MANIFOLD LEARNING AND NONLINEAR HOMOGENIZATION

被引：1

作者：

Chen, Shi ^{[1
]}

Li, Qin ^{[2
,3
]}

Lu, Jianfeng ^{[4
,5
]}

Wright, Stephen J. ^{[6
]}

机构：

[1] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[2] Univ Wisconsin, Math Dept, Madison, WI 53706 USA

[3] Univ Wisconsin, Discovery Inst, Madison, WI 53706 USA

[4] Duke Univ, Dept Math, Dept Phys, Durham, NC 27708 USA

[5] Duke Univ, Dept Chem, Durham, NC 27708 USA

[6] Univ Wisconsin, Dept Comp Sci, 1210 W Dayton St, Madison, WI 53706 USA

来源：

MULTISCALE MODELING & SIMULATION | 2022年 / 20卷 / 03期

基金：

美国国家科学基金会;

关键词：

Key words; nonlinear homogenization; multiscale problems; manifold learning; domain decomposition; FINITE-ELEMENT-METHOD; HETEROGENEOUS MULTISCALE METHOD; ASYMPTOTIC-PRESERVING SCHEMES; RADIATIVE HEAT-TRANSFER; KINETIC-EQUATIONS; ELLIPTIC PDES; DIMENSIONALITY REDUCTION; DIFFUSION-APPROXIMATION; POSITIVE SOLUTIONS; GEOMETRIC METHODS;

D O I：

10.1137/20M1377771

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We describe an efficient domain decomposition-based framework for nonlinear mul-tiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our frame-work is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on a detailed analytical understanding of multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.

引用

页码：1093 / 1126

页数：34

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