A multigrid method for PDE-constrained optimization with uncertain inputs

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.Comment: 24, 2 figure

H-matrix based second moment analysis for rough random fields and finite element discretizations

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an H- matrix, in particular if the correlation length is rather short or the correlation kernel is nonsmooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the H- matrix format, we can solve the correspondent H- matrix equation in essentially linear time by using the H -matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations

AbstractThis work is motivated by the preconditioned iterative solution of linear systems that arise from the discretization of uniformly elliptic partial differential equations. Iterative methods with bounds independent of the discretization are possible only if the preconditioning strategy is based upon equivalent operators. The operators A, B: W → V are said to be V norm equivalent if ∥Au∥v∥Bu∥v is bounded above and below by positive constants for u ϵ D, where D is “sufficiently dense.” If A is V norm equivalent to B, then for certain discretization strategies one can use B to construct a preconditioned iterative scheme for the approximate solution of the problem Au = F. The iteration will require an amount of work that is at most a constant times the work required to approximately solve the problem Bû = \̂tf to reduce the V norm of the error by a fixed factor. This paper develops the theory of equivalent operators on Hubert spaces. Then, the theory is applied to uniformly elliptic operators. Both the strong and weak forms are considered. Finally, finite element and finite difference discretizations are examined

Homogenization of Parabolic Equations with a Continuum of Space and Time Scales

This paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in $\Omega \subset \mathbb{R}^n$ with $L^\infty(\Omega \times (0,T))$-coefficients. It appears that the inverse operator maps the unit ball of $L^2(\Omega\times (0,T))$ into a space of functions which at small (time and space) scales are close in $H^1$ norm to a functional space of dimension $n$. It follows that once one has solved these equations at least $n$ times it is possible to homogenize them both in space and in time, reducing the number of operation counts necessary to obtain further solutions. In practice we show under a Cordes-type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in $L^2$ (instead of $H^{-1}$ with Euclidean coordinates). If the medium is time-independent, then it is sufficient to solve $n$ times the associated elliptic equation in order to homogenize the parabolic equation

Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds