CMG Collaborative Research: Fast and Efficient Radial Basis Function Algorithms for Geophysical Modeling on Arbitrary Geometries

在过去的几十年里，地球科学中的数值方法创新无法与计算机硬件发展的爆炸性相提并论。然而，为了推进科学计算，关键是要开发新的数值方法，在利用这场硬件技术革命的同时，提高算法的简单性、灵活性和准确性。目前的项目正是针对这一目标：开发快速、高效和可并行的径向基函数(RBF)方法，并通过新的图形处理单元(GPU)技术来增强，用于计算地球科学中的应用。为此，已经建立了一个由计算数学家和地球物理学家组成的合作小组。径向基函数法对于n维空间中的任意几何形状具有很高的精度，自然允许局部加密，无网格(无网格)，算法复杂度不随维增加，对于给定数目的节点，通常比传统谱方法具有更高的精度和更长的时间步长。要解决的关键问题是：1)在n维空间中任意几何形状的非规则节点布局上开发局部高阶RBF-有限差分(RBF-FD)模板，具有稳定的时间步长；2)RBF-FD的自适应局部节点求精方案；3)开发混合频谱RBF和FD格式以最大限度地发挥RBF的优势并最小化其计算代价；以及4)在硬件加速器(如GPU)上实现该方法。应用的科学对象包括具有不同性质的三维地幔对流，椭圆几何的地球发电机模型，以及不规则海岸线地形的海啸模拟。通过推进计算数学的前沿，这将利用当今蓬勃发展的科技产业，使社会更深入地了解与地幔对流、地球发电机和海啸相关的地球物理过程。这些现象在大陆运动、地球极性反转、S磁场、地震和海岸洪灾中发挥了关键作用。提出的创新的数学计算方法可能是解开这些地球物理领域中长期存在的重要问题的关键。一个由计算数学家和地球物理学家组成的跨学科协作小组由一个国家实验室和四所大学(科罗拉多州大学、博伊西州立大学、爱达荷州大学、爱达荷州大学)组成。明尼苏达州，佛罗里达州立大学和大学。加州大学戴维斯分校)开发计算数学的新方法，以解决这些关键的地球物理问题。该项目支持处于职业生涯不同阶段的研究人员以及4个州的博士、硕士和本科生以及众多科学和数学学科的合作参与。这些学生将有机会作为跨学科团队的成员参加，该团队由对教育和研究表现出承诺的高级人员组成。NCAR不仅将作为科学努力的整合中心，还将为来自全国各地的学生提供一个媒介，让他们在提案的多个方面共同努力。通过在NCAR的SciPARCS 10周实习计划，学生们将有机会参与联合合作研究，为他们在21世纪的计算地球科学职业生涯做好准备。

Numerical method innovations in the geosciences have not paralleled theexplosion in computer hardware development over the last decades. Yet, forscientific computing to advance, it is crucial that novel numericalapproaches are developed that improve the simplicity, flexibility, andaccuracy of algorithms while taking advantage of this revolution in hardwaretechnology. The current project is aimed at exactly that objective: todevelop fast, efficient and parallelizable radial basis function (RBF)methodologies, enhanced by novel graphics processing unit (GPU) technology,for applications in computational geosciences. A collaborative team ofcomputational mathematicians and geophysicists has been established for thispurpose. The RBF approach is very attractive in that it achieves high-orderaccuracy for arbitrary geometries in n-dimensional space, naturally permitslocal refinement, is mesh-free (no grids), algorithmic complexity does notincrease with dimension and, generally offers higher accuracy with longertime steps than traditional spectral methods for a given number of nodes.However, RBFs are still in an early developmental stage; key issues to beaddressed are: 1) development of localized high-order RBF-finite difference(RBF-FD) stencils on irregular node layouts in n-dimensional space forarbitrary geometries with stable time-stepping, 2) adaptive local noderefinement schemes for RBF-FD, 3) development of hybrid spectral RBF and FDschemes to maximize advantages of RBFs and while minimizing theircomputational cost, and 4) implementation of the method on hardwareaccelerators, such as GPUs. Scientific targets for application include 3Dmantle convection with varying properties, models of the geodynamo in anelliptic geometry, and tsunami modeling with irregular coastline topography.By advancing the frontier of computational mathematics that will takeadvantage of today?s booming technology industry, society can be offered amore in depth understanding of geophysical processes related to mantleconvection, the geodynamo, and tsunamis. These phenomena play a key role incontinental movement, polarity reversal of the earth?s magnetic field,earthquakes, and coastal flooding. The proposed innovative mathematicalcomputational approach may hold the key to unlocking long-standing questions of fundamental importance in such geophysics areas. An interdisciplinary collaborative team of computational mathematicians and geophysicists has been assembled from a national lab and four universities (NCAR-Colorado, Boise State Univ.-Idaho, Univ. of Minnesota, Florida State Univ., and the Univ. of California?Davis) to develop new methods in computational mathematics for addressing these critical geophysical problems. The project supports collaborative participation among researchers at different stages in their careers as well as Ph.D, Masters, and undergraduate students in 4 states and in a myriad of scientific and mathematical disciplines. These students will have the opportunity to participate as members of an interdisciplinary team composed of senior personnel with a demonstrated commitment to education and research. NCAR will serve not only as an integrating hub for scientific endeavors but provide a medium for students from across the country to work together on multiple facets of the proposal. Through the SciPARCS 10 week internship program at NCAR, students will have the opportunity to engage in joint collaborative research, preparing them for a career in computational geosciences for the 21st century.