Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost functional with an additional penalty on the variance of the state. The expressions for the gradient and Hessian corresponding to either problem contain expected value operators. Due to the large number of uncertainties considered in our model, we suggest to evaluate these expectations using a multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that this results in the gradient and Hessian corresponding to the MLMC estimator of the original cost functional. Furthermore, we show that the use of certain correlated samples yields a reduction in the total number of samples required. Two optimization methods are investigated: the nonlinear conjugate gradient method and the Newton method. For both, a specific algorithm is provided that dynamically decides which and how many samples should be taken in each iteration. The cost of the optimization up to some specified tolerance $\tau$ is shown to be proportional to the cost of a gradient evaluation with requested root mean square error $\tau$. The algorithms are tested on a model elliptic diffusion problem with lognormal diffusion coefficient. An additional nonlinear term is also considered.Comment: This work was presented at the IMG 2016 conference (Dec 5 - Dec 9, 2016), at the Copper Mountain conference (Mar 26 - Mar 30, 2017), and at the FrontUQ conference (Sept 5 - Sept 8, 2017

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.Comment: 29 page

Wavelet and Multiscale Methods

[no abstract available

Numerical Techniques for Optimization Problems with PDE Constraints

The development, analysis and implementation of efficient and robust numerical techniques for optimization problems associated with partial differential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, significant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize first, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reflected the progress made in the field. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identification of parameters in multi-scale physical and physiological processes

Cumulative reports and publications through December 31, 1989

A complete list of reports from the Institute for Computer Applications in Science and Engineering (ICASE) is presented. The major categories of the current ICASE research program are: numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; control and parameter identification problems, with emphasis on effectual numerical methods; computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, structural analysis, and chemistry; computer systems and software, especially vector and parallel computers, microcomputers, and data management. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Summary of research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period October 1, 1988 through March 31, 1989 is summarized