Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

Abstract

In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F : R n → R n . When applied to such a problem the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F . The advantage of the proposed generalized trapezoidal rule is threefold: Firstly we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.

Sign controlled solvers for the absolute value equation with an application to support vector machines

“…in the numerical solution of ordinary differential equations with nonsmooth righthand side [8]. Such system can be, with a one-to-one solution correspondence, transformed into an AVE [7,Lem.…”