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

Hou, Thomas Y.; Ma, Dingjiong; Zhang, Zhiwen

doi:10.1137/18m1205844

Cited by 24 publications

(14 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…that optimally approximates the input solution snapshots. Given any integer m, the POD basis functions minimize the following error 20) subject to the constraints that…”

Section: Construction Of Multiscale Reduced Basis Functionsmentioning

confidence: 99%

“…Therefore, one can develop certain sparsity or data-driven basis functions to solve the SPDEs and RPDEs efficiently. In [7][8][9]20,39,40], Hou et al explored the Karhunen-Loève expansion of the stochastic solution, and constructed problem-dependent stochastic basis functions to solve these SPDEs and RPDEs. In [11,26], the compressive sensing technique is employed to identify a sparse representation of the solution in the stochastic direction.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

Zhang¹

2020

CMR

View full text Add to dashboard Cite

We propose a multiscale multilevel Monte Carlo (MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

show abstract

“…that optimally approximates the input solution snapshots. Given any integer m, the POD basis functions minimize the following error 20) subject to the constraints that…”

Section: Construction Of Multiscale Reduced Basis Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

Zhang¹

2020

CMR

View full text Add to dashboard Cite

show abstract

“…The functions ϕ i (x) are called measurement functions, which are chosen as the characteristic functions on each coarse element in [24,36] and piecewise linear basis functions in [31]. In [29,22], it is found that the usage of FEM nodal basis functions reduces the approximation error and thus the same setting is adopted in the current work.…”

Section: Construction Of Multiscale Basis Functionsmentioning

confidence: 99%

“…[24] extend these works such that localized basis functions can also be constructed for higher-order strongly elliptic operators. Recently, Hou, Ma, and Zhang propose to build localized multiscale stochastic basis to solve elliptic problems with multiscale and random coefficients [22].…”

Section: Introductionmentioning

confidence: 99%

A Multiscale Finite Element Method for the Schrödinger Equation with Multiscale Potentials

Chen¹,

Ma²,

Zhang³

2019

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schrödinger equation with correlated random potentials in both 1D and 2D.

show abstract

“…For linear problems, many multiscale methods have been developed. These include homogenization-based approaches [5,6], multiscale finite element methods [5,7,8], generalized multiscale finite element methods (GMsFEM) [9,10], constraint energy minimizing GMsFEM (CEM-GMsFEM) [11,12], nonlocal multi-continua (NLMC) approaches [13], metric-based upscaling [14], the heterogeneous multiscale method [15], localized orthogonal decomposition (LOD) [16], equation-free approaches [17,18], multiscale stochastic approaches [19][20][21] and the hierarchical multiscale method [22]. For high-contrast problems, approaches such as GMsFEM and NLMC have been developed.…”

Section: Introductionmentioning

confidence: 99%

Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

et al. 2021

View full text Add to dashboard Cite

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast.

show abstract

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

Cited by 24 publications

References 39 publications

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

A Multiscale Finite Element Method for the Schrödinger Equation with Multiscale Potentials

Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

Contact Info

Product

Resources

About