Approximation and Analysis of Selected Nonsmooth, Nonlinear, and Nonlocal Equations

选定的非光滑、非线性和非局部方程的逼近和分析

基本信息

批准号：
2111228
负责人：
Abner Salgado
金额：
$ 36.75万
依托单位：
University of Tennessee Knoxville
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111228&HistoricalAwards=false
关键词：
Approximation Analysis Selected Nonsmooth Nonlinear

项目摘要

The numerical approximation of partial differential equations, and the analysis of schemes to approximate the solution of classical models in the pure and applied sciences, is a well-established topic. There are general theories to deduce convergence and accuracy of approximations. However, new classes of models have recently appeared that do not lend themselves to the general treatment and require new techniques and ideas. In this project the PI aims to develop, together with students and a postdoctoral associate, rigorous analyses of approximation techniques for nonsmooth, nonlinear, and nonlocal systems that describe a wide range of phenomena. The analysis will require a careful interplay between subtle smoothness properties of the solutions and the fine structure of the problems and schemes at hand. The analysis will borrow techniques, not among the standard tools invoked in numerical analysis, from other fields of mathematics. The research will enhance modeling and prediction capabilities for this important class of models, and early-career researchers will be trained through involvement in the project.Systems under study in this project include a) initial value problems where the presence of singular data or the inherent nature of the equation make the solution very rough; b) nonlinear systems where the strength of the nonlinearity is such that even for smooth data one cannot immediately assert the smoothness of the solution; c) nonlocal problems, those which describe long range interactions or memory effects and, thus, the determination of the state of the system at one point requires global knowledge; and d) nonvariational equations, that is, those that do not arise from conservation principles. In all these examples, standard arguments invoked to establish stability and convergence of numerical methods are not effective. As an outcome of this work, new numerical techniques will be developed, and the existing ones will be strengthened by solid mathematical analysis of their approximation properties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

偏微分方程的数值近似以及对纯科学和应用科学中经典模型解的近似方案的分析是一个成熟的主题。有一些通用理论可以推导出近似值的收敛性和准确性。然而，最近出现的新型模型不适合一般处理，需要新的技术和想法。在这个项目中，PI 旨在与学生和博士后同事一起开发对描述各种现象的非光滑、非线性和非局部系统的近似技术的严格分析。分析需要解决方案的微妙平滑特性与当前问题和方案的精细结构之间的仔细相互作用。该分析将借鉴其他数学领域的技术，而不是数值分析中引用的标准工具。该研究将增强这一类重要模型的建模和预测能力，早期职业研究人员将通过参与该项目接受培训。该项目正在研究的系统包括：a）初始值问题，其中存在奇异数据或固有的方程的性质使得解非常粗糙； b) 非线性系统，其非线性强度使得即使对于平滑数据也无法立即断言解的平滑性； c) 非局部问题，即那些描述长程相互作用或记忆效应的问题，因此，确定系统在某一点的状态需要全局知识； d) 非变分方程，即那些不是由守恒定律产生的方程。在所有这些例子中，为建立数值方法的稳定性和收敛性而调用的标准论证并不有效。作为这项工作的成果，新的数值技术将被开发出来，并且现有的数值技术将通过对其近似特性的可靠数学分析得到加强。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。