Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation

Randomized Local Model Order Reduction

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith

Adaptivity and Online Basis Construction for Generalized Multiscale Finite Element Methods

Many problems in application involve media with multiple scale, for example, in composite materials, porous media. These problems are usually computationally challenging since fine grid computation is extremely expensive. Therefore, one may need to develop a coarse grid model reduction for this type of problems. In this dissertation, we will consider a multiscale method called generalized multiscale finite element method (GMsFEM). GMsFEM follows the framework of multiscale finite element method. Instead of using one basis function per coarse grid node, GMsFEM uses several basis functions for one coarse grid node. Since the media is highly heterogeneous and may involves high contrast, having more than one basis function per node is important to reduce the error significantly. Due to the varying heterogeneity in the domain, we may require different numbers of basis functions in different regions. Then the question is how to determine the number of basis functions in each region. In this dissertation, we will discuss an adaptive enrichment algorithm for enriching basis functions for the regions with large error. We will consider two different types of basis function for enrichment. One is using the pre-computed offline basis functions. We call this method offline adaptive enrichment. The other method uses online constructed basis functions called online adaptive enrichment. In applications, non-conforming basis functions can give us more flexibility on gridding. The discontinuous Galerkin method also makes the mass matrix block diagonal, which enhances the computation speed in solving time-dependent problem with an explicit scheme. In this dissertation, we will discuss offline and online adaptive methods for the generalized multiscale discontinuous Galerkin method (GMsDGM). We will also discuss using GMsDGM for simulating wave propagation in heterogeneous media

Global-Local Nonlinear Model Reduction for Flows in Heterogeneous Porous Media

Many problems in engineering and science are represented by nonlinear partial differential equations (PDEs) with high contrast parameters and multiple scales. Solving these equations involves expensive computational cost because the fine-grid needs to resolve smallest scales and high contrast. In such cases, reduced-order methods are often needed. Reduced-order methods can be divided into local reduction methods and global reduction methods. Local reduced-order methods such as upscaling, Multiscale Finite Element Method (MsFEM) and Generalized Multiscale Finite Element Method (GMsFEM) divide the computational domain into coarse grids, where each grid contains small-scale heterogeneities and high contrast, and represents the computations for macroscopic simulations. In local model reduction, reduced-order models are constructed in each coarse region. Some known approaches, such as homogenization and numerical homogenization, are developed for problems with and without scale separation, respectively. Global reduced-order models, such as Proper Orthogonal Decomposition (POD), construct the reduced-order models via global finite element basis functions. These basis functions are constructed by solving many forward problems that can be expensive. In this dissertation, we propose global-local model reduction methods. The idea of global-local model reduction methods is to approximate the global basis functions locally and adaptively. However, in the case of nonlinear systems, additional interpolation techniques are required, such as Discrete Empirical Interpolation Method (DEIM). We propose a general global-local approach that uses the GMsFEM to construct adaptive approximation for the global basis functions. The developments of these methods require adaptive offline and adaptive online reduced-order model strategies, which we pursue in this work. We consider the applications to nonlinear ow problems, such as nonlinear Forch-heimer flow. In this case, we construct multiscale basis functions for the velocity field following mixed GMsFEM. In addition, we present a local online adaptive method for the basis enrichment of the function space based on an error indicator depending on the local residual norm. Finally, we propose a global online adaptive method to add new global basis functions to the POD subspace. We use local error indicators and solve the global residual problem using the GMsFEM, with local online adaptation