A variational principle for adaptive approximation of ordinary differential equations

Abstract

A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error = local error · weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms. (2000): 65L70, 65G50 Mathematics Subject Classification

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

“…The result was inspired by a corresponding a priori analysis derived in Talay and Tubaro [41], with the main difference that the weight for the local error contribution to the global error can be computed efficiently by stochastic flows and discrete dual backward problems, extending Moon et al [31] to SDEs. These a posteriori error expansions can be used in adaptive algorithms, in order to control the approximation error.…”

A Posteriori Error Estimation and Global Error Control for Ordinary Differential Equations by the Adjoint Method

“…Further details on this form of the condition number for ODEs can be found in [26]. Note that λ is a function determined by the differential equation and the derived function.…”

“…Recently, much work has been done towards this end for ODEs and partial differential equations (PDEs) (see [9,10,11,12,13,15,21,26,27]). An extensive review is given in [12] of the theoretical framework of a posteriori error estimation, which includes a discussion of implementation, particularly for reaction-diffusion equations.…”

“…The analysis is based on finite elements and formulating the error through the dual (adjoint) problem. Goal-oriented a posteriori error estimation for a single derived function has been proposed for error estimation and adaptive error control for PDEs (see [1,2,3,15,19]) and ODEs (see [10,21,26,27]). Implementation details for adaptive stepsize control for ODEs can also be found in [11,12,13,25].…”

“…It can be applied easily to all the known one-step and multistep methods, such as BDF, Adams, and Runge-Kutta methods. The overall strategy for global error control is similar to the approaches in [12,26,27,33]. In this paper we focus mainly on the BDF method, where adjoint sensitivity software [6,7] is readily available.…”