Title: Modern Krylov Subspace Methods for Parabolic Control Problems

Daniel B Szyld
Temple University, Department of Mathematics

Abstract: We discuss the use of state-of-the-art Krylov Subspace
Methods for the solution of certain large scale optimization
problems governed by parabolic partial differential equations.
We present recent results on inexact Krylov subspace methods,
in which the matrix-vector product at each step does not need
to be computed exactly. In fact, these products can be more and
more inexact, as the iteration progresses.  When the matrix
in question is in product form with one or more implied solution
of (inner) linear systems, these systems can be approximated less
and less accurately as the iteration progresses. Computable inner
stopping criteria were developed to guarantee convergence of
the overall method.  We will discuss these criteria, and
illustrate its application to several problems.
In particular, we apply these ideas to parabolic control problems,
where the reduced Hessian has two different inverses; and thus two
inner iteration criteria are needed.  Truncated methods are also
discussed.