# Tutorial on Optimization of PDEs with Uncertain Inputs

Monday, June 6, 2016 - 1:30pm - 2:45pm

Lind 305

Drew Kouri (Sandia National Laboratories)

Uncertainty is ubiquitous in science and engineering applications. In such applications, it is critical to determine optimal solutions that are resilient to uncertainty. Thus, in formulating these infinite-dimensional stochastic programs, one must encode subjective risk preference. In this tutorial, we will develop a general theory for PDE-constrained optimization problems in which inputs or coefficients of the PDE are uncertain. We will discuss numerous approach to incorporate risk preferences and conservativity into the optimization problem formulation. Included here, we will motivate engineering risk preferences through concrete application problems. Finally, we will conclude with a brief discussion of nonintrusive solution methods.

(1) Motivating applications

(a) Compliance minimization

(b) Reservoir management

(2) General problem formulation

(a) Conditions of random variable objective function

(b) Conditions on PDE solution

(c) Conditions on numerical cost surrogate

(d) Existence of solutions

(e) First-order necessary conditions (under certain assumptions)

(3) Numerical cost surrogates

(a) Risk neutral

(b) Distributionally robust

(c) Risk averse

(d) Probabilistic including buffered probability

(4) Solution methods

(a) Stochastic approximation

(b) Sample average approximation

(c) Adaptive stochastic collocation

(1) Motivating applications

(a) Compliance minimization

(b) Reservoir management

(2) General problem formulation

(a) Conditions of random variable objective function

(b) Conditions on PDE solution

(c) Conditions on numerical cost surrogate

(d) Existence of solutions

(e) First-order necessary conditions (under certain assumptions)

(3) Numerical cost surrogates

(a) Risk neutral

(b) Distributionally robust

(c) Risk averse

(d) Probabilistic including buffered probability

(4) Solution methods

(a) Stochastic approximation

(b) Sample average approximation

(c) Adaptive stochastic collocation