Parallel, Adaptive Cartesian Grid Algorithms for Natural Hazards Modeling

洪水、海啸、泥石流、山体滑坡、风暴潮和潜在的水坝倒塌等地球物理灾害威胁着美国和全球各地的社区。 该项目开发了能够有效模拟这些灾害的计算工具，使不同的研究人员和应急规划人员能够开发最有可能受到这些灾害影响的地区的灾害地图。 为了效率，计算框架使用自适应的“深度平均”数学模型，其仅需要二维平面网格，而不是完全三维网格。 该项目的一个主要目标是纠正我们的深度平均模型，以捕捉局部波，可能溢出洪水屏障或超过港口防波堤。 这些校正条款将为当前用户的计算框架的关键附加功能，模拟浅层地球物理灾害，并允许他们创建更强大的灾害地图。 计算工具还可以充分利用桌面工作站、中等规模的计算集群以及NSF资助的超级计算站点上可用的大规模并行计算设施上的新兴硬件趋势。 该项目还向用户提供工具，利用谷歌地球浏览器等开放源码软件将结果可视化。 最终，计算建模可以帮助响应者预测如何分配应急资源，在不可避免的危险的情况下，并告知开发商，立法代表，和公民的潜在风险在他们的community.The研究将集中在实现一个直接求解器的变系数椭圆问题的自适应细化四叉树网格。 目标解算器是由A. Gillman和P. Martinsson。 满足四个关键属性，这个求解器（1）具有无矩阵方法的易用性，（2）可以快速求解附近的系统，（3）具有最佳的O（N）效率，（4）提供可以调整的参数，以减少计算成本与精度要求成比例。 此外，该方法使用低秩近似来压缩稠密矩阵和加速矩阵计算。 在拟议的工作中，PI将修改原始HPS求解器，以用于二阶有限体积格式，并在ForestClaw（并行、基于补丁的笛卡尔自适应四叉树代码）中实现求解器。 PI还将报告HPS方法实施的可扩展性和并行效率。将会出现的两个技术挑战是开发有效的程序来跨处理器边界合并Dirichlet到Neumann映射，并增量更新动态演变网格的求解器因式分解。 有针对性的应用程序是解决方案的Serre-Green Naghdi方程建模色散校正浅水波方程。 这些更正将包含在ForestClaw的GeoClaw扩展中。 GeoClaw（D.乔治河J. LeVeque，M. Berger）是一个广泛使用的求解水深平均水流方程的软件包。在GeoClaw扩展中添加这些校正项将为GeoClaw用户提供建模海啸、洪水、泥石流、风暴潮和其他浅层地球物理流的关键能力。 最终，建议的求解器可以在ForestClaw框架内作为一个通用的椭圆形求解器的各种物理流phenomenon. This奖项反映了NSF的法定使命，并已被认为是值得通过使用该基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Geophysical hazards such as flooding, tsunamis, debris flows, landslides, storm surges and potential dam failures threaten communities across the United States and globally. This project develops computational tools that can efficiently simulate these hazards, enabling a diverse group of researchers and emergency planners to develop hazard maps of areas most likely to be impacted by these disasters. For efficiency, the computational framework uses adaptive, "depth-averaged" mathematical models that only require two dimensional planar grids, rather than fully three dimensional meshes. A primary goal of the project is to correct our depth-averaged model to capture localized waves that may spill over flood barriers or overtop harbor breakwaters. These correction terms will give current users of the computational framework critical additional capabilities for modeling shallow geophysical hazards and allow them to create more robust hazard maps. The computational tools can also take full advantage of emerging hardware trends available on desktop workstations, moderate sized compute clusters, as well as massively parallel computing facilities available at NSF funded supercomputing sites. The project also provides users with tools for visualizing results using open source software such as the Google Earth browser. Ultimately, computational modeling can aid responders in predicting how to distribute emergency resources in the event of unavoidable hazards and serve to inform developers, legislative representatives, and citizenry of potential risks in their communities.The research will focus on the implementation of a direct solver for variable coefficient elliptic problems on adaptively refined quad-tree meshes. The targeted solver is the Hierarchical Poincare Steklov (HPS) solver, developed by A. Gillman and P. Martinsson. Satisfying four crucial properties, this solver (1) has the ease of use of matrix-free methods, (2) can solve nearby systems quickly, (3) has optimal O(N) efficiency, and (4) provides parameters that can be tuned to reduce computational cost in proportion to accuracy requirements. Furthermore, the method uses low rank approximations to compress dense matrices and accelerate matrix computations. In the proposed work, the PI will modify the original HPS solver for use with second order, finite volume schemes and implement the solver in ForestClaw, the parallel, patch-based Cartesian adaptive quad-tree code. The PI will also report on the scalability and parallel efficiency of the implementation of the HPS method. Two technical challenges that will arise are to develop effective procedures for merging Dirichlet-to-Neumann maps across processor boundaries and incrementally updating the solver factorization for dynamically evolving meshes. The targeted application is the solution to the Serre-Green Naghdi equations for modeling dispersive corrections to the shallow water wave equations. These corrections will be included in the GeoClaw extension of ForestClaw. GeoClaw (D. George, R. J. LeVeque, M. J. Berger) is a widely used software package for solving depth-averaged flow equations. The addition of these correction terms to the GeoClaw extension will provide GeoClaw users with critical capabilities for modeling tsunamis, flooding, debris flows, storm surges and other shallow geophysical flows. Ultimately, the proposed solver can be used within the ForestClaw framework as a general purpose elliptic solver for a variety of physical flow phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.