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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