High-performance computational methods for Partial Differential Equations and applications

偏微分方程（PDE）是许多物理/技术现象的数学模型的基础。我的研究计划的长期目标涉及新的数值方法的发展和分析偏微分方程，和数学软件的开发，测试和评估的解决方案的偏微分方程的各种计算机架构。 该研究在金融和医学领域具有实际应用价值，如评估违约风险和改进治疗策略。偏微分方程的计算方案的两个主要挑战是连续问题的离散化技术，以及由此产生的离散代数方程组的求解方法。物理现象的模型通常涉及线性椭圆偏微分方程，其离散化产生大型稀疏线性方程组。其他模型涉及时间相关的偏微分方程，这往往需要在时间离散化问题的每个时间步长的大型稀疏线性系统的解决方案。在开发和研究求解大规模偏微分方程问题的计算方法时，必须解决两个关键问题--计算的精度和效率。解决这些问题主要取决于四个因素：（i）离散化方法的收敛特性;（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;（iv）利用并行性与模型大小成比例的能力。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。这些关键问题将使用以下方法来解决：（a）高阶PDE离散化方法：样条配置;和低计算复杂性求解器：FFT方法、交替方向隐式方法和区域分解技术，具有可扩展的并行度。离散化方法和求解器将首先为简单的模型问题开发，然后扩展到更困难的问题，例如分层，不连续性和非线性问题。(b)所提出的方法在金融衍生品估值和胶质瘤侵袭医学中的应用。我们将针对当前的挑战，包括方法的稳定性，所产生的线性方程组的有效解决方案，以及适应的方法来处理问题的解决方案的特殊属性。(c)分析和测试在具有多个处理器的并行机上求解大型模型的方法;在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上）、加速、利用率、负载平衡和可扩展性方面对求解PDE的方法和机器进行性能评估。这项研究将对经济、卫生和相关科学领域的发展产生直接和重大的影响。

Partial Differential Equations (PDEs) are the basis of many mathematical models of physical/technological phenomena. The long-term goal of my research program involves the development and analysis of novel numerical methods for PDEs, and the development, testing and evaluation of mathematical software for the solution of PDEs on a variety of computer architectures.  This research has practical applications in finance and medicine, such as valuation of default risk and improvement of treatment strategies. Two of the main challenges of a computational scheme for PDEs are the discretization 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 PDEs, the discretisation of which gives rise to large sparse linear systems of equations. Other models involve time-dependent PDEs, which often require the solution of large sparse linear systems at each timestep of the time-discretized problem. In developing and studying computational methods for solving large-scale PDE problems, two key issues have to be addressed -- the accuracy and the efficiency of the computations. Addressing these issues mainly depends on four factors (i) the convergence properties of the discretisation method; (ii) the computational complexity of the linear solver; (iii) the implementation of the discretization method and solver; (iv) the ability to exploit parallelism to a degree proportional to the model size. This last factor becomes particularly important when the size of the mathematical model, i.e. the number of discrete equations, is very large. These key issues will be addressed using the following methodologies: (a) High-order PDE discretisation methods: spline collocation; and low computational complexity solvers: FFT methods, Alternating Direction Implicit methods and domain decomposition techniques, with a scalable degree of parallelism. Discretization methods and solvers will be first developed for simple model problems, then extended to more difficult ones, e.g. problems with layers, discontinuities and nonlinearities. (b) Application of the proposed methods to financial derivatives valuation and glioma invasion in medicine. We will target current challenges including the stability of the methods, the efficient solution of the resulting linear systems of equations, and the adaptation of the methods to handle special properties of the problems' solutions. (c) Analysis and testing of the proposed methods for solving large models on parallel machines with many processors; 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 GPUs), speedup, utilisation, load balancing and scalability. This research will have a direct and significant impact on the economy, health and the development of related fields of science.