Efficient Methods for Random Field Approximation with Application to Nonlinear Schrodinger Equation

随机场逼近的有效方法及其在非线性薛定谔方程中的应用

• 批准号: 1016047
• 负责人: Qian-Yong Chen
• 金额: $ 11.67万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016047&HistoricalAwards=false
• 关键词: Efficient; Methods; Random; Field; Approximation

This project studies the formation and evolution of the soliton waves in the 1D and 2D nonlinear Schrodinger equation (NLSE) with a random potential (also called the Gross-Pitaevskii equation), which governs the evolution of the mean-field wave function in Bose-Einstein condensate (BEC). The main focus is to investigate the impact of three parameters: the strength and the correlation length of the disorder, and the norm of the solution (i.e., the number of atoms in the condensate). But first, the random field approximation will be investigated within a more general context in the sense that the methodology can be applied in any applications involving uncertainties, not limited to the random potential approximation of the NLSE. In practical problems, the variables with uncertainty are often described as a second-order stochastic process (or random field/function), i.e., its second-order moment is finite. The marginal distribution and covariance function are typically the available information. One prominent way of discretizing a second order random field is through the Karhunen-Loeve (KL) series expansion. The approximation with truncated KL expansion is optimal in terms of mean square error, and the errors for the first two moments are fixed for any specific truncated KL expansion. But there are still many issues needed to be addressed. In particular, the proposed research will focus on the following topics. 1.) More efficient computation of the KL expansion by introducing two adaptive meshes, when the analytical formulas are unavailable (true for most cases). 2.) Minimize the error of the marginal distribution by determining the distribution of the random variables in the KL expansion through the minimization of higher order moments. 3. Compare the efficiency of the KL expansion and the 'direct sampling' technique paired with correlation control technique, when the correlation length is short.Either due to the randomness in nature or the insufficiency of knowledge, uncertainty is nearly observed in all the disciplines to some degree. Petroleum reservoir a few miles under the earth's surface, traffic flow on state highways, measurement of gas flow in turbine, and the stock and futures market are several such examples. To gain a better understanding of the intrinsic dynamics, such uncertainty should be modeled and analyzed. The proposed research will enable faster and more efficient calculations of the involved uncertainties, provide unprecedented predictive capabilities. It will bring profound impact across all the scientific and engineering disciplines that involve uncertainty.

本项目研究了具有随机势的一维和二维非线性薛定谔方程（也称为Gross-Pitaevskii方程）中孤子波的形成和演化，该方程控制着玻色-爱因斯坦凝聚体（BEC）中平均场波函数的演化。主要焦点是研究三个参数的影响：无序的强度和相关长度，以及解的范数（即，冷凝物中的原子数）。但首先，随机场近似将在一个更一般的背景下，在这个意义上，该方法可以应用于任何涉及不确定性的应用程序，不限于随机潜在的NLSE近似的调查。在实际问题中，具有不确定性的变量通常被描述为二阶随机过程（或随机场/函数），即，它的二阶矩是有限的。边缘分布和协方差函数通常是可用的信息。离散二阶随机场的一个突出方法是通过Karhunen-Loeve（KL）级数展开。截断KL展开的近似在均方误差方面是最优的，并且对于任何特定的截断KL展开，前两个矩的误差是固定的。但仍有许多问题需要解决。具体而言，拟议的研究将侧重于以下主题。1.）的人。当解析公式不可用时，通过引入两个自适应网格来更有效地计算KL展开（在大多数情况下是真的）。2.）的情况。通过高阶矩的最小化确定KL展开式中随机变量的分布，从而最小化边际分布的误差。3.当相关长度较短时，比较KL展开和直接抽样技术与相关控制技术的效率。由于自然界的随机性或知识的不足，几乎所有学科都存在不同程度的不确定性。地表下几英里处的石油储藏、国道上的交通流量、涡轮机中的气体流量测量以及股票和期货市场都是这样的例子。为了更好地理解内在动力学，应该对这种不确定性进行建模和分析。拟议的研究将能够更快，更有效地计算所涉及的不确定性，提供前所未有的预测能力。它将对所有涉及不确定性的科学和工程学科产生深远的影响。