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

偏微分方程组是许多重要物理和技术现象的数学模型的基础。*这个项目涉及开发和分析偏微分方程组的数值方法，以及在各种计算机体系结构上开发、测试和评价求解偏微分方程组的数学软件。偏微分方程组的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组的求解方法。*物理现象模型通常涉及偏微分方程组的线性椭圆边值问题，其离散化依次为：产生了大型稀疏线性代数方程组。*其他模型可能涉及依赖时间的偏微分方程组，这往往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方程组。*在开发和研究大型偏微分方程组的计算方法时，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；*(Iii)离散化方法和求解器的实现；以及*(Iv)利用并行性到与模型大小成比例的程度的能力。*当数学模型的大小(即，离散方程的数量)非常大时，最后一个因素变得特别重要。*本研究包括以下部分：*(A)*高阶PDE离散化方法的发展和分析，例如Spline配置法，*和低计算复杂度的求解器，例如FFT方法、多重网格格式、*区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是针对简单的模型问题开发的，然后扩展到处理更困难的问题，如分层问题、粗糙行为、病态、间断等。*(B)*在具有多个处理器的并行机上实现和测试所提出的求解大型模型的方法。*这包括从并行时间和内存复杂性、通信复杂性(在分布式内存机上)、*内存访问延迟(在GPU机器上)、加速比、利用率、负载均衡和可扩展性。*(C)*所提出的方法在解决金融衍生品估值和医疗应用等问题中的应用和测试。*这些领域具有重要的战略意义，直接影响经济和其他相关科学领域的发展。**

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.**