Adaptive model reduction to accelerate optimization problems governed by partial differential equations
- Optimization problems constrained by Partial Differential Equations (PDEs) are ubiquitous in modern science and engineering. They play a central role in optimal design and control of multiphysics systems, as well as nondestructive evaluation and detection, and inverse problems. Methods to solve these optimization problems rely on potentially many numerical solutions of the underlying equations. For complicated physical interactions taking place on complex domains, these solutions will be computationally expensive---in terms of both time and resources---to obtain, rendering the optimization procedure difficult or intractable. This dissertation introduces a globally convergent, error-aware trust region algorithm for leveraging inexpensive approximation models to greatly reduce the cost of solving PDE-constrained optimization problems in increasingly complex scenarios. While the trust region theory is general, in that it is agnostic to the particular form of the approximation model, provided it possesses certain properties, this work employs reduced-order models based on the method of snapshots and Proper Orthogonal Decomposition (POD). The trust region algorithm proceeds by progressively refining the fidelity of the reduced-order model while converging to the optimal solution. Thus, the reduced-order model is trained exactly along the optimization trajectory, circumventing the task of training in a potentially high-dimensional parameter space. The proposed method is shown to find the optimal aerodynamic shape of a full aircraft configuration in about half the time required by accepted methods. The proposed error-aware trust region algorithm is extended to handle the case where uncertainties are present in the governing equations. In such situations, the goal is to find a design or control that is risk-averse with respect to some quantity of interest. The objective function and constraints in these problems usually correspond to integrals of quantities of interest over the stochastic space, which will inevitably require many solutions of the underlying partial differential equation. For this reason, dimension-adaptive sparse grids are combined with reduced-order models to define an inexpensive approximation model, which is wrapped in the error-aware trust region framework to ensure convergence to the optimal risk-averse solution. This framework is demonstrated on a model problem from computational mechanics and shown to be several orders of magnitude faster than existing methods.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Zahr, Matthew Joseph
|Stanford University, Institute for Computational and Mathematical Engineering.
|Saunders, Michael A
|Saunders, Michael A
|Statement of responsibility
|Matthew Joseph Zahr.
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Matthew Joseph Zahr
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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