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, where the exponential integrator is equivalent to approximating the product of the matrix exponential and a vector. In this paper, we consider linear inhomogeneous ODEs, $y'(t)=Ay(t)+g(t)$, where the function $g(t)$ is assumed to satisfy certain regularity conditions. We derive an algorithm for this problem which is equivalent to approximating the product of the matrix exponential and a vector using Arnoldi's method. The construction is based on expressing the function $g(t)$ as a linear combination of given basis functions $[ϕ_i]_{i=0}^\infty$ with particular properties. The properties are such that the inhomogeneous ODE can be restated as an infinite-dimensional linear homogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE has properties that directly allow us to extend a Krylov method for finite-dimensional linear ODEs. Although the construction is based on an infinite-dimensional operator, the algorithm can be carried out with operations involving matrices and vectors of finite size. This type of construction resembles in many ways the infinite Arnoldi method for nonlinear eigenvalue problems. We prove convergence of the algorithm under certain natural conditions, and illustrate properties of the algorithm with examples stemming from the discretization of partial differential equations.
High order exponential integrators require computing linear combination of exponential like $φ$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix is $A$ is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis $V_m$. In parallel, GS orthogonalization requires \textit{global …
Exponential Adams Bashforth integrators for stiff ODEs, application to cardiac electrophysiology
INRIA Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5241, Université de Bordeaux（法国国家信息与自动化研究所波尔多中心，波尔多数学研究所）；Université de Pau et des Pays de l’Adour（波尔多大学）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142, Université de Pau et des Pays de l’Adour（波尔多大学）
Local Linearization-Runge Kutta Methods: a class of A-stable explicit integrators for dynamical systems
IMPA（巴西数学研究所）；Universidad de Ciencias Informáticas（古巴信息科学大学）；CIMFAV-DEUV, Facultad de Ciencias, Universidad de Valparaiso（瓦尔帕莱索大学理学院）；Instituto de Cibernética, Matemática y Física（古巴计算机、数学与物理研究所）；Montreal Neurological Institute, McGill University（麦吉尔大学蒙特利尔神经学研究所）
National University of Singapore（新加坡国立大学）；TU München（慕尼黑工业大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Forschungszentrum Jülich GmbH（于利希研究中心）；RWTH Aachen University（亚琛工业大学）
We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge-Kutta (RK) integrator architecture…
Rosenbrock-Krylov Methods for Large Systems of Differential Equations
Virginia Polytechnic Institute and State University（弗吉尼亚理工大学）；Computational Science Laboratory, Department of Computer Science, Virginia Tech.（弗吉尼亚理工学院计算机科学与计算科学实验室）
