Linear systems of the form A x = b arise in almost all disciplines of
scientific computing, in particular as subproblems for the solution of
nonlinear problems. For large-scale systems, iterative methods are an
attractive alternative to direct methods, which use decompostitions of the
matrix A. These decompositions might be prohibitively expensive in terms of
memory and computational effort due to the generation of fill-in, i.e., former
zero entries of A turning non-zero in the decomposition. Iterative methods do
not require (exact) decompositions of A.
Participants of this seminar will study fixed-point iterations, nonlinear acceleration through Krylov subspace methods, and aspects of preconditioning. Particular emphasis will be given on linear problems arising after discretization of partial differential equations or in optimization problems.