A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

Abstract

Abstract. In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449-474] and the datadriven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452-493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen-Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM 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. 1. Introduction. In recent years, there has been an increased interest in the simulation of systems with uncertainties. Many physical and engineering applications involving uncertainty quantification can be described by stochastic partial differential equations (SPDEs). Several numerical methods have been developed in the literature to solve SPDEs, such as the stochastic finite element method [24], Wiener chaos expansion or generalized polynomial chaos (gPC) method [31,25,40,41,42,39,33,32], and stochastic collocation method [43,5,30,6]. These methods can effectively solve the SPDEs when the dimension of stochastic input variables is low. However, their performance deteriorates dramatically when the dimension of stochastic input variables is high. This so-called curse of dimensionality is one of the essential challenges in uncertainty quantification. Recently, some progress has been made to alleviate this difficulty by exploring the sparse structure of the solutions and constructing a problem-dependent stochastic basis to solve these SPDEs; see, e.g., the data-driven stochastic method [11,44] and dynamically biorthogonal method [9,10].

Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

“…For stochastic problems, one can use multiscale finite element framework with Monte Carlo techniques or separation of variables. A promising approach for solving multiscale stochastic problems is to use data driven stochastic method concepts as shown in [69].…”

A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach

“…We found that many high-dimensional SPDE and RPDE problems have certain low-dimensional structures, in the sense of Karhunen-Loève expansion (KLE) [20,23], which suggest the existence of reduced-order models and better formulations for efficient numerical methods. Motivated by this observation, we have made progress in developing data-driven stochastic method (DSM) to solving high-dimensional RPDE problems [10,39,38,19].…”

“…We also make some progress in developing numerical methods for multiscale PDEs with random coefficients by exploring the low-dimensional structure of the solutions and constructing problemdependent basis functions to solve these RPDEs. In [10,38,19], we proposed the data-driven stochastic methods to solve partial differential equations with high-dimensional random input and/or multiscale coefficients. We found that the data-driven stochastic basis functions can be used to solve the RPDEs with many different force functions.…”

Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach

“…In [21], Peterseim used the numerical upscaling techniques to compute eigenvalues for a class of linear second-order selfadjoint elliptic partial differential operators. Using similar methodology to construct lowdimensional generalized finite element spaces is pioneered by the generalized finite element method (GFEM) [1] and the multiscale finite element method (MsFEM) [11,14,15], and is pervasive in the recent developments in the numerical methods for multiscale problems and elliptic PDEs with random coefficients, see [8,34] and references therein.…”