Related papers: From Memory Model to CPU Time: Exponential Integra…
The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of…
The computation of the Log-determinant of large, sparse, symmetric positive definite (SPD) matrices is essential in many scientific computational fields such as numerical linear algebra and machine learning. In low dimensions, Cholesky is…
Error estimates for the numerical solution of the master equation are presented. Estimates are based on adjoint methods. We find that a good estimate can often be computed without spending computational effort on a dual problem. Estimates…
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial…
High order exponential integrators require computing linear combination of exponential like $\varphi$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for…
With the widespread adoption of Large Language Models (LLMs), the demand for high-performance LLM inference services continues to grow. To meet this demand, a growing number of AI accelerators have been proposed, such as Google TPU, Huawei…
We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space…
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation.…
Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati…
This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial…
Traditional step size controllers make the tacit assumption that the cost of a time step is independent of the step size. This is reasonable for explicit and implicit integrators that use direct solvers. In the context of exponential…
We consider Arnoldi like processes to obtain symplectic subspaces for Hamiltonian systems. Large systems are locally approximated by ones living in low dimensional subspaces; we especially consider Krylov subspaces and some extensions. This…
Exponential integrators that use Krylov approximations of matrix functions have turned out to be efficient for the time-integration of certain ordinary differential equations (ODEs). This holds in particular for linear homogeneous ODEs,…
Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this…
We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as…
This paper presents a hybrid CPU-GPU framework for solving combinatorial scheduling problems formulated as Integer Linear Programming (ILP). While scheduling underpins many optimization tasks in computing systems, solving these problems…
Exponential integrators are explicit methods for solving ordinary differential equations that treat linear behaviour exactly. The stiff-order conditions for exponential integrators derived in a Banach space framework by Hochbruck and…
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…
This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show…
This paper gives an analysis and an evaluation of linear algebra operations on Graphics Processing Unit (GPU) with complex number arithmetics with double precision. Knowing the performance of these operations, iterative Krylov methods are…