Random Processes and Fields: Discrete Approximations, Special Wavelet-Based Decompositions and Simulation

主要研究者研究随机过程和随机场的特殊小波分解。这些表示具有多分辨率结构，并具有标准小波分解的其他优良特性。然而，与标准情况相反，特殊小波分解中的高频（小尺度）项具有基本的随机结构。在广泛的假设下考虑随机过程和场：高斯，高斯从属，稳定，平稳，自相似，一些非平稳，等等。除了建立和理解特殊的小波分解外，还讨论了有关其在仿真，分析和建模中的使用的问题。该项目增加了最近增长的多尺度（多分辨率）方法，在不同的尺度（分辨率）上考虑观察到的现象。这种以小波为核心的方法已经成功地应用于许多领域，如医学成像、湍流、去噪、天文和互联网流量数据分析。该项目的目标是开发一种适合随机现象的新的多分辨率框架。尽管取得了一些进展，但这种多分辨率框架在基础层面上仍存在缺失。开发的框架应该提供新的分析工具和概念视角，例如，在离散时间和连续时间的模型之间建立新的联系，揭示模拟，这已经成为任何科学研究不可或缺的工具，或者通过多尺度方法建模有用。该项目也应直接感兴趣，并加强许多科学领域的研究人员之间的联系，如应用和纯数学，物理学，信号处理，概率和统计。

The principal investigator studies special wavelet decompositions ofrandom processes and fields. These representations have a multi-resolutionstructure and share other nice properties of standard waveletdecompositions. In contrast to the standard case, however, the highfrequency (small scale) terms in special wavelet decompositions have anelementary stochastic structure. Random processes and fields areconsidered under a wide range of assumptions: Gaussian, Gaussiansubordinated, stable, stationary, self-similar, some non-stationary, andothers. In addition to establishing and understanding special waveletdecompositions, questions regarding their use in simulation, analysis andmodeling are also addressed.The project adds to the recent growth of the multi-scale(multi-resolution) approach where observed phenomena are considered atdifferent scales (resolutions). This approach where wavelets play acentral role, has found successful applications in numerous fields such asmedical imaging, turbulence, de-noising, astronomical and Internet trafficdata analysis. The goal of the project is to develop a newmulti-resolution framework that is suitably adapted to random phenomena.Despite some progress, such multi-resolution framework has been missing ata fundamental level. The developed framework should provide novel analysistools and conceptual perspectives, for example, establish new connectionsbetween models in discrete and continuous times, shed light on simulation,which has become an indispensable tool of any scientific research, or beuseful in modeling through the multi-scale approach. The project shouldalso be of direct interest to and enhance ties between researchers in manyareas of Science such as Applied and Pure Mathematics, Physics, SignalProcessing, Probability and Statistics.