Inexact Objective Function Evaluations in a Trust-Region
Algorithm for PDE-Constrained Optimization under Uncertainty
D. P. Kouri
Optimization and Uncertainty Quantification
Sandia National Laboratories
Matthias Heinkenschloss
Department of Computational and Applied Mathematics
Rice University
D. Ridzal
Optimization and Uncertainty Quantification
Sandia National Laboratories
B. G van Bloemen Waanders
Numerical Analysis and Applications
Sandia National Laboratories
SIAM
Journal on Scientific Computing, 2014, Vol. 36, No. 6, pages A3011-A3029.
Abstract
This paper improves the trust-region algorithm with adaptive sparse grids introduced
in our previous paper 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 typically 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.
Keywords. PDE optimization, uncertainty, stochastic collocation, trust regions, sparse grids, adaptivity