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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=656040
• 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和准确的积分方程方法，用于调查流体动力学或其他物理和生物系统中产生的应用。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快速算法的发展，这些算法可以用于加速求解过程。一般来说，快速积分方程法（FIEM）首先需要为偏微分方程（PDE）制定一个条件良好的积分方程，然后选择一个合适的求积方法进行离散化，最后实现一个快速算法，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解决方案。与基于差分的方法相比，FIEMS的优点是显著的：通过降维和使用快速算法获得效率;避免了与直接离散控制方程相关的病态;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解决台式计算机上的复杂问题，而不是在集群上花费数小时。上级稳定性加上高阶精度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性质、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。该提案中讨论的两个长期目标如下：**1。* 不可压缩Navier-Stokes方程（INSE）是一个高度非线性、复杂的偏微分方程系统，在描述从游动微生物到天气系统的各种流体现象时无处不在。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一系列线性高阶椭圆方程，这些方程必须在每个时间步长求解。拟议的研究将开发方法，通过将解表示为适当选择的层势和体积势的总和，将这些方程重新转换为积分方程。然后，适当的快速算法将被用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。我们还将从三个方面考虑问题。** 二.曲面上边值问题的有限元法涉及表面上偏微分方程的解决方案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图案形成。当前最先进的方法不包括基于积分方程的求解器。然而，最近的工作开发一个快速多极加速求解拉普拉斯-贝尔特拉米方程的复杂的子流形上的表面上的一个球表明，这是一个非常有前途的探索途径。