### Andy Wathen (University of Oxford)

#### Preconditioning for self-adjointness

*Joint work with Martin Stoll.*

*Tuesday 18 March 2008 at 11.00, JCMB 6309*

##### Abstract

Linear Algebra is a major component of many areas of Numerical Analysis
from PDEs to Optimization. In the latter area in particular,
saddle-point (or KKT) systems are ubiquitous. As the desire for
problems of larger and larger dimension arises (for example from
PDE Optimization), iterative linear system solution methods provide
the main possible solution approaches. No one wants slow convergence, so
preconditioning is almost always needed to ensure adequately rapid
convergence.

For symmetric matrices, the Conjugate Gradient and MINRES algorithms
are methods of choice which are underpinned by descriptive
convergence theory that gives clear guidance on desirable
preconditioners to enable rapid convergence. For these methods
symmetric and positive definite preconditioning is usually
employed to preserve symmetry.

In this talk I will describe and illustrate with examples
useful situations where nonsymmetric preconditioning can be
used to give self-adjoint preconditioned matrices in
non-standard inner products. The most long-standing and widely
used example is the method of Bramble and Pasciak which arose in
the context of the Stokes problem in PDEs - a saddle point system.
In these situations symmetric iterative methods can be employed
in such inner products with the consequent benefits.

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