Mathematical Sciences: Wavelet Analysis: Numerical Algorithms and Turbulence

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This project will focus on mathematical research into analysis-based algorithms for the compression and rapid manipulation of numerical data. The work continues efforts to improve compression of acoustical and video data, to pattern and structure detection and in the design of numerical algorithms for the solution of differential and integral equations. Applications to the study of turbulent phenomena as well as forward and inverse scattering problems are also expected. The primary tool of this work is the fast wavelet and wave packet transformations. Wavelet analysis has opened the way to the design of classes of fast numerical algorithms similar in usage to the Fast Fourier Transform, but more versatile. Its strength lies in its applicability to problems which do not have translational invariance and for certain classes of nonlinear computations. Recent progress in these areas has pointed to the need for designing special purpose orthonormal bases. In particular, it has become clear that the analysis of locally modulated signals, as well as the study of oscillatory integral operators requires the introduction of bases capable of more precise frequency analysis and good space localization. Various classes of bases have been designed to achieve both numerical efficacy and specific features. Present plans include development of large-scale calculations which can be implemented in two-dimensional structures (such as modeling radar scattering by a whole aircraft including the multizone dielectrics and layered surface effects). Work will also be done developing new computational tools for the analysis of turbulence data in which multiscale, multifrequencey and localization analysis will be provided by wavelet and wavepacket correlations aimed at identifying turbulence mechanisms. Further work will also be carried out in developing modulated wave form libraries of orthonormal waveforms which permit high compression rates for voice transmission and analysis.