Adaptive Wavelet Element Methods for Highly Parallel Computations

自适应小波元方法是一种新的数学方法，从根本上讲，它比现有的方法更适合于高度并行计算。本项目研究基于自适应小波方法的可扩展并行算法和模拟，用于研究燃烧流动问题。由于工业应用的内在物理特性对大规模模拟能力的需求，本课题的研究重点是燃烧流动问题的并行小波模拟，包括相关的算法/数学开发和软件实现。在小波和区域分解方法的基础上，提出了一种新的方法--小波元方法。该方法将小波方法的适用范围扩展到燃烧问题和其他复杂几何形状的粘性流动问题，并提供了基于区域分解的粗粒度并行。它高效、稳定、高精度、高并行性，并具有良好的数据局部性。小波支持空域和频域的局部化，使得自适应网格生成成为可能。小波系数唯一地反映了子域之间的工作量变化，而小波元方法中的并行性来自两个层次--区域分解层次和每个子域内的Spline小波ADI的并行求解层次。主要工作包括：1)提出了一种两层并行小波元方法，将区域分解和小波ADI方法相结合，以最大限度地提高并行性、局部性和可扩展性。2)研究了小波元背景下单元(子域)之间的界面条件及其对求解精度、工作量平衡和通信代价的影响。3)研究了基于区域分解层小波系数和各子域ADI小波解的动态负载均衡。4)对多维预混火焰燃烧转爆轰问题的可伸缩并行自适应小波元方法进行了验证。

The Adaptive Wavelet element method is a new mathematical method which is fundamentally more appropriate for highly parallel computing than existing methods. This project studies scalable parallel algorithms and simulations based on adaptive wavelet methods for the investigation of combustion flow problems. The project isa driven by the need for large scale simulation capability required by the intrinsic physical property of industrial applications.The focus of this research is on parallel wavelet simulation of combustion flow problems, which includes the associated algorithmic/mathematical development and software implementation. A new method - the Wavelet Element Method - is introduced based on wavelet and domain decomposition methods. The new method extends the applicability of wavelet methods to combustion problems and other viscous flow problems in more complex geometries and provides a coarse grain parallelism based on domain decomposition. It is efficient, stable, highly accurate, and highly parallel, and it has good data locality. Wavelets support localization in both space and frequency domain which makes adaptive grid generation possible. Wavelet coefficients uniquely indicate the workload variation across subdomains, and parallelism in the wavelet element method comes from two levels - the level of domain decomposition and the level of parallel solution of Spline Wavelet ADI in each subdomain.Four main tasks will be performed: 1) Develop a two level parallelized wavelet element method, which combines the domain decomposition and wavelet ADI methods, to maximize parallelism, locality, and scalability. 2) Investigate interface conditions among elements (subdomains) in the context of wavelet element and investigate their effects on solution accuracy, workload balance, and communication cost. 3) Study dynamic workload balance based on wavelet coefficients on both domain decomposition level and on ADI wavelet solution in each subdomain. 4) Test the scalable parallel adaptive wavelet element methods on the Deflagration to Detonation Transition Problems for multidimensional premixed flames.