A generalized framework for block Krylov subspace methods with restarts, shifts, custom Ritz values, and many applications

  • Date:
  • Time: 15:30 - 16:30
  • Address:
    Sokolovská 83, Praha
  • Room: Seminar room of Department of Numerical Mathematics
  • Speaker: Kathryn Lund (EPFL)

Abstract: Krylov subspace methods have played an enormous role in numerical linear algebra and scientific computing.  This talk focuses on block Krylov methods and summarizes the attempt to describe as many varieties as possible under a common framework, which is made possible by block inner products and matrix polynomials.  The driving application is the computation of f(A)B, where f is a scalar-valued function, A is a large and sparse matrix, and B is a block vector, or the concatenation of multiple vectors into a tall-and-skinny matrix.  Because it is necessary to store a full basis to compute f(A)B, we only consider methods that allow for restarts.  Moreover, we only consider functions with Cauchy-Stieltjes representations, so that we can take advantage of theory for families of shifted linear systems.  The resulting framework thus accounts for many features at once, including the possibility to customize the Ritz values of the Arnoldi decomposition.  In addition to linear systems, families of shifted linear systems, and univariate matrix functions, we also touch on how this framework can be applied to tensor-valued functions and bivariate functions of matrices.