Multi-Scaling Theory and Methods for Random Fields

9803391 Waymire这项研究的主要重点是数学理论和方法，这些理论和方法直接关系到涉及乘法结构中多尺度现象的问题，这些问题在各种各样的现代应用中自然出现，如流域水文学，流体湍流，金融证券市场，基本的数学成分是由乘法级联随机过程定义的时空随机场和由随机树图表示的随机网络，以及指导演化的方程组。 关于这些结构收集和报告的大部分数据是以某种数量与长度尺度的双对数图的形式。 这导致了新的自相似的空间/时间模型，其缩放结构推断从经验观察到的样本实现类的介绍和改进。 因此，我们试图计算某些大样本（精细尺度）的指数和限制法律波动的统计估计的限制。 此外，自相似性和标度指数的级联和网络模型寻求。 然后可以研究流动过程的标度指数与级联和网络标度指数之间的关系。 计算结构函数的理论的前景，（即，环境科学的一个基本问题是在给定当地气候（如降雨）和地形（河网结构、土壤湿度）的情况下确定流域的河流流量结构（如极值）。 在世界大多数地方，可用于水坝规划、洪水保险、军事战术等的信息是以遥感当地气候和地形的形式提供的。 的数学公式是基于一个随机系统的守恒方程（质量，动量）的流量乘法随机降雨输入和复杂的网络地形，通过缩放和多尺度指数，估计从遥感数据。 此类结果的实用方面之一是帮助水文学家和工程师将局部观测结果外推到更大的尺度，并对预测流量进行区域化。 然而，广泛的数学框架有助于我们理解各种自然随机现象，如流体湍流，随机投资收益率，可再生自然资源分布，自旋玻璃磁体等，其在空间和/或时间上本质上是乘法的。

9803391WaymireThe main focus of this research is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena in multiplicative structures that arise naturally in a wide variety of modern applications, such as river basin hydrology, fluid turbulence, financial securities markets, spin glasses etc. The essential mathematical ingredients are a space-time random field defined by a multiplicative cascade random process and a random network represented by a random tree graph, together with a system of equations directing the evolution. Much of the data collected and reported on these structures is in the form of log-log plots of some quantity versus a length scale. This leads to the introduction and refinements of new classes of self-similar spatial/temporal models whose scaling structure is inferred from empirically observed sample realizations. Thus we seek to calculate certain large-sample (fine scale) limits of statistical estimators of exponents and limit laws governing fluctuations. In addition, self-similarities and scaling exponents are sought for the cascade and network models. Then connections between the scaling exponents of the flow processes and those of the cascade and network exponents may be investigated. The prospect of a theory which computes structure functions, (i.e., multiscaling exponents) for extreme flows from corresponding structure function calculations on the inputs and network defines the frontiers of this research.A fundamental problem from environmental science is to determine the structure of river flows (e.g. extremes) from a basin given data on the local climate (e.g. rainfall) and topography (river network structure, soil moisture). In most parts of the world the information available for the planning of dams, flood insurance, military tactics etc. is in the form of remotely sensed local climate and topography. The mathematical formulation is based on a stochastic system of conservation equations (mass, momentum) which relate the flows to the multiplicative stochastic rainfall inputs and complex network topography via scaling and multiscaling exponents which are estimated from remotely sensed data. One of the practical aspects of results of this type is to assist hydrologists and engineers in extrapolating localized observations to larger scales, and to regionalize predicted flows. However, the broad mathematical framework contributes to our understanding of diverse natural stochastic phenomena such as fluid turbulence, stochastic investment yields, renewable natural resource distributions, spin glass magnets etc., which are intrinsically multiplicative in space and/or time.