Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty

Abstract

This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847-A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function. These adapted sparse grids often contain significantly fewer points than the high-fidelity grids, which leads to a dramatic reduction in the computational cost. This is demonstrated numerically using two examples. Moreover, the numerical results indicate that the new algorithm rapidly identifies the stochastic variables that are relevant to obtaining an accurate optimal solution. When the number of such variables is independent of the dimension of the stochastic space, the algorithm exhibits near dimension-independent behavior. Introduction.The solution of large-scale optimization problems in science and engineering must accommodate model uncertainties, such as unknown material properties and boundary conditions. The coupling of traditional optimization methods with uncertainty quantification faces significant computational challenges due to the potentially large number of stochastic variables. To address this issue, we have developed an algorithm that shows promising results for optimization problems governed by partial differential equations (PDEs) with random coefficients [18]. This algorithm uses a trust-region framework to manage models that are based on sparse grids. In [18] we use adaptive sparse grids to compute the optimization step and a fixed high-fidelity sparse grid to determine whether to accept the step. As dis-

A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization

“…The model function M k (µ) changes at each trust region iteration and is often a local quadratic Taylor expansion. Other surrogates, however, have also been considered in the literature [1,2,19,21,33].…”

“…However, its reliance on heuristic estimators means that only heuristic convergence could be demonstrated and realized in practice. Heuristic error indicators have also been applied to trust region optimization for POD models [34], and in a stochastic context, an approximation based on sparse grids [19].…”

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

“…Algorithms proposed in Refs. [18,19] have shown promise in reducing the cost of optimization under uncertainty compared to previously proposed methods. These algorithms enable computational efficiency via adaptive sparse-grid stochastic collocation and a practical trust-region framework to manage the resulting models.…”

“…The first algorithm constructs a sparse grid and reduced basis such that an accuracy condition on the gradient error indicator [18] is satisfied at the trust-region center. The second algorithm constructs a (possibly different) sparse grid and reduced basis such that an accuracy condition on the objective error indicator [19] is satisfied. Both algorithms combine the dimension-adaptive approach proposed in Ref.…”

“…Section 4 presents the new adaptive algorithm for optimization under uncertainty that manages the inexactness introduced from the sparse grid quadrature and reduced-order-model evaluations using the trust-region method introduced in Ref. [19]. Appendix A introduces and derives residual-based error bounds used to define the error indicators required by the trust-region convergence theory.…”