Computing tensor operator exponentials within low‐rank tensor formats with application to the parameter‐dependent multigrid method

Abstract

AbstractUncertainties in physical models can lead to parameter‐dependent linear systems. The representation and solution of these systems are an important task in numerical mathematics. We summarize our previous results on how to represent these systems using low‐rank tensor methods and how to solve these systems using the parameter‐dependent multigrid method. We propose a new approach to compute the tensor operator exponential, by which we mean the matrix exponential applied to a tensor operator, directly within low‐rank tensor formats. This approach is based on classical matrix methods combined with low‐rank arithmetic. The tensor operator exponential within a low‐rank tensor format is used to approximate the inverse diagonal of a low‐rank operator. This approximation is then used as Jacobi smoother for the parameter‐dependent multigrid method. Using this we observe in numerical experiments a grid size independent convergence rate of the multigrid method. Instead of inverting only diagonals of tensor operators, our approach also allows for the inversion of all symmetric positive definite tensor operators.