# Automatic smoothness detection of the resolvent Krylov subspace method for the approximation of $C_0$-semigroups

## TL;DR

This paper introduces an automatic smoothness detection feature in the resolvent Krylov subspace method, enabling faster convergence for smoother vectors without user intervention, demonstrated through numerical experiments.

## Contribution

It presents a novel property of the resolvent Krylov method that accelerates convergence based on vector smoothness, eliminating the need for user adjustments.

## Key findings

- Faster convergence for smoother vectors
- No user modifications needed for smoothness adaptation
- Numerical experiments validate the theoretical results

## Abstract

The resolvent Krylov subspace method builds approximations to operator functions $f(A)$ times a vector $v$. For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is automatically faster when the vector $v$ is smoother. The user of the method does not need to know the presented theory and alterations of the method are not necessary in order to adapt to the (possibly unknown) smoothness of $v$. The findings are illustrated by numerical experiments.