Mathematical Sciences: Wavelet Analysis: Numerical Algorithms and Turbulence

这个项目将集中在数学研究， 基于分析的算法的压缩和快速 操纵数字数据。 工作继续努力， 改进声音和视频数据压缩，以形成图案， 结构检测和数值算法的设计， 微分和积分方程的解。 应用于湍流现象的研究以及 还预期有前向和反向散射问题。 本文的主要工具是快速小波和小波变换 数据包转换。 小波分析开辟了 设计类快速数值算法类似于 使用快速傅立叶变换，但更通用。 其 它的优点在于适用于那些没有 平移不变性和某些类的非线性 计算。 这些领域的最新进展表明， 需要设计特殊用途的标准正交基。 在 特别是，很明显，分析当地的 调制信号，以及振荡积分的研究 运营商需要引进能够更好地 精确的频率分析和良好的空间定位。 已经设计了各种类型的基底来实现这两个目标 数字功效和具体特征。 目前的计划包括 开发可实施的大规模计算 在二维结构中（例如建模雷达散射 包括多区域飞行器在内的整个飞机， 分层表面效果）。 还将开展工作，开发新的 用于分析湍流数据的计算工具， 多尺度、多频率和局部化分析将是 由小波和波包相关性提供， 识别湍流机制。 进一步的工作还将 在开发调制波形库中进行， 标准正交波形允许高压缩率， 语音传输和分析。

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.