In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good coarse-scale approximation of the solution for small $p$, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.
In this paper, we propose offline and online adaptive enrichment algorithms for the generalized multiscale approximation of a mixed finite element method with velocity elimination to solve the subsurface flow problem in high-contrast and heterogeneous porous media. We give the theoretical analysis for the convergence of these two adaptive methods, which shows that sufficient initial basis functions (belong to the offline space) leads to a faster convergence rate. A series of numerical examples are provided to highlight the performance of both these two adaptive methods and also validate the th…
In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize an…
In this paper, we propose a multiscale method for the Darcy-Forchheimer model in highly heterogeneous porous media. The problem is solved in the framework of generalized multiscale finite element methods (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. %In the MFMFE methods, appropriate mixed finite element spaces and suitable quadrature rules are employed, which allow for local velocity elimination and lead to a cell-centered system for the pressure. We consider the MFMFE method that utilizes the lowest order Brezzi-Douglas-Marini ($\textrm{BDM}_1$) mixed finite e…
Re-iterated multiscale model reduction using the GMsFEM
The Chinese University of Hong Kong（香港中文大学）；Texas A&M University（德克萨斯农工大学）；Institute of Mathematics and Informatics, North-Eastern Federal University（北联邦大学数学与信息学研究所）
In this paper, an online multiscale model reduction method is presented for stochastic partial differential equations (SPDEs) with multiplicative noise, where the diffusion coefficient is spatially multiscale and the noise perturbation nonlinearly depends on the diffusion dynamics. It is necessary to efficiently compute all possible trajectories of the stochastic dynamics for quantifying model's uncertainty and statistic moments. The multiscale diffusion and nonlinearity may cause the computation intractable. To overcome the multiscale difficulty, a constraint energy minimizing generalized mul…
In this paper, we present an Online Generalized Multiscale Finite Element Method(Online GMsFEM) for heat and mass transfer problem in heterogeneous media with artificial ground freezing pipes. The mathematical model of the process is based on the classical Stefan model, which describes heat transfer with a phase transition and takes into account filtration in a porous medium. The model is described by a system of equations for temperature and pressure. For fine grid solution, we use a finite element method using the fictitious domain method. To derive a solution on the coarse grid, we use a mo…
Conventional offline training of reduced-order bases in a predetermined region of a parameter space leads to parametric reduced-order models that are vulnerable to extrapolation. This vulnerability manifests itself whenever a queried parameter point lies in an unexplored region of the parameter space. This paper addresses this issue by presenting an in-situ, adaptive framework for nonlinear model reduction where computations are performed by default online, and shifted offline as needed. The framework is based on the concept of a database of local Reduced-Order Bases (ROBs), where locality is …
In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are theref…
The Localized Reduced Basis Multiscale method for two-phase flows in porous media
Institute for Computational and Applied Mathematics（计算与应用数学研究所）；Institute for Applied Analysis and Numerical Simulation（应用分析与数值模拟研究所）；Institute for Modelling Hydraulic and Environmental Systems（水与环境系统建模研究所）；SINTEF ICT（辛特夫信息与通信研究所）
A residual driven multiscale method for Darcy's flow in perforated domains
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, National Center for Applied Mathematics in Hunan, Xiangtan University（湘潭大学湖南省计算与仿真科学工程重点实验室、湖南应用数学中心）；Eastern Institute for Advanced Study, Eastern Institute of Technology（宁波东部理工学院先进研究院）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学教育部智能计算与信息处理重点实验室、湖南省基础学科基本算法理论与新型计算方法研究中心）；Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan University（湘潭大学计算科学国际科技创新合作基地）
In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media using Koopman operators. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale …
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the under…
Subspace decomposition with defect diffusion coefficient
Institute for Numerical Simulation, University of Bonn（波恩大学数值模拟研究所）；Department of mathematical sciences, Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学数学科学系）
A concurrent global-local numerical method for multiscale parabolic equations
National University of Singapore（新加坡国立大学）；SKLMS, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
In this paper, we propose a coupled Discrete Empirical Interpolation Method (DEIM) and Generalized Multiscale Finite element method (GMsFEM) to solve nonlinear parabolic equations with application to the Allen-Cahn equation. The Allen-Cahn equation is a model for nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. The GMsFEM allows solving multiscale problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. In arXiv:1301.2866, it was shown that the GMsFEM provi…
In this paper, we propose a local model reduction approach for subsurface flow problems in stochastic and highly heterogeneous media. To guarantee the mass conservation, we consider the mixed formulation of the flow problem and aim to solve the problem in a coarse grid to reduce the complexity of a large-scale system. We decompose the entire problem into a training and a testing stage, namely the offline coarse-grid multiscale basis generation stage and online simulation stage with different parameters. In the training stage, a parameter-independent and small-dimensional multiscale basis funct…
In this paper, we consider the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the method offers some advantages such as flexibility in coarse grid construction and sparsity of resulting discrete systems. Moreover, to our best knowledge, this is the first time where the proof of th…
Homogenization in terms of multiscale limits transforms a multiscale problem with $n+1$ asymptotically separated microscales posed on a physical domain $D \subset \mathbb{R}^d$ into a one-scale problem posed on a product domain of dimension $(n+1)d$ by introducing $n$ so-called "fast variables". This procedure allows to convert $n+1$ scales in $d$ physical dimensions into a single-scale structure in $(n+1)d$ dimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with…
In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis funct…
Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use of approximate local solution maps constructed offline via a random sampling strategy. Numerical examples demonstrate the accuracy, robustness, and efficiency of the proposed approach.
Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to th…
In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference coefficient. The proposed method is based on the Petrov--Galerkin Localized Orthogonal Decomposition (PG-LOD) which allows for straightforward parallelization with low communcation overhead and memory consumption. We focus on two types of perturbations: local defects which we treat …
