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

拟议研究的目标是开发更快、更准确的算法来计算一类广泛的方程的近似解，这些方程模拟物理现象，如热传输、弹性物体的变形、电磁波的散射等。求解这些方程的任务通常是计算模拟中最耗时的部分，也是决定哪些问题可以通过计算建模，哪些不可以的部分。处理复杂形状(如复杂几何形状的散射或复杂形状的流动)增加了计算任务的难度。从技术上讲，现有的求解复杂几何上的偏微分方程组和积分方程组的大型数值算法大多基于所谓的迭代方法，即构造一系列逐渐逼近精确解的近似解。这项拟议的研究旨在开发求解方程的“直接方法”。“直接法”一次就能从给定数据计算出未知数据。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“黑箱”的方式使用。因此，这些方法更适合合并到通用软件中，并且在许多情况下可以处理用现有迭代方法无法解决的重要问题。它们今天通常不被使用的原因是，现有的解决许多问题的直接方法往往昂贵得令人望而却步。然而，PI和其他研究人员最近的结果已经证明，构建在速度方面与现有最快迭代求解器具有竞争力的直接方法是可能的。新算法将应用于流体流动的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展示。

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.