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

协作研究：可扩展且精确的曲面积分方程直接求解器

• 批准号: 1320621
• 负责人: Denis Zorin
• 金额: $ 22万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1320621&HistoricalAwards=false
• 关键词: Collaborative; Research; Scalable; accurate; direct

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.

拟议的研究的目的是开发更快，更准确的算法，以计算对广泛的方程式进行近似解决方案，以建模物理现象，例如热传输，弹性体的变形，电磁波的散射等。求解此类方程的任务通常是计算模拟中最耗时的部分，并且是确定可以通过计算对哪些问题进行建模的部分，哪些不能进行建模。处理复杂的形状（例如，从复杂的几何形状或流过复杂形状的通道的散射）为计算任务增加了难度。技术上来说，大多数现有的大规模数字算法求解了求解部分差异和积分方程的复杂几何形状上的部分差分方程和积分方程，这些算法是基于所谓的“迭代方法”，这些方法构建了一个逐渐构建的解决方案，以构建了一个逐渐构建的解决方案。拟议的研究试图开发解决方程的“直接方法”。一个“直接方法”一次拍摄从给定数据中计算未知数据。 如果有的话，直接方法通常比迭代方法更适合迭代，因为它们更强大，并且可以以“黑盒”方式使用。 结果，这些更适合于通用软件中掺入，并且在许多情况下，这些问题用于使用现有迭代方法无法解决的重要问题。他们今天不使用它们的原因是，许多问题的现有直接方法通常非常昂贵。但是，PIS和其他研究人员的最新结果证明，可以使用最快的现有迭代求解器来构建在速度方面具有竞争力的直接方法。新算法将应用于流体流和生物分子模拟的模拟，并且通过对复杂几何形状的模拟执行将证明它们的性能。