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

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（NLSE）（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（BEC）中平均场波函数的演化。主要重点是研究三个参数的影响：无序的强度和相关长度，以及溶液的范数（即凝聚物中的原子数）。但首先，随机场近似值将在更一般的背景下进行研究，因为该方法可以应用于任何涉及不确定性的应用，而不仅仅局限于NLSE的随机势近似值。在实际问题中，具有不确定性的变量通常被描述为一个二阶随机过程（或随机场/函数），即其二阶矩是有限的。边际分布和协方差函数是典型的可用信息。二阶随机场离散化的一个重要方法是通过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。特别地，建议的研究将集中在以下主题。1)。当解析公式不可用时（大多数情况下），通过引入两个自适应网格更有效地计算KL展开。2)。通过最小化高阶矩来确定KL展开中随机变量的分布，从而最小化边际分布的误差。3. 比较了相关长度较短时KL展开与“直接抽样”技术与相关控制技术相结合的效率。由于自然界的随机性或知识的不足，几乎在所有学科中都存在一定程度的不确定性。地下几英里处的石油储层、州际公路上的交通流量、涡轮机中气体流量的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。提出的研究将使所涉及的不确定性计算更快，更有效，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。

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.