TY - JOUR
T1 - Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations
AU - Alexanderian, Alen
AU - Petra, Noemi
AU - Stadler, Georg
AU - Ghattas, Omar
N1 - Funding Information:
∗Received by the editors February 26, 2016; accepted for publication (in revised form) September 7, 2017; published electronically November 21, 2017. http://www.siam.org/journals/juq/5/M106306.html Funding: This work was partially supported by NSF grants 1508713 and 1507009, and by DOE grants DE-FC02-13ER26128, DE-SC0010518, and DE-FC02-11ER26052. †Department of Mathematics, North Carolina State University, Raleigh, NC 27695 ([email protected]). ‡Applied Mathematics, School of Natural Sciences, University of California, Merced, CA 95340 (npetra@ ucmerced.edu). §Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 ([email protected]). ¶Institute for Computational Engineering and Sciences, Department of Mechanical Engineering, and Department of Geological Sciences, The University of Texas at Austin, Austin, TX 78712 ([email protected]).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics and American Statistical Association.
PY - 2017
Y1 - 2017
N2 - 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, leading to a risk-averse optimal control problem. Conventional numerical methods for optimization under uncertainty are prohibitive when applied to this problem. 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 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. To overcome this difficulty, we employ trace estimators, which only require a modest number of Hessian-vector products. We illustrate our approach with two specific problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Along with the quadratic approximation and trace estimation, 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, and depends only on the number of random vectors employed in the trace estimation, leading to an efficient quasi-Newton method for solving the optimal control problem. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with an uncertain permeability field.
AB - 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, leading to a risk-averse optimal control problem. Conventional numerical methods for optimization under uncertainty are prohibitive when applied to this problem. 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 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. To overcome this difficulty, we employ trace estimators, which only require a modest number of Hessian-vector products. We illustrate our approach with two specific problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Along with the quadratic approximation and trace estimation, 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, and depends only on the number of random vectors employed in the trace estimation, leading to an efficient quasi-Newton method for solving the optimal control problem. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with an uncertain permeability field.
KW - Gaussian measure
KW - Hessian
KW - Optimal control
KW - Optimization under uncertainty
KW - PDE-constrained optimization
KW - PDEs with random coefficients
KW - Risk-aversion
KW - Trace estimators
UR - http://www.scopus.com/inward/record.url?scp=85054715762&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85054715762&partnerID=8YFLogxK
U2 - 10.1137/16M106306X
DO - 10.1137/16M106306X
M3 - Article
AN - SCOPUS:85054715762
SN - 2166-2525
VL - 5
SP - 1166
EP - 1192
JO - SIAM-ASA Journal on Uncertainty Quantification
JF - SIAM-ASA Journal on Uncertainty Quantification
IS - 1
ER -