A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction

Abstract

In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projection onto a reduced space, which combines the full solution space in the dominant direction with a reduction space in the transverse direction. The latter is spanned by modal orthonormal basis functions. While so far the basis functions in the HMR approach have been chosen a priori [S. Perotto, A. Ern, and A. Veneziani, Multiscale Model. Simul., 8 (2010), pp. 1102--1127], for instance, as Legendre or trigonometric polynomials, in this work a highly nonlinear approximation is employed for the construction of the reduction space. To this end we first derive a lower dimensional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction. Exploiting the good approximation properties of RB methods, we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation. For an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article. We introduce our method for general elliptic problems such as advection-diffusion equations in two space dimensions. Numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach.