Complexity Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters

ESAIM Mathematical Modelling and Numerical Analysis, vol. 55, pp. 1599–1633

Abstract

We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squaredL2misfit between the state (i.e.solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with aStochastic Gradientmethod on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.

Authors 3

1. Matthieu Martin

   Criteo (France)

   Affiliation as printed

   Criteo AI Lab, 4 Rue des Méridiens, 38130 Échirolles, France

2. Sebastian Krumscheid Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen, 52062 Aachen, Germany

3. Fabio Nobile

   École Polytechnique Fédérale de Lausanne

   Affiliation as printed

   CSQI, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland