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

偏微分方程（PDEs）是许多重要物理和技术现象的数学模型的基础。***本项目涉及偏微分方程数值方法的开发和分析，以及在各种计算机体系结构上求解偏微分方程的数学软件的开发、测试和评估。***偏微分方程计算方案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结果集的求解方法。***物理现象的模型通常涉及偏微分方程的线性椭圆边值问题，其离散化反过来又产生大型代数方程的稀疏线性系统。***其他模型可能涉及时间相关偏微分方程，这通常需要在时间离散问题的每个时间步上求解大型稀疏线性系统。***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。这些主要取决于***(i)离散化方法的收敛性；***(ii)线性求解器的计算复杂度；***(iii)离散化方法和求解器的实现；以及***(iv)利用与模型大小成比例的并行性的能力。***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。***本研究包括以下组成部分：***(a)***开发和分析高阶PDE离散化方法，如样条搭配方法，***和低计算复杂度的求解方法，如FFT方法，多网格方案，***域分解技术和混合方法，具有可扩展的并行度。***离散化方法和求解器首先是为简单的模型问题而开发的，然后扩展到处理更困难的问题，***例如有层、粗糙行为、病态、不连续等问题***(b)***在具有许多处理器的并行机器上实现和测试所提出的解决大型模型的方法。这包括在并行时间和内存复杂性、通信复杂性（在分布式内存机器上）、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方面对解决pde的方法和机器进行性能评估。***(c)***拟议方法在解决金融衍生品估值和医疗应用等问题方面的应用和测试。这些地区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的影响

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.**