Gross, Christian; Krause, Rolf On the convergence of recursive trust-region methods for multiscale nonlinear optimization and applications to nonlinear mechanics. (English) Zbl 1410.90198 SIAM J. Numer. Anal. 47, No. 4, 3044-3069 (2009). Summary: We prove new convergence results for a class of multiscale trust-region algorithms originally introduced by S. Gratton et al. [SIAM J. Optim. 19, No. 1, 414–444 (2008; Zbl 1163.90024)] to solve unconstrained minimization problems within the Euclidean space \(\mathbb{R}^n\). We will state less restrictive assumptions on the objective function and on the iteratively computed trust-region corrections, which allow for proving first-order convergence. Moreover, we propose a novel projection approach for obtaining the initial coarse level iterates, which are needed within the nonlinear multiscale iteration. We show the efficiency and robustness of our approach by means of numerical examples from nonlinear continuum mechanics, where stored energy functions for materials of Ogden-type and for materials with visco-plastic behavior are minimized. Cited in 11 Documents MSC: 90C30 Nonlinear programming 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65K05 Numerical mathematical programming methods 90C26 Nonconvex programming, global optimization 35A15 Variational methods applied to PDEs 35J60 Nonlinear elliptic equations 74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) 74P10 Optimization of other properties in solid mechanics Keywords:nonlinear programming; nonlinear multilevel methods; nonlinear elasticity; convergence theory Citations:Zbl 1163.90024 Software:UG; NewtonLib; Ipopt PDFBibTeX XMLCite \textit{C. Gross} and \textit{R. Krause}, SIAM J. Numer. Anal. 47, No. 4, 3044--3069 (2009; Zbl 1410.90198) Full Text: DOI