Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces

TL;DR

This paper introduces a framework for locally adaptive greedy algorithms in reduced basis methods, which tailor local approximation spaces to anisotropic parameter behaviors, reducing online computational costs.

Contribution

The work develops a novel approach that adapts local approximation spaces to anisotropic features in parameter space, improving efficiency of reduced basis models.

Findings

Effective control of online computational cost.

Adaptation to local anisotropic behavior improves accuracy.

Numerical tests demonstrate efficiency gains.

Abstract

Reduced order models, in particular the reduced basis method, rely on empirically built and problem dependent basis functions that are constructed during an off-line stage. In the on-line stage, the precomputed problem-dependent solution space, that is spanned by the basis functions, can then be used in order to reduce the size of the computational problem. For complex problems, the number of basis functions required to guarantee a certain error tolerance can become too large in order to benefit computationally from the model reduction. To overcome this, the present work introduces a framework where local approximation spaces (in parameter space) are used to define the reduced order approximation in order to have explicit control over the on-line cost. This approach also adapts the local approximation spaces to local anisotropic behavior in the parameter space. We present the algorithm and numerous numerical tests.