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

The objective of this paper is to design novel multi-layer neural network architectures for multiscale simulations of flows taking into account the observed data and physical modeling concepts. Our approaches use deep learning concepts combined with local multiscale model reduction methodologies to predict flow dynamics. Using reduced-order model concepts is important for constructing robust deep learning architectures since the reduced-order models provide fewer degrees of freedom. Flow dynamics can be thought of as multi-layer networks. More precisely, the solution (e.g., pressures and saturations) at the time instant $n+1$ depends on the solution at the time instant $n$ and input parameters, such as permeability fields, forcing terms, and initial conditions. One can regard the solution as a multi-layer network, where each layer, in general, is a nonlinear forward map and the number of layers relates to the internal time steps. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. In each layer, our reduced-order models will provide a forward map, which will be modified ("trained") using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system. We will present main ingredients of our approach and numerical results. Numerical results show that using deep learning and multiscale models, we can improve the forward models, which are conditioned to the available data.

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.

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.

Eric T. Chung, Shubin Fu, Yanfang Yang

Mortar methods are widely used techniques for discretizations of partial differential equations and preconditioners for the algebraic systems resulting from the discretizations. For problems with high contrast and multiple scales, the standard mortar spaces are not robust, and some enrichments are necessary in order to obtain an efficient and robust mortar space. In this paper, we consider a class of flow problems in high contrast heterogeneous media, and develop a systematic approach to obtain an enriched multiscale mortar space. Our approach is based on the constructions of local multiscale basis functions. The multiscale basis functions are constructed from local problems by following the framework of the Generalized Multiscale Finite Element Method (GMsFEM). In particular, we first create a local snapshot space. Then we select the dominated modes within the snapshot space using an appropriate Proper Orthogonal Decomposition (POD) technique. These multiscale basis functions show better accuracy than polynomial basis for multiscale problems. Using the proposed multiscale mortar space, we will construct a multiscale finite element method to solve the flow problem on a coarse grid and a preconditioning technique for the fine scale discretization of the flow problem. In particular, we develop a multiscale mortar mixed finite element method using the mortar space. In addition, we will design a two-level additive preconditioner and a two-level hybrid preconditioner based on the proposed mortar space for the iterative method applied to the fine scale discretization of the flow problem. We present several numerical examples to demonstrate the efficiency and robustness of our proposed mortar space with respect to both the coarse multiscale solver and the preconditioners.

Eric T. Chung, Maria Vasilyeva, Yating Wang

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy.

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.

Joshua Joy, Eric Chung, Zengwen Yuan et al.

This paper presents a secure communication application called DiscoverFriends. Its purpose is to securely communicate to a group of online friends while bypassing their respective social networking servers under a mobile ad hoc network environment. DiscoverFriends leverages Bloom filters and a hybrid encryption technique with a self-organized public-key management scheme to securely identify friends and provide authentication. Additionally, DiscoverFriends enables anonymous location check-ins by utilizing a new cryptographic primitive called Function Secret Sharing. Finally, to the best of our knowledge, DiscoverFriends implements and evaluates the first Android multi-hop WiFi direct protocol using IPv6.

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.

Ho Yuen Chan, Eric T. Chung, Yalchin Efendiev

In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.

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.