×

An adaptive projected Newton non-conforming dual approach for trust-region reduced basis approximation of PDE-constrained parameter optimization. (English) Zbl 1510.90268

Summary: In this contribution we device and analyze improved variants of the non-conforming dual approach for trust-region reduced basis (TR-RB) approximation of PDE-constrained parameter optimization that has recently been introduced in [T. Keil et al., ESAIM, Math. Model. Numer. Anal. 55, No. 3, 1239–1269 (2021; Zbl 1527.90221)]. The proposed methods use model order reduction techniques for parametrized PDEs to significantly reduce the computational demand of parameter optimization with PDE constraints in the context of large-scale or multi-scale applications. The adaptive TR approach allows to localize the reduction with respect to the parameter space along the path of optimization without wasting unnecessary resources in an offline phase. The improved variants employ projected Newton methods to solve the local optimization problems within each TR step to benefit from high convergence rates. This implies new strategies in constructing the RB spaces, together with an estimate for the approximation of the Hessian. Moreover, we present a new proof of convergence of the TR-RB method based on infinite-dimensional arguments, not restricted to the particular case of an RB approximation and provide an a posteriori error estimate for the approximation of the optimal parameter. Numerical experiments demonstrate the efficiency of the proposed methods.

MSC:

90C31 Sensitivity, stability, parametric optimization
35J20 Variational methods for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
90C06 Large-scale problems in mathematical programming

Citations:

Zbl 1527.90221
PDFBibTeX XMLCite
Full Text: arXiv Link