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.

This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847-A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function. These adapted sparse grids often contain significantly fewer points than the high-fidelity grids, which leads to a dramatic reduction in the computational cost. This is demonstrated numerically using two examples. Moreover, the numerical results indicate that the new algorithm rapidly identifies the stochastic variables that are relevant to obtaining an accurate optimal solution. When the number of such variables is independent of the dimension of the stochastic space, the algorithm exhibits near dimension-independent behavior. Introduction.The solution of large-scale optimization problems in science and engineering must accommodate model uncertainties, such as unknown material properties and boundary conditions. The coupling of traditional optimization methods with uncertainty quantification faces significant computational challenges due to the potentially large number of stochastic variables. To address this issue, we have developed an algorithm that shows promising results for optimization problems governed by partial differential equations (PDEs) with random coefficients [18]. This algorithm uses a trust-region framework to manage models that are based on sparse grids. In [18] we use adaptive sparse grids to compute the optimization step and a fixed high-fidelity sparse grid to determine whether to accept the step. As dis-

We develop and analyze a trust-region sequential quadratic programming (SQP) method for the solution of smooth equality constrained optimization problems, which allows the inexact and hence iterative solution of linear systems. Iterative solution of linear systems is important in large-scale applications, such as optimization problems with partial differential equation constraints, where direct solves are either too expensive or not applicable. Our trust-region SQP algorithm is based on a composite-step approach that decouples the step into a quasi-normal and a tangential step. The algorithm includes critical modifications of substep computations needed to cope with the inexact solution of linear systems. The global convergence of our algorithm is guaranteed under rather general conditions on the substeps. We propose algorithms to compute the substeps and prove that these algorithms satisfy global convergence conditions. All components of the resulting algorithm are specified in such a way that they can be directly implemented. Numerical results indicate that our algorithm converges even for very coarse linear system solves.1. Introduction. Sequential quadratic programming (SQP) methods are used successfully for the solution of smooth nonlinear programming problems. Each iteration of an SQP method requires the solution of linear systems that involve the constraint Jacobian or its transpose. Most convergence theories for SQP methods and most SQP implementations require that these linear systems be solved exactly. For many large-scale problems, especially problems with partial differential equation (PDE) constraints, this is not possible. In many such applications, constraint Jacobians are not formed explicitly and only their action and the action of their transpose on a vector are available. Even if these matrices are formed explicitly the solution of the linear systems using direct linear algebra is prohibitively expensive. In these cases, iterative linear system solvers must be applied. As a consequence, all linear systems are solved inexactly and it is crucial to account for this inexactness in the design and in the convergence analysis of SQP methods.In this paper we introduce a general trust-region SQP algorithm for nonconvex equality constrained optimization that incorporates inexact linear system solves, we prove its global convergence, we propose subalgorithms to fully define its implementation, and we construct easily implementable stopping criteria for iterative

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Intrepid is a Trilinos package for advanced discretizations of Partial Differential Equations (PDEs). The package provides a comprehensive set of tools for local, cell-based construction of a wide range of numerical methods for PDEs. This paper describes the mathematical ideas and software design principles incorporated in the package. We also provide representative examples showcasing the use of Intrepid both in the context of numerical PDEs and the more general context of data analysis.

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