Numerical Methods for Partial Differential Equations: Algorithms and Software on Innovative Computer Architectures

偏微分方程的数值方法：创新计算机架构的算法和软件

• 批准号: RGPIN-2015-05648
• 负责人: Christara, Christina
• 金额: $ 1.31万
• 依托单位: University of Toronto
• 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=613669
• 关键词: Numerical; Methods; Partial; Differential; Equations

Partial Differential Equations (PDEs) are the basis of many mathematical models of important physical and technological phenomena. This project involves the development and analysis of numerical methods for PDEs, and the development, testing and evaluation of mathematical software for the solution of PDEs on a variety of computer architectures. Two of the main components of a computational scheme for PDEs are the discretisation technique for the continuous problem and the solution method for the resulting set of discrete algebraic equations. Models of physical phenomena often involve linear elliptic Boundary Value Problems for PDEs, the discretisation of which, in turn, gives rise to large sparse linear systems of algebraic equations. Other models may involve time-dependent PDEs, which often require the solution of large sparse linear systems at each time step of the time-discretized problem. In developing and studying computational methods for solving large-scale PDE problems, two key issues have to be addressed -- namely, the accuracy and the efficiency of the computations.These mainly depend on (i) the convergence properties of the discretisation method; (ii) the computational complexity of the linear solver; (iii) the implementation of the discretisation method and solver; and (iv) the ability to exploit parallelism to a degree proportional to the size of the model. This last factor becomes particularly important when the size of the mathematical model (i.e., the number of discrete equations) is very large. This research includes the following components: (a) Development and analysis of high-order PDE discretisation methods, such as spline collocation methods, and low computational complexity solvers, such as FFT methods, multigrid schemes, domain decomposition techniques and hybrid approaches, with a scalable degree of parallelism. Discretisation methods and solvers are first developed for simple model problems, then extended to handle more difficult problems, such as problems with layers, rough behaviour, ill-conditioning, discontinuities, etc. (b) Implementation and testing of the proposed methods for solving large models on parallel machines with many processors. This includes the performance evaluation of methods and machines for solving PDEs in terms of parallel time and memory complexity, communication complexity (on distributed memory machines), memory access latency (on GPU machines), speedup, utilisation, load balancing and scalability. (c) Application and testing of the proposed methods in the solution of problems such as financial derivatives valuation and medical applications. These areas are strategically important having a direct impact on the economy and the development of other related fields of science.

部分微分方程（PDE）是许多重要物理和技术现象的数学模型的基础。 该项目涉及PDE的数值方法的开发和分析，以及用于在各种计算机架构上解决PDE的数学软件的开发，测试和评估。 PDE的计算方案的两个主要组成部分是连续问题的离散技术，也是由此产生的离散代数方程组集的解决方案方法。 物理现象的模型通常涉及PDE的线性椭圆边界价值问题，其离散化反过来又产生了代数方程的稀疏线性系统。 其他模型可能涉及时间依赖性PDE，通常需要在时间限制问题的每个时间步骤中解决大型稀疏线性系统的解决方案。 在开发和研究解决大规模PDE问题的计算方法时，必须解决两个关键问题 - 即计算的准确性和效率。 （i）离散方法的收敛属性； （ii）线性求解器的计算复杂性； （iii）实施离散方法和求解器；和 （iv）将并行性与模型大小成正比的程度相称的能力。 当数学模型的大小（即离散方程数）非常大时，最后一个因素变得尤为重要。 这项研究包括以下组成部分： （一个） 高阶PDE离散方法的开发和分析，例如样条搭配方法， 和低计算复杂性求解器，例如FFT方法，跨部方案， 域分解技术和混合方法，具有可扩展的并行度。 离散方法和求解器首先是针对简单模型问题开发的，然后扩展以处理更困难的问题， 例如层次，粗暴行为，不良条件，不连续性等问题。 （b） 使用许多处理器上的平行机上求解大型模型的建议方法的实施和测试。 这包括解决PDE的方法和机器的性能评估 在平行时间和内存复杂性方面，通信复杂性（在分布式内存机上）， 内存访问延迟（在GPU机器上），加速，利用率，负载平衡和可扩展性。 （C） 在解决金融衍生品评估和医疗应用等问题中的应用和测试。 这些领域在战略上对经济和其他相关科学领域的发展产生直接影响。