Online adaptive basis construction for nonlinear model reduction through local error optimization

TL;DR

This paper introduces an online adaptive basis construction method for nonlinear model reduction that enhances accuracy and reduces computational costs by optimizing local residuals without inverse full-model operations.

Contribution

It proposes a novel low-rank update approach with auxiliary vectors for online adaptive basis construction, improving ROM quality efficiently.

Findings

Significantly improved ROM accuracy in nonlinear problems.

Reduced computational costs compared to conventional methods.

Effective basis spanning process demonstrated in numerical examples.

Abstract

The accuracy of the reduced-order model (ROM) mainly depends on the selected basis. Therefore, it is essential to compute an appropriate basis with an efficient numerical procedure when applying ROM to nonlinear problems. In this paper, we propose an online adaptive basis technique to increase the quality of ROM while decreasing the computational costs in nonlinear problems. In the proposed method, the adaptive basis is defined by the low-rank update formulation, and two auxiliary vectors are set to implement this low-rank condition. To simultaneously tackle the issues of accuracy and the computational cost of the ROM basis, the auxiliary vectors are algebraically derived by optimizing a local residual operator. As a result, the reliability of ROM is significantly improved with a low computational cost because the error information can be contained without inverse operations of the full model dimension required in conventional approaches. The other feature of the proposed iterative algorithm is that the number of the initial incremental ROM basis could be varied, unlike in the typical online adaptive basis approaches. It may provide a fast and effective spanning process of the high-quality ROM subspace in the iteration step. A detailed derivation process of the proposed method is presented, and its performance is evaluated in various nonlinear numerical examples.