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.
Highlights
Many problems in engineering and science, e.g., shape optimization in aerodynamics or heat transfer in thermal conduction problems, deal with optimization problems constrained by partial differential equations (PDEs) [8,13,25,27,34]
We have analyzed and compared the complexity of three gradient based methods for the numerical solution of a risk-averse optimal control problem involving an elliptic PDE with random coefficients, where a Finite Element discretization is used to approximate the underlying PDEs and a Monte Carlo sampling is used to approximate the expectation in the risk measure
The second version is a Stochastic Gradient method in which the finite element discretization is still kept fixed over the iterations, but the expectation in the objective function is re-sampled independently at each iteration, with a small sample size
Summary
We follow the second modeling category and consider a similar risk averse OCP as in [7, 49] which consists in minimizing the expected squared L2 misfit between the state and a given target function as objective function, equipped with an L2 regularization on the (deterministic) control For this setting we consider numerical gradient based methods, either deterministic or stochastic, where adjoint calculus is used to represent the gradient of the objective function. Both the primal problem and the adjoint problem are discretized by finite elements and Monte Carlo estimators are used to approximate expectations defining the risk measure.
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