Large-scale stochastic PDE-constrained optimization

Omar Ghattas

Oden Institute for Computational Engineering and Sciences Departments of Geological Sciences and Mechanical Engineering

University of Texas at Austin, USA

We consider optimization problems governed by PDEs with infinite dimensional random parameter fields. Such problems arise in numerous applications: optimal design and control of systems with stochastic forcing or uncertain material properties or geometry; inverse problems with stochastic forward problems; or Bayesian optimal experimental design problems with the goal of minimizing the uncertainty or maximizing the information gain in the inferred parameters.

Randomness in the PDEs implies randomness in the control/design objective (and any state constraints). This is addressed by formulating the optimization objective in terms of moments of the control/design objective or various risk measures. Monte Carlo evaluation of the objective as per the popular Sample Average Approximation (SAA) algorithm results in an optimization problem that is constrained by N PDE systems, where N is the number of samples. This results in an optimization problem that is prohibitive to solve, especially when the PDEs are expensive to solve and discretization of the infinite-dimensional parameter field results in a high-dimensional parameter space.

We discuss high-order derivative-based approximations of the parameter-to-objective maps that exploit the structure of these maps, in particular their smoothness, geometry, and low effective dimensionality. Their use as a basis for variance reduction, in combination with randomized linear algebra algorithms, is demonstrated to accelerate Monte Carlo sampling by up to three orders of magnitude and permit efficient solution of large scale stochastic PDE-constrained optimization problems with up to with O(10^6) uncertain parameters and O(10^6) optimization variables. Applications to optimal control of turbulent flow, optimal design of acoustic metamaterials, and chance-constrained optimal control of groundwater flow are presented.

This work is joint with Peng Chen (UT Austin) and Umberto Villa (Washington University).

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