Randomized sketching for Krylov approximations of large-scale matrix functions
The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is very large and hence computing f(A) explicitly is unfeasible. Here we discuss a new approach to overcome memory limitations of Krylov approximation methods for this task by combining randomized sketching with an integral representation of f(A)b. Two different approximations are introduced, one based on sketched FOM and another based on sketched GMRES. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the sketching approaches. This is joint work with Marcel Schweitzer (Bergische Universität Wuppertal).