Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method

TL;DR

This paper reviews and proposes hybrid strategies to improve the efficiency of greedy algorithms in the reduced basis method, aiming to mitigate the curse of dimensionality in parameter domain sampling.

Contribution

It introduces two new hybrid strategies for greedy algorithms in RBM, combining successive refinement and multilevel maximization to reduce computational costs.

Findings

Hybrid strategies outperform traditional greedy algorithms in efficiency.

Proposed methods effectively delay the curse of dimensionality.

Experimental results on thermal block and Helmholtz problems demonstrate improved performance.

Abstract

The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline-online decomposition, a.k.a. a learning-execution process. A key feature of the method is a greedy algorithm repeatedly scanning the training set, a fine discretization of the parameter domain, to identify the next dimension of the parameter-induced solution manifold along which we expand the surrogate solution space. Although successfully applied to problems with fairly high parametric dimensions, the challenge is that this scanning cost dominates the offline cost due to it being proportional to the cardinality of the training set which is exponential with respect to the parameter dimension. In this work, we review three recent attempts in effectively delaying this curse of dimensionality, and propose two new hybrid strategies through successive refinement and multilevel maximization of the error estimate over the training set. All five offline-enhanced methods and the original greedy algorithm are tested and compared on {two types of problems: the thermal block problem and the geometrically parameterized Helmholtz problem.