Matthias Heinkenschloss

Large-scale PDE-constrained optimization

I Introduction -- Large-Scale PDE-Constrained Optimization: An Introduction -- II Large-Scale CFD Applications -- Nonlinear Elimination in Aerodynamic Analysis and Design Optimization -- Optimization of Large-Scale Reacting Flows using MPSalsa and Sequential Quadratic Programming -- III Multifidelity Models and Inexactness -- First-Order Approximation and Model Management in Optimization -- Multifidelity Global Optimization Using DIRECT -- Inexactness Issues in the Lagrange-Newton-Krylov-Schur Method for PDE-constrained Optimization -- IV Sensitivities for PDE-based Optimization -- Solution Adapted Mesh Refinement and Sensitivity Analysis for Parabolic Partial Differential Equation Systems -- Challenges and Opportunities in Using Automatic Differentiation with Object-Oriented Toolkits for Scientific Computing -- Piggyback Differentiation and Optimization -- V NLP Algorithms and Inequality Constraints -- Assessing the Potential of Interior Methods for Nonlinear Optimization -- An Interior-Point Algorithm for Large Scale Optimization -- SQP SAND Strategies that Link to Existing Modeling Systems -- Interior Methods For a Class of Elliptic Variational Inequalities -- Hierarchical Control of a Linear Diffusion Equation -- VI Time-Dependent Problems -- A Sequential Quadratic Programming Method for Nonlinear Model Predictive Control -- Reduced Order Modelling Approaches to PDE-Constrained Optimization Based on Proper Orthogonal Decomposition -- Adaptive Simulation, the Adjoint State Method, and Optimization -- VII Frameworks for PDE-Constrained Optimization -- 18 The SIERRA Framework for Developing Advanced Parallel Mechanics Applications -- rSQP++: An Object-Oriented Framework for Successive Quadratic Programming -- Sundance Rapid Prototyping Tool for Parallel PDE Optimization -- Color Plates.

Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems

In this paper, a family of trust-region interior-point sequential quadratic programming (SQP) algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise, e.g., from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed, for a different class of problems, by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a first-order Karush--Kuhn--Tucker (KKT) limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent, conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q-quadratic. The results given here include, as special cases, current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.

Shape optimization in steady blood flow: A numerical study of non-Newtonian effects

We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.

Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations

We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with time-independent advection and the linearized Navier-Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI/Smith-type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.

Analysis of Inexact Trust-Region SQP Algorithms

In this paper we extend the design of a class of composite-step trust-region SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear system solves within the trust-region SQP method or from approximations of first-order derivatives. Accuracy requirements in our trust-region SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrix-free implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, El-Alem, and Maciel [SIAM J. Optim., 7 (1997), pp. 177--207]. If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trust-region methods with inexact gradient information for unconstrained optimization.

Global Convergence of Trust-region Interior-point Algorithms for Infinite-dimensional Nonconvex Minimization Subject to Pointwise Bounds

A class of interior-point trust-region algorithms for infinite-dimensional nonlinear optimization subject to pointwise bounds in L p-Banach spaces,

Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

2\le p\le\infty

Airfoil Design by an All-At-Once Method

, is formulated and analyzed. The problem formulation is motivated by optimal control problems with L p-controls and pointwise control constraints. The interior-point trust-region algorithms are generalizations of those recently introduced by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] for finite-dimensional problems. Many of the generalizations derived in this paper are also important in the finite-dimensional context. All first- and second-order global convergence results known for trust-region methods in the finite-dimensional setting are extended to the infinite-dimensional framework of this paper.

Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

Preconditioners for Karush-Kuhn-Tucker Matrices Arising in the Optimal Control of Distributed Systems

Abstract The all-at-once approach is implemented to solve an optimum airfoil design problem. The airfoil design problem is formulated as a constrained optimization problem in which flow variables and design variables arc viewed as independent and the coupling steady state Euler equation is included as a constraint, along with geometry and other constraints. In this formulation, the optimizer computes a sequence of points which tend toward feasibility and optimalily at the same lime (all-at-once). This decoupling of variables typically makes the problem less non-linear and can lead to more efficient solutions. In this paper an existing optimization algorithm is combined with an existing flow code. The problem formulation, its discretization, and the underlying solvers arc described. Implementation issues arc presented and numerical results are given which indicate that the cost of solving the design problem is appropriately six times the cost of solving a single analysis problem.