Multi-element stochastic reduced basis methods

Stochastic Reduced Basis Methods

Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

JAN S. HESTHAVEN ∗, SHUN ZHANG † , AND XUEYU ZHU ‡ Abstract. In this paper, we propose reduced basis multiscale finite element methods (RB-MsFEM) for elliptic problems with highly oscillating coefficients. The method is based on multiscale finite element methods with local test functions that encode the oscillatory behavior ([4, 14]). For uniform rectangular meshes, the local oscillating test f...

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...

Reduced basis methods with adaptive snapshot computations

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the...

Reduced Basis Finite Element Modelling of Electrical Machines with Multi-Conductor Windings

Finite element analysis of electrical machines with multi-conductor windings can be computationally costly. This paper proposes a solution to this problem, using a reduced basis approach. The field-circuit problem is first solved in a single slot only, with a set of different boundary conditions. These pre-computed solutions are then used as shape functions to approximate the solution in all sl...