Mathematical Sciences: Wavelet Analysis: Numerical Algorithms and Turbulence

该项目将侧重于基于分析的算法的数学研究，用于压缩和快速处理数字数据。这项工作继续努力改进声音和视频数据的压缩、模式和结构检测以及设计求解微分方程式和积分方程式的数值算法。在湍流现象以及前向和反向散射问题的研究中也有望得到应用。这项工作的主要工具是快速的小波和波包变换。小波分析为设计类似于快速傅立叶变换的快速数值算法开辟了道路，但更具通用性。它的长处在于它适用于不具有平移不变性的问题和某些类型的非线性计算。这些领域的最新进展表明，有必要设计特殊用途的正交化基准面。特别是，已经清楚地表明，分析局部调制信号以及研究振荡积分算子需要引入能够进行更精确的频率分析和良好的空间局部化的基。设计了各种类型的基地，以实现数字效果和特定功能。目前的计划包括开发可在二维结构中实施的大规模计算(例如模拟整架飞机的雷达散射，包括多区域电介质和分层表面效应)。还将开展工作，开发用于分析湍流数据的新计算工具，其中将通过小波和波包相关性提供多尺度、多频率和局部化分析，目的是识别湍流机制。还将开展进一步的工作，开发可用于语音传输和分析的高压缩比的正交化波形的调制波形库。

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.