A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

TL;DR

This paper introduces a novel non-conforming dual approach combined with adaptive Trust-Region methods and Reduced Basis models to efficiently solve large-scale PDE-constrained optimization problems, with proven convergence and reduced computational costs.

Contribution

It develops a new NCD approach integrated into adaptive Trust-Region Reduced Basis methods, providing rigorous error bounds and convergence analysis for PDE-constrained optimization.

Findings

Significant reduction in computational demand for large-scale PDE problems.

Successful convergence proof of the NCD-corrected adaptive Trust-Region RB algorithm.

Numerical experiments demonstrate improved efficiency and accuracy.

Abstract

In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.