Optimal operator preconditioning for pseudodifferential boundary problems

Operator preconditioning for a class of constrained optimal control problems

We propose and analyze two strategies for preconditioning linear operator equations that arise in PDE constrained optimal control in the framework of conjugate gradient methods. Our particular focus is on control or state constrained problems, where we consider the question of robustness with respect to critical parameters. We construct a preconditioner that yields favorable robustness properti...

Pseudodifferential Boundary Problems and Applications

Pseudodifferential Methods for Boundary Value Problems

In these lecture notes we introduce some of the concepts and results from microlocal analysis used in the analysis of boundary value problems for elliptic differential operators, with a special emphasis on Dirac-like operators. We first consider the problem of finding elliptic boundary conditions for the ∂̄operator on the unit disk. The rather explicit results in this special case delineate the ...

Equivalent operator preconditioning for linear elliptic problems

Finite element or finite difference approximations to linear partial differential equations of elliptic type lead to algebraic systems, normally of very large size. However, since the matrices are sparse, even extremely large-sized systems can be handled. To save computer memory and elapsed time, such equations are normally solved by iteration, most commonly using a preconditioned conjugate gra...

Optimal Operator preconditioning for weakly singular operator over 3D screens

In this supplement to [14], we propose a new Calderón-type preconditioner for the weakly singular integral operator for −∆ on screens in R. We introduce a modified hypersingular operator, which is the exact inverse of the weakly singular operator on the unit disk. It forms the foundation for dual-mesh-based operator preconditioning. Applied to low-order boundary element Galerkin discretizations...