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

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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.