Fast Integral Equation Methods: Algorithms and Applications

拟议研究计划的总主题是开发快速和准确的积分方程式方法，以研究流体动力学或其他物理和生物系统中的应用。基于积分方程组的数值方法已经变得越来越流行，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。一般而言，快速积分方程法首先需要建立一个条件良好的偏微分方程组，然后选择合适的求积方法对其进行离散，最后实现快速算法，如快速多极子方法或快速直接求解器，以加速所得到的线性方程组的求解。与基于差分的方法相比，有限元方法的优点是显著的：通过降维和使用快速算法来获得效率；避免了直接离散控制方程所带来的病态；以及更容易获得高阶精度。FIEM的计算效率可能意味着能够在台式计算机上在几分钟内解决复杂问题，而不是在群集上花费数小时。优越的稳定性与高阶精度相结合，意味着可以达到接近机器精度的精度。当使用这些工具研究偏微分方程组的分析性质、解析解中的复杂特征并为其他计算方法提供基准数据时，这种高精度是无价的。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场不可压缩的Navier-Stokes方程(INSE)是一个高度非线性、复杂的偏微分方程组系统，它普遍存在于描述从游泳微生物到天气系统的广泛流体现象中。虽然积分方程不能直接求解，但适当的时间离散化将产生在每个时间步必须求解的线性、高阶椭圆型方程的集合。建议的研究将通过将解表示为适当选择的层势和体积势的总和来发展将这些方程重塑为积分方程组的方法。然后，将使用合适的快速算法来求解和评估这些潜力。在开发二维解算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。2.表面边值问题的场：涉及求解表面偏微分方程组的应用包括行星尺度流动的计算流体力学、图像处理、电磁散射和生物系统中的图案形成。目前最先进的方法不包括基于积分方程的求解器。然而，最近关于开发球面上复子流形的Laplace-Beltrami方程的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。

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 DynamicsThe 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.