Determination of Optimal Parameters in a Wavelet Collocation Method
Marco Schuchmann, M. Rasguljajew · 2013
Abstract — This article describes the setting of the parameters 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. Parameters are the number of collocation points, the number of base elements, which will be considered in the approximation and the resolution index j (also known as the detail parameter). An important question is how to assess an approximation, if we don't know the exact solution. By using the Shannon wavelet we have additional information about the parameter j with respect to the Fourier space. The advantage of the wavelet collocation is its universal application possibility on different types of differential equations. In this article we show in examples, how we can detect a too small parameter j and how to recognize if we have not enough collocation points. With the fast discrete wavelet collocation we can- under certain conditions- assess the change of the approximation from the resolution j-1 to the resolution j. In examples we show how to detect a too small j or a too small number of collocation points. The last chapter deals with the description of the algorithm. Index Terms—ODE, sinc collocation, Shannon wavelet, wavelet collocation, I.