The phenomenal advance in computer technology in terms of processing speed and capacity, closely described by Moore's law, in the last four decades has been outpaced by the explosive amount of data that are used to describe more realistic models in scientific computing. For instance, the number of unknowns in a linear system has grown from hundreds in the past to tens of millions nowadays. Fast algorithms such as the celebrated fast multipole method (FMM) have provided a computational tool for narrowing the gap. At the same time, there is a great need and challenge to develop better computation techniques and utilize the present and emerging computers, with the gain in speed up to a couple of orders of magnitude. The goal of the proposed research is to advance computational theories and techniques, in order to meet the demand and challenge for large scale simulations of complex systems in scientific, medical and engineering studies.
The research team proposed to investigate, innovate and integrate the key simulation steps, from analytic re-formulation of system models with complex geometries to combinatorial optimization in mapping numerical algorithms to computing architectures. Many traditional models are formulated in terms of linear or nonlinear partial differential equations (PDEs) with boundary conditions on complex geometries. By the work of other researchers and principal investigators, integral equation (IE) formulations have lead to better numerical algorithms in both efficiency and stability, and more importantly enabled certain important large-scale simulations. It is proposed first to study the reformulation of traditional PDE models into IE models, as a direct and analytical approach to innovative algorithm design. Next, preconditioning techniques will be studied as an indirect and stabilization approach. Furthermore, Graph-theoretic methods will be applied to optimize the FMM-based algorithms on various modern computer architectures, especially, parallel architectures. These key components will be studied in conjunction, not in isolation.
The intellectual merits of the proposed work are three-fold. It sheds lights on (1) the model reformulation into IEs of the second kind as a fundamental analytic-algorithmic approach to accelerating and stabilizing numerical computation, (2) the connection between reformulation and preconditioning, and (3) on the mutual dependence of numerical algorithms and computer architectures. The proposed work will have broader impacts on various applications through timely dissemination with demonstration of case studies. Three application areas of specific concern are electrostatics calculation in molecular dynamics simulations, computational fluid dynamics, and the study of oxygen delivery in tissues and tumors via microvascular networks. The proposed work involves interdisciplinary research collaboration and cultivation of young and new researchers with multi-disciplinary backgrounds. Finally, the findings and algorithms will be embodied in open source high performance software to facilitate research computing by and large and to be used in classrooms.
