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

偏微分方程式的数值逼近，以及在纯科学和应用科学中逼近经典模型解的方案的分析，是一个广为流传的话题。有一些一般的理论可以推导出近似的收敛和精度。然而，最近出现了新的模型类别，它们不适合一般的治疗，需要新的技术和想法。在这个项目中，PI的目标是与学生和博士后助理一起，对描述广泛现象的非光滑、非线性和非局部系统的近似技术进行严格的分析。分析将需要解决方案的微妙平滑特性与手头问题和方案的精细结构之间的谨慎相互作用。分析将借用其他数学领域的技术，而不是数值分析中引用的标准工具。这项研究将加强对这类重要模型的建模和预测能力，并将通过参与该项目来培训职业生涯早期的研究人员。本项目研究的系统包括a)初值问题，其中奇异数据的存在或方程的固有性质使解非常粗糙；b)非线性系统，其中非线性的强度使得即使对于平滑的数据也不能立即断言解的光滑性；c)非局部问题，这些问题描述了长期相互作用或记忆效应，因此，在一点确定系统的状态需要全局知识；和d)非变分方程，即那些不是由守恒原理产生的方程。在所有这些例子中，为建立数值方法的稳定性和收敛而引用的标准论点都是无效的。作为这项工作的成果，将开发新的数值技术，并将通过对其近似性质的坚实数学分析来加强现有技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.