Computational aspects of the geometric mean of two matrices: a survey

DA Bini, B Iannazzo - Acta Scientiarum Mathematicarum, 2024‏ - Springer
Algorithms for the computation of the (weighted) geometric mean G of two positive definite
matrices are described and discussed. For large and sparse matrices the problem of …

[HTML][HTML] A rational preconditioner for multi-dimensional Riesz fractional diffusion equations

L Aceto, M Mazza - Computers & Mathematics with Applications, 2023‏ - Elsevier
We propose a rational preconditioner for an efficient numerical solution of linear systems
arising from the discretization of multi-dimensional Riesz fractional diffusion equations. In …

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

M Benzi, M Rinelli, I Simunec - Numerische Mathematik, 2023‏ - Springer
We consider the problem of approximating the von Neumann entropy of a large, sparse,
symmetric positive semidefinite matrix A, defined as tr (f (A)) where f (x)=-x log x. After …

Challenges in computing matrix functions

M Fasi, S Gaudreault, K Lund, M Schweitzer - arxiv preprint arxiv …, 2024‏ - arxiv.org
This manuscript summarizes the outcome of the focus groups at" The f (A) bulous workshop
on matrix functions and exponential integrators", held at the Max Planck Institute for …

Rational Krylov methods for fractional diffusion problems on graphs

M Benzi, I Simunec - BIT Numerical Mathematics, 2022‏ - Springer
In this paper we propose a method to compute the solution to the fractional diffusion
equation on directed networks, which can be expressed in terms of the graph Laplacian L as …

A unified rational Krylov method for elliptic and parabolic fractional diffusion problems

T Danczul, C Hofreither… - Numerical Linear Algebra …, 2023‏ - Wiley Online Library
We present a unified framework to efficiently approximate solutions to fractional diffusion
problems of stationary and parabolic type. After discretization, we can take the point of view …

Improved ParaDiag via low-rank updates and interpolation

D Kressner, S Massei, J Zhu - Numerische Mathematik, 2023‏ - Springer
This work is concerned with linear matrix equations that arise from the space-time
discretization of time-dependent linear partial differential equations (PDEs). Such matrix …

Adaptive rational Krylov methods for exponential Runge–Kutta integrators

K Bergermann, M Stoll - SIAM Journal on Matrix Analysis and Applications, 2024‏ - SIAM
We consider the solution of large stiff systems of ODEs with explicit exponential Runge–
Kutta integrators. These problems arise from semidiscretized semilinear parabolic PDEs on …

Krylov subspace restarting for matrix Laplace transforms

A Frommer, K Kahl, M Schweitzer, M Tsolakis - SIAM Journal on Matrix …, 2023‏ - SIAM
A common way to approximate—the action of a matrix function on a vector—is to use the
Arnoldi approximation. Since a new vector needs to be generated and stored in every …

[PDF][PDF] Improved parallel-in-time integration via low-rank updates and interpolation

D Kressner, S Massei, J Zhu - arxiv preprint arxiv:2204.03073, 2022‏ - researchgate.net
This work is concerned with linear matrix equations that arise from the space-time
discretization of time-dependent linear partial differential equations (PDEs). Such matrix …