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: 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=647659
• 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许多数学模型的基础。 ***物理现象的模型通常涉及PDE的线性边界值问题，PDE的椭圆形值问题又会引起大型的代数方程式稀疏线性系统。解决大规模PDE问题的方法，必须解决两个关键问题 - 即计算的准确性和效率。这些主要取决于***（i）离散方法的收敛属性； ***（ii）线性溶液的计算复杂性；和***（iv）将并行性利用与模型大小成比例的程度的能力。诸如FFT方法，多移民方案，***域分解技术和混合方法，具有可扩展程度的并行性。***离散方法和求解器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，然后处理更困难的问题，***，例如，用于层次，不良行为，不良行为，实施的差异，差异，等等。 *** ***这包括对求解PDE的方法和机器的性能评估*** ***在并行时间和记忆复杂性，通信复杂性（在分布式存储器上），***内存访问延迟（在GPU机器上）（在GPU机器上），加速，速度，利用率，负载平衡和伸缩率和稳定性的问题（C）应用。***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。**