High-performance computational methods for Partial Differential Equations and applications

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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.