Extensions of Boundary Integro-Differential Operators and the Associated Computational Methods

该研究项目将推进基于计算机的模拟，以便在广泛的应用中实现更好的过程控制和预测;这些应用包括地震成像，石油工程，包含细线的领域中的电磁学，以及分子生物学应用，其中需要分子静电能量来理解蛋白质的结构-功能关系。学生在拟议的研究领域的培训，无论是通过PI的直接监督或通过新的研究生课程开发，将使他们能够在高度跨学科的项目中进行研究，并将最先进的数值分析和计算算法带到相关领域。该研究项目将开发一个通用框架，用于制定一类边界积分的扩展，数值计算的微分算子。该框架的主要优点是，简单而准确的数值算法可以很容易地设计在各种网格的几何形状和计算方法。研究将集中在（i）边界积分微分方程，从包含不规则形状的散射表面的无界域中的波散射问题，以及（ii）微积分的变化问题上提出的不同余维流形。关于（ii），将考虑定义在流形上的凸能量最小化和最小作用原理。最小化问题导致椭圆方程的表面，而最小作用原理导致双曲方程。一个主要的重点将是衍生这些边界运营商的扩展“到散装”，同时保留尽可能多的分析性能（运营商和解决方案），使用额外的自由度，来自边界的余维。具体的应用，包括电磁波传播耦合细导线在空间中将进行研究和模拟的框架下。

This research project will advance computer-based simulations for better process control and prediction in a wide range of applications; these include seismic imaging, petroleum engineering, electromagnetism in domains containing thin-wires, and molecular biology applications where molecular electrostatics energies are needed to understand the structure-function relations of proteins. The training of students in the proposed research areas, both through direct supervision of the PI or through the new graduate course to be developed, will allow them to conduct research in highly inter-disciplinary projects and bring state-of-the-art numerical analysis and computational algorithms to the related areas.This research project will develop a general framework for formulating extension of a class of boundary integro-differential operators for numerical computation. The main advantage of the framework is that simple and accurate numerical algorithms can easily be designed on a variety of grid geometries and computational methodologies. The research will concentrate on (i) boundary integro-differential equations that arise from wave scattering problems in unbounded domains containing irregularly shaped scattering surfaces, and (ii) calculus of variation problems posed on manifolds of different codimensions. Regarding to (ii), minimization of convex energies and least-action principles defined on the manifolds will be considered. The minimization problems lead to elliptic equations on surfaces while the least-action principles lead to hyperbolic equations. A major focus will be on deriving extensions of these boundary operators "to the bulk", while preserving as much analytical properties (of the operators and of the solutions) as possible, using the additional degrees of freedom that come from the codimensions of the boundary. Concrete applications involving electromagnetic wave propagation coupled with thin-wires in space will be studied and simulated under the framework.