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，它比现有的方法更适合于高度并行计算。 本计画研究以自适应小波方法为基础之可量测平行演算法与模拟，以研究燃烧流动问题。 该项目伊萨由需要大规模的模拟能力所需的固有物理性质的工业应用，本研究的重点是并行小波模拟燃烧流问题，其中包括相关的算法/数学开发和软件实现。 在小波分析和区域分解方法的基础上，提出了一种新的方法--小波单元法. 新方法将小波方法的适用性扩展到更复杂几何形状的燃烧问题和其他粘性流动问题，并提供了基于区域分解的粗粒度并行性。 它具有高效、稳定、高精度、高并行性和良好的数据局部性等特点。 Wavelet支持空间域和频域的局部化，这使得自适应网格生成成为可能。 小波系数唯一地指示子域之间的工作量变化，并且小波元素方法中的并行性来自两个级别-区域分解级别和样条小波ADI在每个子域中的并行解级别。将执行四个主要任务：1）提出了一种两级并行小波元方法，该方法结合了区域分解和小波ADI方法，最大限度地提高了并行性、局部性和可扩展性。2)研究小波元素上下文中元素（子域）之间的接口条件，并研究它们对解决方案精度、工作负载平衡和通信成本的影响。3)研究了基于域分解层小波系数和基于各子域内ADI小波解的动态负载平衡问题。4)在多维预混火焰的爆燃转爆轰问题上测试可扩展并行自适应小波元方法。

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.