An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficients
Kouri, Drew P.
In this thesis, I develop and analyze a general theoretical framework for optimization problems governed by partial differential equations (PDEs) with random inputs. This theoretical framework is based on the adjoint calculus for computing derivatives of the objective function. I develop an efficient discretization and numerical optimization algorithm for the solution of these PDE constrained optimization problems. Using derivative based numerical optimization algorithms to solve these PDE constrained optimization problems is computationally expensive due to the large number of PDE solves required at each iteration. I present a stochastic collocation discretization for these PDE constrained optimization problems and prove the convergence of this discretization method for a specific class of problems. The stochastic collocation discretization technique described here requires many decoupled PDE solves to compute gradient and Hessian information. I develop a novel optimization theoretic framework based on dimension adaptive sparse grid quadrature to reduce the total number of PDE solves. My adaptive framework employs basic or retrospective trust regions to manage the adapted stochastic collocation models. In addition, I prove global first order convergence of the retrospective trust region algorithm under weakened assumptions on the modeled gradients. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the retrospective trust region algorithm follows. Finally, I describe a high performance implementation of my adaptive collocation and trust region framework. This framework can be efficiently implemented in the C++ programming language using the Message Passing Interface (MPI). Due to the large number of PDE solves required for derivative computations, it is essential to choose inexpensive approximate models and appropriate large-scale nonlinear programming techniques throughout the optimization routine to obtain an efficient algorithm. Numerical results for the adaptive solution of these optimization problems are presented.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/64617
Citable link to this pagehttps://hdl.handle.net/1911/102199
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