Interactive Simulations Using Localized Reduced Basis Methods★

Stochastic Reduced Basis Methods

A reduced basis localized orthogonal decomposition

In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. ...

The Localized Reduced Basis Multiscale Method

In this paper we introduce the Localized Reduced Basis Multiscale (LRBMS) method for parameter dependent heterogeneous elliptic multiscale problems. The LRBMS method brings together ideas from both Reduced Basis methods to efficiently solve parametrized problems and from multiscale methods in order to deal with complex heterogeneities and large domains. Experiments on 2D and real world 3D data ...

The localized reduced basis multi-scale method with online enrichment

We are interested in the efficient and reliable numerical solution of parametric multi-scale problems, the multi-scale (parametric) character of which is indicated by ε (μ) if expressed in the general notation of (1). It is well known that solving parametric multi-scale problems accurately can be challenging and computationally costly for small scales ε and for a strong dependency of the soluti...

The Theoretical Foundation of Reduced Basis Methods

The main theme of this volume is the efficient solution of families of stochastic or parametric partial differential equations. This article focuses on the theoretical underpinnings of such methods. It shows how concepts from approximation theory, such as entropy or widths, can help quantify a priori how well such methods can perform. This theory is then used to analyze one of the primary numer...