Model's sparse representation based on reduced mixed GMsFE basis methods

• Published in: Journal of computational physics Vol. 338; pp. 285 - 312
• Main Authors: Jiang, Lijian, Li, Qiuqi
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.06.2017; Elsevier Science Ltd
• Subjects: ALGORITHMS; Approximation; APPROXIMATIONS; Basis functions; Basis-oriented cross-validation method; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational physics; Computer simulation; COMPUTERIZED SIMULATION; Computing time; DECOMPOSITION; DEGREES OF FREEDOM; EFFICIENCY; Finite element analysis; FINITE ELEMENT METHOD; FLOW MODELS; FUNCTIONS; Greedy algorithms; GRIDS; LEAST SQUARE FIT; Least-squares method of snapshots; MASS; Mathematical models; Mixed generalized multiscale finite element method; Multiscale analysis; Parameters; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS; Porous media; Proper Orthogonal Decomposition; Random variables; RANDOMNESS; Representations; SAMPLING; Simulation; Sparse tensor approximation; Sparsity; TRAINING; TWO-PHASE FLOW; VALIDATION; VELOCITY; Mixed generalized multiscale finite element method; Least-squares method of snapshots; Sparse tensor approximation; Basis-oriented cross-validation method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2017.02.055

In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. A typical application for the elliptic PDEs is the flow in heterogeneous random porous media. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve the flow problem in a coarse grid and obtain the velocity with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned by the reduced mixed GMsFE basis functions is much smaller than the dimension of the original full order model, the online computation still depends on the number of coarse degree of freedoms. To significantly improve the online computation, we integrate the reduced mixed GMsFE basis methods with sparse tensor approximation and obtain a sparse representation for the model's outputs. The sparse representation is very efficient for evaluating the model's outputs for many instances of parameters. To illustrate the efficacy of the proposed methods, we present a few numerical examples for elliptic PDEs with multiscale and random inputs. In particular, a two-phase flow model in random porous media is simulated by the proposed sparse representation method.