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

在过去的几十年里，地球科学中数值方法的创新并没有与计算机硬件发展的爆炸式增长相提并论。然而，为了科学计算的进步，在利用硬件技术革命的同时，开发新的数值方法来提高算法的简单性、灵活性和准确性是至关重要的。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，通过新型图形处理单元（GPU）技术增强，用于计算地球科学的应用。为此目的，一个由计算数学家和地球物理学家组成的合作小组已经成立。RBF方法非常有吸引力，因为它可以在n维空间中实现任意几何的高阶精度，自然允许局部细化，无网格（无网格），算法复杂性不随维度增加，并且对于给定数量的节点，通常比传统的谱方法提供更高的精度和更长的时间步长。然而，rbf仍处于早期发展阶段；需要解决的关键问题是：1)开发具有稳定时间步进的n维空间任意几何的不规则节点布局的局部高阶RBF-有限差分（RBF- fd）模板；2)RBF- fd的自适应局部节点定义方案；3)开发混合频谱RBF和fd方案，以最大限度地发挥RBF的优势，同时最小化其计算成本；4)在硬件加速器（如gpu）上实现该方法。应用的科学目标包括具有不同性质的三维地幔对流，非椭圆几何的地球动力学模型，以及不规则海岸线地形的海啸模型。通过推进计算数学的前沿将利用今天？随着科技行业的蓬勃发展，社会可以更深入地了解与地幔对流、地球动力学和海啸有关的地球物理过程。这些现象在大陆运动、地球极性反转中起着关键作用。S磁场、地震和沿海洪水。提出的创新数学计算方法可能是解开这些地球物理领域长期存在的重要基础问题的关键。一个由计算数学家和地球物理学家组成的跨学科合作团队已经从一个国家实验室和四所大学(NCAR-Colorado、博伊西州立大学-爱达荷分校、明尼苏达大学、佛罗里达州立大学和加州大学？戴维斯)开发新的计算数学方法来解决这些关键的地球物理问题。该项目支持4个州和无数科学和数学学科的不同职业阶段的研究人员以及博士、硕士和本科生之间的合作参与。这些学生将有机会作为一个跨学科团队的成员，该团队由具有教育和研究承诺的高级人员组成。NCAR不仅将作为科学努力的综合中心，而且将为来自全国各地的学生提供一个在提案的多个方面共同努力的媒介。通过SciPARCS在NCAR为期10周的实习项目，学生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领域的职业生涯做好准备。

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.