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

The Complexity of the Poisson Problem for Spaces of Bounded Mixed Derivatives

We are interested in the complexity of the Poisson problem with homogeneous Dirichlet boundary conditions on the d-dimensional unit cube Ω. Error is measured in the energy norm, and only standard information (consisting of function evaluations) is available. In previous work on this problem, the standard assumption has been that the class F of problem elements has been the unit ball of a Sobolev space of fixed smoothness r, in which case the ϵ-complexity is proportional to ϵ to -d/r. Given this exponential dependence on d, the problem is intractable for such classes F. In this paper, we seek to overcome this intractability by allowing F to be the unit ball of a space Hp(Ω) of bounded mixed derivatives, with p a fixed multi-index with positive entries. We find that the complexity is proportional to c(d)(1/ϵ) 1/pmin[ln(1/ϵ)] and we give bounds on b = bp,d. Hence, the problem is tractable in 1/ϵ with exponent at most 1/ϵmin. The upper bound on the complexity (which is close to the lower bound) is attained by a modified finite element method (MFEM) using discrete blending spline spaces; we obtain an explicit bound (with no hidden constants) on the cost of using this MFEM to compute ϵ-approximations. Finally, we show that for any positive multi-index p, the Poisson problem is strongly tractable, and that the MFEM using discrete blended piecewise polynomial splines of degree p is a strongly polynomial time algorithm. In particular, for the case p=1, the MFEM using discrete blended piecewise linear splines produces an ϵ-approximation with cost at most 0.839262(c(d)+2)(1/ϵ)to 5.07911

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications