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.

Abstract. Optimal design, optimal control, and parameter estimation of systems governed by partial differential equations (PDE) give rise to a class of problems known as PDE-constrained optimization. The size and complexity of the discretized PDEs often pose significant challenges for contemporary optimization methods. Recent advances in algorithms, software, and high performance computing systems have resulted in PDE simulations that can often scale to millions of variables, thousands of processors, and multiple physics interactions. As PDE solvers mature, there is increasing interest in industry and the national labs in solving optimization problems governed by such large-scale simulations. This article provides a brief introduction and overview to the Lecture Notes in Computational Science and Engineering volume entitled Large-Scale PDE-Constrained Optimization.This volume contains nineteen articles that were initially presented at the First Sandia Workshop on Large-Scale PDE-Constrained Optimization. The articles in this volume assess the state-of-the-art in PDE-constrained optimization, identify challenges to optimization presented by modern highly parallel PDE simulation codes and discuss promising algorithmic and software approaches to address them. These contributions represent current research of two strong scientific computing communities, in optimization and PDE simulation. This volume merges perspectives in these two different areas and identifies interesting open questions for further research. We hope that this volume leads to greater synergy and collaboration between these communities. Algorithmic challenges for PDE-constrained optimizationPDE simulation is widespread in science and engineering applications. Moreover, with increasing development and application of supercomputing hardware and advances in numerical methods, very large-scale and detailed simulations can now be considered. An essential sequel to simulation is its application in design, control, data assimilation, and inversion. Most of these tasks are naturally stated as continuous variable optimization problems (i.e.,t Sandia is a multi program laboratory operated by Sandia Corporation, a

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.

Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

In this paper we study the global convergence behavior of a class of composite-step trust-region SQP methods that allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trust-region SQP method or from approximations of first-order derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. In the absence of inexactness our global convergence theory is equal to that of Dennis, El-Alem, 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.