Numerical Methods for PDE Constrained Optimization with Uncertain Data

Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization. The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods

Combining Parameterizations, Sobolev Methods and Shape Hessian Approximations for Aerodynamic Design Optimization

Aerodynamic design optimization, considered in this thesis, is a large and complex area spanning different disciplines from mathematics to engineering. To perform optimizations on industrially relevant test cases, various algorithms and techniques have been proposed throughout the literature, including the Sobolev smoothing of gradients. This thesis combines the Sobolev methodology for PDE constrained flow problems with the parameterization of the computational grid and interprets the resulting matrix as an approximation of the reduced shape Hessian. Traditionally, Sobolev gradient methods help prevent a loss of regularity and reduce high-frequency noise in the derivative calculation. Such a reinterpretation of the gradient in a different Hilbert space can be seen as a shape Hessian approximation. In the past, such approaches have been formulated in a non-parametric setting, while industrially relevant applications usually have a parameterized setting. In this thesis, the presence of a design parameterization for the shape description is explicitly considered. This research aims to demonstrate how a combination of Sobolev methods and parameterization can be done successfully, using a novel mathematical result based on the generalized Faà di Bruno formula. Such a formulation can yield benefits even if a smooth parameterization is already used. The results obtained allow for the formulation of an efficient and flexible optimization strategy, which can incorporate the Sobolev smoothing procedure for test cases where a parameterization describes the shape, e.g., a CAD model, and where additional constraints on the geometry and the flow are to be considered. Furthermore, the algorithm is also extended to One Shot optimization methods. One Shot algorithms are a tool for simultaneous analysis and design when dealing with inexact flow and adjoint solutions in a PDE constrained optimization. The proposed parameterized Sobolev smoothing approach is especially beneficial in such a setting to ensure a fast and robust convergence towards an optimal design. Key features of the implementation of the algorithms developed herein are pointed out, including the construction of the Laplace-Beltrami operator via finite elements and an efficient evaluation of the parameterization Jacobian using algorithmic differentiation. The newly derived algorithms are applied to relevant test cases featuring drag minimization problems, particularly for three-dimensional flows with turbulent RANS equations. These problems include additional constraints on the flow, e.g., constant lift, and the geometry, e.g., minimal thickness. The Sobolev smoothing combined with the parameterization is applied in classical and One Shot optimization settings and is compared to other traditional optimization algorithms. The numerical results show a performance improvement in runtime for the new combined algorithm over a classical Quasi-Newton scheme

A class of augmented Lagrangian algorithms for infinite-dimensional optimization with equality constraints

We consider a class of augmented Lagrangian algorithms for the solution of optimization problems posed in an infinite-dimensional setting. This class extends the augmented Lagrangian algorithm developed in [2] (when considering equality constraints only) which is motivated by solving a sequence of subproblems in which the augmented Lagrangian is approximately minimized { i.e., each subproblem is terminated as soon as a stopping condition is satisfied. Global and local convergence results are outlined. A study of the behavior of the extended class of algorithms is further presented when the sequence of approximately solved infinite-dimensional subproblems is replaced by a sequence of finite-dimensional subproblems obtained by a more and more rened discretization of their infinite-dimensional counterpart

A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations

We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained

Some robust optimization methods for inverse problems.

Wang, Yiran.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 70-73).Abstract also in Chinese.Chapter 1 --- Introduction --- p.6Chapter 1.1 --- Overview of the subject --- p.6Chapter 1.2 --- Motivation --- p.8Chapter 2 --- Inverse Medium Scattering Problem --- p.11Chapter 2.1 --- Mathematical Formulation --- p.11Chapter 2.1.1 --- Absorbing Boundary Conditions --- p.12Chapter 2.1.2 --- Applications --- p.14Chapter 2.2 --- Preliminary Results --- p.17Chapter 2.2.1 --- Weak Formulation --- p.17Chapter 2.2.2 --- About the Unique Determination --- p.21Chapter 3 --- Unconstrained Optimization: Steepest Decent Method --- p.25Chapter 3.1 --- Recursive Linearization Method Revisited --- p.25Chapter 3.1.1 --- Frechet differentiability --- p.26Chapter 3.1.2 --- Initial guess --- p.28Chapter 3.1.3 --- Landweber iteration --- p.30Chapter 3.1.4 --- Numerical Results --- p.32Chapter 3.2 --- Steepest Decent Analysis --- p.35Chapter 3.2.1 --- Single Wave Case --- p.36Chapter 3.2.2 --- Multiple Wave Case --- p.39Chapter 3.3 --- Numerical Experiments and Discussions --- p.43Chapter 4 --- Constrained Optimization: Augmented Lagrangian Method --- p.51Chapter 4.1 --- Method Review --- p.51Chapter 4.2 --- Problem Formulation --- p.54Chapter 4.3 --- First Order Optimality Condition --- p.56Chapter 4.4 --- Second Order Optimality Condition --- p.60Chapter 4.5 --- Modified Algorithm --- p.62Chapter 5 --- Conclusions and Future Work --- p.68Bibliography --- p.7