On the application of a greedy reduced basis scheme to the multi‐frequency solution of acoustic boundary element equations

Abstract

AbstractIn this work, a greedy reduced basis scheme for the multi‐frequency solution of linear time‐harmonic acoustic problems is investigated. The scheme allows to express the approximate solution within a frequency range of interest as a linear combination of a few basis vectors. These basis vectors are given by the solutions of the underlying frequency dependent linear system at iteratively chosen frequency samples within the frequency band. In each step of the algorithm, the next basis vector is evaluated at a frequency point for which the solution is worst approximated by the current reduced basis. In order to approximate the solution between the frequency points, a least square solver is used and an a posteriori error estimator is applied for assessing the quality of the approximation. The boundary element method is used for discretizing the underlying Helmholtz problem and an iterative solver is applied for the solution of the high‐fidelity system. In view of medium to large‐scale problems, a data sparse representation of the system matrix based on the ℋ2‐format is introduced. This reduces both the memory requirements of the scheme as well as the computational effort of the matrix‐vector products. The performance of the proposed scheme is shown based on interior and exterior acoustic problems. Comparisons to conventional multi‐frequency strategies verify the efficiency of the method.