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On the convergence of recursive trust-region methods for multiscale nonlinear optimization and applications to nonlinear mechanics. (English) Zbl 1410.90198

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.

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

Citations:

Zbl 1163.90024

Software:

UG; NewtonLib; Ipopt
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