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An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids. (English) Zbl 1448.65057

The authors introduce and analyze an efficient method for solving optimization problems constrained by large-scale nonlinear systems of equations with uncertain parameters. For this challenging purpose they join together methods for gradient-based optimization, stochastic collocation, and efficient approximation of large-scale systems. The method effectively acts on two sources of inexactness that trade accuracy for speed, namely the stochastic collocation based on dimension-adaptive sparse grids and the projection-based reduced-order models. These two sources of inexactness lead to inexact objective function and gradient evaluations. A trust-region method, that guarantees global convergence, gets control on both sources. The method is applicable to a wide range of problems, including those where sharp computable error bounds are not available.

MSC:

65K10 Numerical optimization and variational techniques
65D40 Numerical approximation of high-dimensional functions; sparse grids
65K05 Numerical mathematical programming methods
90C15 Stochastic programming
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
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