Dynamical properties of numerically truncated trajectories are discussed for periodically driven chaotic systems with largely converging dynamics. Various trajectories having different initial conditions can be contracted so close to each other that they are truncated into a single pseudotrajectory. It is more easily observed numerically if the driving frequency is lower. Driven by an exactly p...

A novel and eecient generalized feature extraction method is presented based on the Expansion Matching (EXM) method and the Karhunen-Loueve (KL) transform. The EXM method is used to design optimal detectors for diierent features. The KL representation is used to deene an optimal basis for representing these EXM feature detectors with minimum truncation error. Input images are then analyzed with...

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on n...

Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation Nediako Stoyanov Nedialkov Doctor of Philosophy Graduate Department of Computer Science University of Toronto 1999 Compared to standard numerical methods for initia1 value problems (IVPs) for ordinary differential equations (ODEs), validated (aiso called interval) methods for IVPs for ODE...

Abstract A new class of high order weighted essentially non-oscillatory (WENO) schemes [J. Comput. Phys., 318 (2016), 110-121] is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes [J. Comput. Phys., 126 (1996), 202-228] might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude ...

Abstract ScaLAPACK contains a pair of routines for solving systems which are narrow banded and diagonally dominant by rows. Mathematically, the algorithm is block cyclic reduction. The ScaLAPACK implementation can be improved using incomplete, rather than complete block cyclic reduction. If the matrix is strictly dominant by rows, then the truncation error can be bounded directly in terms of th...

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the deri...

We focus on the analysis of variance (ANOVA) method for high dimensional function approximation using Jacobi polynomial chaos to represent the terms of the expansion. First, we develop a weight theory inspired by quasi-Monte Carlo theory to identify which functions have low effective dimension using the ANOVA expansion in different norms. We then present estimates for the truncation error in th...

1. Introduction. The problem of finding rational approximations to functions has received a considerable amount of attention recently, and many methods exist for finding such approximations. A brief survey of the most widely used methods has been given by Cheney and Southard [1]. In all problems of approximation, it is essential to know the truncation error which arises when the function is rep...