### 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, 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1166-1192 |

Number of pages | 27 |

Journal | SIAM-ASA Journal on Uncertainty Quantification |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Gaussian measure
- Hessian
- Optimal control
- Optimization under uncertainty
- PDE-constrained optimization
- PDEs with random coefficients
- Risk-aversion
- Trace estimators

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*SIAM-ASA Journal on Uncertainty Quantification*,

*5*(1), 1166-1192. https://doi.org/10.1137/16M106306X

**Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations.** / Alexanderian, Alen; Petra, Noemi; Stadler, Georg; Ghattas, Omar.

Research output: Contribution to journal › Article

*SIAM-ASA Journal on Uncertainty Quantification*, vol. 5, no. 1, pp. 1166-1192. https://doi.org/10.1137/16M106306X

}

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

PY - 2017/1/1

Y1 - 2017/1/1

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

VL - 5

SP - 1166

EP - 1192

JO - SIAM-ASA Journal on Uncertainty Quantification

JF - SIAM-ASA Journal on Uncertainty Quantification

SN - 2166-2525

IS - 1

ER -