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

偏微分方程是许多重要物理和技术现象的数学模型的基础。 该项目包括开发和分析偏微分方程组的数值方法，以及开发、测试和评估在各种计算机体系结构上求解偏微分方程组的数学软件。 偏微分方程组计算格式的两个主要组成部分是连续问题的离散化技术和离散代数方程组的求解方法。 物理现象的模型通常涉及偏微分方程组的线性椭圆型边值问题，其离散化又产生大型稀疏线性代数方程组。 其他模型可能涉及与时间相关的偏微分方程组，这通常需要在时间离散化问题的每个时间步长求解大型稀疏线性系统。 在开发和研究求解大规模偏微分方程组问题的计算方法时，必须解决两个关键问题--即计算的精度和效率。这主要取决于 (I)离散化方法的收敛性质； (Ii)线性求解器的计算复杂性； (3)离散化方法和解算器的实施； (4)利用并行性达到与模型大小成正比的程度的能力。 当数学模型的规模(即离散方程的数量)非常大时，最后一个因素变得特别重要。 本研究包括以下几个部分： (A) 高阶偏微分方程离散化方法的发展与分析 以及计算复杂度低的求解器，例如FFT方法、多重网格格式、 域分解技术和混合方法，具有可伸缩的并行性。 离散方法和求解器首先被开发用于简单的模型问题，然后扩展到处理更困难的问题， 如层数问题、行为粗暴、病态、不连续等。 (B) 在具有多处理机的并行机上实现和测试了所提出的求解大型模型的方法。 这包括对求解偏微分方程组的方法和机器的性能评估 在并行时间和存储复杂性、通信复杂性(在分布式存储机器上)、 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) 对所提出的方法在金融衍生品估值和医疗应用等问题的解决中进行了应用和测试。 这些领域具有重要的战略意义，对经济和其他相关科学领域的发展有直接影响。

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.