Fast Direct Solvers for Boundary Value Problems on Evolving Geometries

演化几何边值问题的快速直接求解器

• 批准号: 1522631
• 负责人: Adrianna Gillman
• 金额: $ 15万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522631&HistoricalAwards=false
• 关键词: Fast; Direct; Solvers; Boundary; Value

With the ability to reduce the cost of testing theories and ideas, numerical simulations will continue to play a growing role in scientific discovery and device development. Frequently, these simulations involve the solution of problems that are prescribed by physics. In many cases, the speed and accuracy with which such problems can be solved is a key limiting factor to what can and cannot be modeled numerically. Fast direct solvers have recently shown great promise for solving a large number of problems involving the same geometry by reusing the most expensive part of the solution technique. This situation happens often in settings such as product development where each problem involving the geometry corresponds to a different physical situation. When a large number of problems are under consideration, the fast direct solvers can show hundreds of times speed up over other techniques. While this speed up is great, many engineering situations involve a large number of problems with slightly different geometries. Each different geometry requires the most expensive part of the solver to be recomputed. This project will focus on the development and application of fast direct solvers to problems with evolving geometries. The new technique will recycle information obtained in the construction of the fast direct solver for one geometry to build solvers for the evolved geometries. As a result, the cost of the most expensive step in the fast direct solution technique will be substantially reduced while retaining the benefit of being able to solve multiple problems for each geometry quickly. This work should accelerate many numerical simulations and will have a technological impact on society through applications such as solar cell design, meta-material design, sonar, radar and simulations of blood flow.The numerical simulations under consideration in the proposed work will involve the solution of linear boundary value problems. Many linear boundary value problems can be recast as integral equations. Solution techniques based on integral equations come with the cost of having to solve a dense linear system upon discretization. Fast direct solvers invert a dense system by exploiting structure in the matrix with a cost that grows linearly (or nearly linearly) with the problem size. The proposed work will adapt the fast linear algebra framework to create efficient direct solvers for a family of problems with similar geometries. The new technique will be the first to reuse the structural information obtained in the construction of a fast direct solver for a single geometry to build new direct solvers for evolved geometries. This recycling of structural information reduces the cost of the most expensive step in constructing fast direct solvers. The new technique should accelerate time-stepping for evolution equations, Stokes' flow, optimal design algorithms, model reduction methods and periodic boundary value problem simulations. The solution technique will be applied to microfluids, inverse scattering, and periodic scattering.

由于能够降低测试理论和想法的成本，数值模拟将继续在科学发现和设备开发中发挥越来越大的作用。通常，这些模拟涉及到物理学规定的问题的解决方案。在许多情况下，解决这些问题的速度和准确性是能够和不能够进行数值模拟的关键限制因素。通过重用求解技术中最昂贵的部分，快速直接求解器最近在解决涉及相同几何的大量问题方面显示出了巨大的前景。这种情况经常发生在产品开发等环境中，其中涉及几何的每个问题对应于不同的物理情况。当考虑大量问题时，快速直接求解器可以显示出比其他技术快数百倍的速度。虽然这种速度是巨大的，但许多工程情况涉及到大量几何形状略有不同的问题。每个不同的几何图形都需要重新计算求解器中最昂贵的部分。该项目将重点开发和应用快速直接解算器来解决不断变化的几何问题。新技术将利用在构建一个几何图形的快速直接求解器中获得的信息来构建演化几何图形的求解器。因此，快速直接求解技术中最昂贵的步骤的成本将大大降低，同时保留了能够快速解决每个几何图形的多个问题的优点。这项工作将加速许多数值模拟，并将通过太阳能电池设计、超材料设计、声纳、雷达和血流模拟等应用对社会产生技术影响。本文所考虑的数值模拟将涉及线性边值问题的求解。许多线性边值问题可以转化为积分方程。基于积分方程的求解技术的代价是必须在离散化时求解密集线性系统。快速直接求解器通过利用矩阵中的结构来反演密集系统，其代价随问题规模线性（或近似线性）增长。提出的工作将适应快速线性代数框架，为具有类似几何形状的一系列问题创建有效的直接求解器。新技术将首次重用在构建单个几何形状的快速直接求解器中获得的结构信息，以构建用于进化几何形状的新直接求解器。这种结构信息的循环利用减少了构建快速直接求解器的最昂贵步骤的成本。新技术将加速进化方程、Stokes流、优化设计算法、模型约简方法和周期边值问题模拟的时间步进。溶液技术将应用于微流体、逆散射和周期散射。