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，以研究流体动力学或其他物理和生物系统中的应用。基于积分方程的数值方法变得越来越流行，这在很大程度上是由于可用于加速求解过程的相关快速算法的发展。一般来说，快速积分方程方法 (FIEM) 首先需要为偏微分方程 (PDE) 制定条件良好的积分方程，然后选择合适的求积方法进行离散化，最后实现快速算法，例如快速多极子方法或快速直接求解器，以加速所得线性系统的求解。 FIEM 相对于基于差异的方法的优点是显着的：通过降维和使用快速算法来获得效率；避免了与直接离散化控制方程相关的病态条件；并且更容易达到高阶精度。 FIEM 的计算效率意味着能够在几分钟内在台式计算机上解决复杂问题，而不是在集群上花费数小时。卓越的稳定性能与高阶精度相结合意味着可以实现接近机器精度的精度。当使用这些工具研究偏微分方程的分析特性、解决解决方案中的复杂特征以及为其他计算方法提供基准数据时，这种高精度是非常宝贵的。 提案中讨论的两个长期目标如下： 1. 不可压缩流体动力学的 FIEM 不可压缩纳维-斯托克斯方程 (INSE) 是一个高度非线性、复杂的偏微分方程组，在描述从游动微生物到天气系统等各种流体现象时普遍存在。虽然 INSE 不能直接通过积分方程求解，但适当的时间离散化将产生必须在每个时间步长求解的线性高阶椭圆方程集合。拟议的研究将开发将这些方程重新转换为积分方程的方法，将解表示为适当选择的层势和体积势的总和。然后，将使用合适的快速算法来求解和评估这些潜力。在开发二维求解器所需的工具方面已经取得了重大进展。我们也会从三个维度来考虑问题。 2. 曲面边界值问题的 FIEM。 涉及表面偏微分方程求解的应用包括行星尺度流动的计算流体动力学、图像处理、电磁散射和生物系统中的模式形成。当前最先进的方法不包括基于积分方程的求解器。然而，最近针对球体表面复杂子流形的 Laplace-Beltrami 方程开发快速多极加速求解器的工作表明，这是一个非常有前途的探索途径。