### Conveners

#### Contributed talks: Krylov subspaces

- Stéphane Gaudreault (Environment and Climate Change Canada)

#### Contributed talks: Exponential integration

- Helmut Podhaisky (Martin-Luther-Universität Halle-Wittenberg)

#### Contributed talks: Randomization and trace(f(A))

- Philipp Birken (Lund University)

We discuss a new augmented Krylov subspace method which allows for the efficient evaluation of a sequence of matrix function applications on a set of vectors using Krylov subspace recycling. If selected appropriately, the recycling subspace can be used to accelerate the convergence of each problem in the sequence, leading to an overall reduction in the computational overhead required to...

We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of $\varphi$-functions in...

In this talk we present some error bounds for the approximation of matrix-vector products $f(A) b$ and quadratic forms $b^T f(A) b$ with a matrix function $f(A)$, for a Hermitian matrix $A$ and a vector $b$ by means a rational Krylov subspace method. The error bounds are obtained by exploiting properties of rational Arnoldi decompositions and the Cauchy integral formula to link the matrix...

In the computation of wall bounded flows, resolving the boundary layer requires a very fine resolution. The CFL condition then makes the use of explicit time integration schemes infeasible. However, these may be used in parts of the domain where the mesh is coarse, and using an implicit method only on the remainder. This gives rise to domain based IMEX methods. In this talk, we consider the...

Due to the importance of simulation in various fields of science and engineering, devising efficient numerical methods for solving high-dimensional evolutionary Partial Differential Equations is of considerable interest.

In this talk, we present an efficient technique to employ exponential integrators for solving evolutionary Advection-Diffusion-Reaction equations with *spatially variable*...

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators [1] form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low-rank approximation [2] is a recent technique for solving high-dimensional...

We consider the problem of estimating the trace of a matrix function $f(A)$. In certain situations, in particular if $f(A)$ cannot be well approximated by a low-rank matrix, combining probing methods [2] based on graph colorings with stochastic trace estimation techniques can yield very accurate approximations. So far, such methods have not been thoroughly analyzed, though, but were rather...

This work is concerned with computing low-rank approximations of a matrix function $f(A)$ for a large symmetric positive semi-definite matrix $A$, a task that arises in, e.g., statistical learning and inverse problems. The application of popular randomized methods, such as the randomized singular value decomposition or the Nyström approximation, to $f(A)$ requires multiplying $f(A)$ with a few...

When analyzing complex networks, an important task is the identification of nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network's communicability is to changes in certain nodes or edges.

Recently, the concept of...

In recent decades, graphs have become widely used for studying technological, biological, and social systems. For this reason, several methods have been proposed to measure the structural and dynamical properties of graphs, aiming to gain insights into the represented system. Among these methods, measures based on matrix functions have been introduced for assessing vertex centrality and...