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

该研究项目将推动基于计算机的模拟，以便在广泛的应用中更好地控制和预测过程；这些领域包括地震成像、石油工程、包含细线的电磁学领域，以及需要分子静电能量来理解蛋白质结构-功能关系的分子生物学应用。通过PI的直接监督或通过即将开发的新研究生课程，对拟议研究领域的学生进行培训，将使他们能够在高度跨学科的项目中进行研究，并为相关领域带来最先进的数值分析和计算算法。本研究计画将发展一般框架，以表述一类数值计算的边界积分-微分算子的扩展。该框架的主要优点是可以很容易地在各种网格几何形状和计算方法上设计简单而精确的数值算法。研究将集中于(i)由包含不规则形状散射面的无界区域中的波散射问题产生的边界积分微分方程，以及（ii）不同余维流形上的变分问题的演算。对于（ii），将考虑在流形上定义的凸能量最小化和最小作用原理。最小问题导致曲面上的椭圆方程，而最小作用原理导致双曲方程。一个主要的焦点将是推导这些边界算子的扩展“到体”，同时使用来自边界余维的额外自由度，尽可能多地保留（算子和解的）解析性质。在该框架下，将研究和模拟涉及电磁波在空间中与细导线耦合传播的具体应用。

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.