Fast Integral Equation Methods: Algorithms and Applications

快速积分方程方法：算法与应用

• 批准号: RGPIN-2014-03576
• 负责人: Kropinski, MaryCatherine
• 金额: $ 1.02万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610525
• 关键词: Fast; Integral; Equation; Methods; Algorithms

The overall theme of the proposed research program is to develop fast and accurate integral equation methods for the purposes of investigating applications arising in fluid dynamics or other physical and biological systems. Numerical methods based on integral equations have become increasingly popular, due in large part to the development of associated fast algorithms that can be used to accelerate the solution procedure. In very general terms, a fast integral equation method (FIEM) first requires formulating a well-conditioned integral equation for a partial differential equation (PDE), then selecting a suitable quadrature method for its discretization, and finally implementing a fast algorithm, such as the fast multipole method or a fast direct solver, to accelerate the solution of the resulting linear system. The advantages of FIEMs over difference-based methods are significant: efficiency is obtained through dimension reduction and the use of fast algorithms; the ill-conditioning associated with directly discretizing the governing equation is avoided; and high-order accuracy is easier to attain. The computational efficiency of FIEMs can mean being able to solve a complex problem on a desktop computer in a matter of minutes instead of taking hours on a cluster. The superior stability properties coupled with high-order accuracy means that near machine-precision accuracy can be achieved. This high precision can be invaluable when using these tools for investigating analytic properties of PDEs, resolving complex features in solutions and providing benchmark data for other computational methods. The two long-term goals discussed in the proposal are the following: 1. FIEM for Incompressible Fluid Dynamics The incompressible Navier-Stokes equations (INSE) are a system of highly nonlinear, complex PDEs that are ubiquitous in describing a wide range of fluid phenomena, from swimming microorganisms to weather systems. While the INSE are not directly amenable to solution via integral equations, a suitable temporal discretization will yield a collection of linear, high-order elliptic equations that must be solved at each time step. The proposed research will develop methods to recast these equations as integral equations, by representing the solution as the sum of suitably chosen layer and volume potentials. Then, suitable fast algorithms will be used to solve for and evaluate these potentials. Significant progress has been made in developing the tools needed for a two-dimensional solver. We will also consider problems in three dimensions. 2. FIEM for Boundary Value Problems on Surfaces. Applications involving the solution to PDEs on surfaces include computational fluid dynamics for planetary-scale flows, image processing, electromagnetic scattering, and pattern formation in biological systems. Current state-of-the-art methods do not include integral-equation based solvers. However, recent work on developing a fast-multipole accelerated solver for the Laplace-Beltrami equation for complex sub-manifolds on the surface of a sphere indicates that this is a very promising avenue of exploration.

拟议的研究计划的总体主题是开发快速准确的积分方程方法，以研究在流体动力学或其他物理和生物系统中引起的应用。基于积分方程的数值方法已经变得越来越流行，这在很大程度上是由于相关的快速算法的开发，可用于加速解决方案程序。 In very general terms, a fast integral equation method (FIEM) first requires formulating a well-conditioned integral equation for a partial differential equation (PDE), then selecting a suitable quadrature method for its discretization, and finally implementing a fast algorithm, such as the fast multipole method or a fast direct solver, to accelerate the solution of the resulting linear system. FIEM比基于差异方法的优点很重要：通过降低尺寸和快速算法的使用获得效率；避免了与直接离散管理方程相关的不良条件；高阶精度更容易获得。 FIEMS的计算效率可能意味着能够在几分钟之内在台式计算机上解决复杂的问题，而不是在集群上花费数小时。相结合的高阶精度，可以实现近乎机器精确的精度。当使用这些工具来研究PDE的分析性能，解决解决方案中的复杂特征并为其他计算方法提供基准数据时，这种高精度可能是无价的。 该提案中讨论的两个长期目标如下： 1。FIEM用于不可压缩的流体动力学 不可压缩的Navier-Stokes方程（INSE）是一个高度非线性，复杂的PDE的系统，它们无处不在，描述从游泳微生物到天气系统的各种流体现象。虽然INSE不能直接通过积分方程进行解，但是合适的时间离散化将产生一系列线性的高阶椭圆方程，必须在每个时间步骤中求解。拟议的研究将通过将解决方案表示为适当选择的层和体积势的总和来开发将这些方程式重塑为积分方程的方法。然后，将使用合适的快速算法来解决和评估这些潜力。在开发二维求解器所需的工具方面取得了重大进展。我们还将考虑三个维度的问题。 2。fiem在表面上的边界价值问题。 在表面上涉及PDE的解决方案的应用包括用于行星尺度流，图像处理，电磁散射以及生物系统中的图案形成的计算流体动力学。当前的最新方法不包括基于积分方程的求解器。然而，最近开发快速多层加速求解器的拉普拉斯 - 贝特拉米方程的求解器的工作，用于球形表面上的复杂子序列，这表明这是一个非常有希望的探索途径。