Multigrid methods for obstacle problems

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set

Nonsmooth Newton methods for set-valued saddle point problems

We present a new class of iterative schemes for large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach

Multilevel Monte Carlo finite element methods for stochastic elliptic variational inequalities

Multilevel Monte Carlo finite element methods (MLMC-FEMs) for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “pathwise” weak formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for the classical MC method and novel MLMC-FEMs. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FEMs then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments