Error Estimation and Assessment of an Approximation in a Wavelet Collocation Method

Marco Schuchmann,Michael Rasguljajew

doi:10.4236/ajcm.2013.32019

Abstract

This article describes how to assess an approximation in a wavelet collocation method which minimizes the sum of squares of residuals. In a research project several different types of differential equations were approximated with this method. A lot of parameters must be adjusted in the discussed method here. For example one parameter is the number of collocation points. In this article we show how we can detect whether this parameter is too small and how we can assess the error sum of squares of an approximation. In an example we see a correlation between the error sum of squares and a criterion to assess the approximation.

Highlights

In the wavelet theory a scaling function is used, which has properties that are defined in the MSA
This article describes how to assess an approximation in a wavelet collocation method which minimizes the sum of squares of residuals
In a research project several different types of differential equations were approximated with this method

Summary

Introduction

In the wavelet theory a scaling function is used, which has properties that are defined in the MSA (multi scale analysis). Many simulations had shown that if Qmin was very small the approximation yj would be good. Simulations have shown that even with m kmax kmin we get good approximations. In the examples we use the Shannon wavelet It has no compact support and no high order, in many examples and simulations we got a much better approximation than using other wavelets (f.e. Daubechies wavelets of order 5 to 8), even with a small n. Example 1: 1) We use the following ODE y t y, y 0 1. We see a regression table (Table 1) of ln sse on ln Q2 , which shows a linear dependency in our example and the graph of the linear regression function.

Error Estimation and Assessment of the Approximation

Conclusions