Figure 4: Four Wavelet grids at final time.
The new method is very straightforward. The numerical values of the matrix P are determined by waveled by wavelet analysis and are simply a function of the magnitude of the wavelet coefficients in thesame area of the domain where the assimilation occurs. In this manner, the wavelet analsys detects the areas of the domain where large computational errors occur and these areas will have relatively large values of P. By contrast, in areas of the domain where the computational errors are small, then the assimilation parameter will be relatively small.
3.2 Estimating Computational Errors
As outlined in the first section, scaling functions are designed to approximate low order polynomials exactly up to a given order, and wavelets are orthogonal to these same low-order polynomials. Any deviation from low-order polynomial structure in a computational domain can then be detected by wavelet analysis. This measure of deviation from low-order polynomial structurc is exactly what is needed to measure computational error. The reason for this is that fundamentally all non-spectal numerical schemes are constructed from low-order algebraic polynomials, and such schemes are exact if the data falls exactly on a low-order polynomial. In practive, however, compuational data will rarely be an exact low-order polynomial, therefore there is always error. The size of this computational error will depend on the deviation from low-order polynomials and can be readily measured with wavelet, analysis.
