Efficient localised model reduction for multiscale PDEs via Grassmannian interpolation

arXiv (Cornell University)

Abstract

Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.

Authors 4

1. Christian Alber

   Heidelberg University · Heidelberg University · Heidelberg Engineering (Germany)

   Affiliation as printed

   Institute for Mathematics , Heidelberg University , 69047 Heidelberg , Germany

2. Markus Bachmayr Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik , RWTH Aachen University , Im Süsterfeld 2 , 52072 Aachen , Germany

3. Robert Scheichl

   Heidelberg University · Heidelberg University · Heidelberg Engineering (Germany)

   Affiliation as printed

   Institute for Mathematics , Heidelberg University , 69047 Heidelberg , Germany

4. Huqing Yang Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik , RWTH Aachen University , Im Süsterfeld 2 , 52072 Aachen , Germany