Collaborative Research: Scalable and accurate direct solvers for integral equations on surfaces

提出的研究目标是开发更快、更准确的算法，用于计算模拟物理现象（如热传输、弹性体变形、电磁波散射等）的一类方程的近似解。求解这类方程的任务通常是计算模拟中最耗时的部分，它决定了哪些问题可以通过计算建模，哪些问题不能。处理复杂的形状（例如，从复杂几何散射或流过复杂形状的通道）增加了计算任务的难度。从技术上讲，大多数现有的求解复杂几何上的偏微分和积分方程的大规模数值算法都是基于所谓的“迭代方法”，即构造一个逐渐接近精确解的近似解序列。拟议的研究旨在开发求解方程的“直接方法”。“直接法”一次从给定数据中计算未知数据。当可用时，直接方法通常比迭代方法更受欢迎，因为它们更健壮，并且可以以“黑盒”方式使用。因此，它们更适合合并到通用软件中，并且在许多情况下可以用于无法用现有迭代方法解决的重要问题。目前不使用这些方法的原因是，现有的直接方法对许多问题来说往往过于昂贵。然而，pi和其他研究人员最近的结果已经证明，有可能构建在速度方面与现有最快的迭代求解器竞争的直接方法。新算法将应用于流体流动模拟和生物分子模拟，其性能将通过对复杂几何形状的模拟来证明。

The goal of the proposed research is to develop faster and more accurate algorithms for computing approximate solutions to a broad class of equations that model physical phenomena such as heat transport, deformation of elastic bodies, scattering of electromagnetic waves, and many others. The task of solving such equations is frequently the most time consuming part of computational simulations, and is the part that determines which problems can be modeled computationally, and which cannot. Dealing with complicated shapes (e.g. scattering from complex geometry or flow through channels of complicated shape) adds difficulty to the computational task.Technically speaking, most existing large-scale numerical algorithms for solving partial differential and integral equations on complex geometries are based on so called "iterative methods" which construct a sequence of approximate solutions that gradually approach the exact solution. The proposed research seeks to develop "direct methods" for solving equations. A "direct method" computes the unknown data from the given data in one shot. When available, direct methods are often preferred to iterative ones since they are more robust, and can be used in a "black-box" way. As a result these are more suitable for incorporation in general purpose software, and in many cases work for important problems that cannot be solved with existing iterative methods. The reason that they are today typically not used is that existing direct methods for many problems are often prohibitively expensive. However, recent results by the PIs and other researchers have proven that it is possible to construct direct methods that are competitive in terms of speed with the very fastest existing iterative solvers. The new algorithms will be applied to the simulation of fluid flows and biomolecular simulations, and their performance will be demonstrated by the execution of simulations on complex geometries.