New trends in algebraic preconditioning

In this chapter, we discuss trends and problems in the design of preconditioned Krylov methods for large-scale problems, particularly when they are formulated with surface integral equations such that dense and large matrices arise. We cover various numerical linear algebra aspects, such as the choice of iterative methods, characteristics and performances of fast integral-equation solvers for the required matrix-vector products, and the design of algebraic preconditioners based on multilevel incomplete LU factorization, sparse approximate inverses, inner-outer methods, and spectral approaches, particularly when they are combined with fast solvers. As shown via examples, the developed numerical linear algebra tools can enable efficient solutions of large electromagnetic problems on moderate number of cores and processors.