Past Event: Babuška Forum

Scalable methods for optimal control of systems governed by PDEs with random coefficient fields

Omar Ghattas, Professor, ICES and Departments of Geological Sciences and Mechanical Engineering, UT Austin

Abstract

We present a method for optimal control of systems governed by partial differential equations(PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective. Conventional numerical methods for optimization under uncertainty are prohibitive when applied to this problem; for example, sampling the (discretized infinite-dimensional) parameter space to approximate the mean and variance would require solution of an enormous number of PDEs, which would have to be done at each optimization iteration. To make the optimal control problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter field. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these uncertain parameter gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian, and are thus prohibitive to evaluate for (discretized) infinite-dimensional parameter fields. To overcome this difficulty, we employ a randomized eigensolver to extract the dominant eigenvalues of the decaying spectrum. The resulting objective functional can now be readily differentiated using adjoint methods along with eigenvalue sensitivity analysis to obtain its gradient with respect to the controls. Along with the quadratic approximation and truncated spectral decomposition, this ensures that the cost of computing the risk-averse objective and its gradient with respect to the control--measured in the number of PDE solves--is independent of the (discretized) parameter and control dimensions, leading to an efficient quasi-Newton method for solving the optimal control problem. Finally, the quadratic approximation can be employed as a control variate for accurate evaluation of the objective at greatly reduced cost relative to sampling the original objective. Several applications with high-dimensional uncertain parameter spaces will be presented. This work is joint with Alen Alexanderian (NCSU), Peng Chen (UT Austin), Noemi Petra (UC Merced), Georg Stadler (NYU), and Umberto Villa (UT Austin). Bio Prof. Ghattas is the John A. and Katherine G. Jackson Chair in Computational Geosciences, professor of geological sciences and of mechanical engineering, and director of the Center for Computational Geosciences in ICES. He also holds courtesy appointments in the Departments of Computer Science, Biomedical Engineering, and in the Texas Advanced Computing Center. Dr. Ghattas is a two times winner of the IEEE/ACM Gordon Bell Prize first in 2003 for Special Accomplishment in Supercomputing and then in 2015 for scalability, and was the recipient of the 2008 TeraGrid Capability Computing Challenge award. He has served on the editorial boards or as associate editor of 16 journals, has been co-organizer of 12 conferences and workshops and served on the scientific or program committees of 53 others, has delivered invited keynote or plenary lectures at 35 international conferences, and has been a member or chair of 29 national or international professional or governmental committees. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM).