Mathematics Colloquia and Seminars

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Wavelets provide a new way to view data, functions, etc. In contrast to Fourier-based methods, wavelets are local in the physical space. Due to the localization in the physical space, wavelets seem like a good candidate for a basis set for numerical calculations in which the computational fields are localized in the physical space. Whereas wavelets are perfect for detecting these local physical space features, they are far from perfect for numerical calculation. The shortcomings are, 1) insufficient boundary accuracy for difference operators, 2) the high cost of evaluating non-linear terms and 3) the increasing complexity of the operators as more and more scales are used. Despite these shortcomings, however, wavelets provide an excellent grid refinement flag. That is, wavelets are defined to approximate local low-order polynomials exactly up to a given order. The truncation error in this approximation is exactly the same as that for finite difference schemes and other numerical methods which depend fundamentally on local low-order polynomial interpolation. A few examples will be given where wavelets are used to choose grids for adaptive grid calculations. Next, the issue of the order of the adaptive scheme will be addressed and the advantages of adapting at high order will be shown. The second part of the talk will give an overview of applications of wavelet analysis to oceanography. The applications will be to data assimilation where wavelets are used to detect computational errors and, hence, build error covariance matrices, and to data analysis where wavelets are used to track and monitor the evolution of eddies.

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