We propose to develop and deploy mathematical software for boundary-value problems in three-dimensional complex geometries. The algorithms in the library will be based on integral equation formulations. The library will be designed to scale on novel computing platforms that comprise special accelerators and manycore architectures.
Integral equations can be used to conduct simulations on many problems in science and engineering with significant societal impact. Three example applications on which the proposed simulation technologies will have an impact in this project are microfluidic chips, biomolecular electrostatics, and plasma physics. First, microfluidic chips are submillimeter-sized devices used for medical diagnosis and drug design. Optimizing the function of such devices at low cost requires efficient computer simulation tools, such as the ones we propose to develop. Second, understanding the structure and function of biomolecules such as DNA and proteins is crucial in biotechnology. The proposed technologies can be used to resolve bimolecular electrostatic interactions. Third, plasma physics, which is related to fusion nuclear reactors, includes electrostatic interactions in complex geometries, and the proposed work will enable large-scale three-dimensional simulations.
The key features of the proposed software are: (1) parallel fast multipole methods, (2) efficient geometric modeling techniques for complex geometries, (3) simple library interfaces that allow use of the proposed software by non-experts, and (4) scalability on heterogeneous architectures.
Along with our research activities, an educational and dissemination program will be designed to communicate the results of this work to students and researchers. Several postdoctoral, graduate, and undergraduate students will be involved with the project. Additional educational activities will include research experiences for undergraduates, leveraging ongoing programs such as NSF REUs. We will encourage participation by women, minorities, and underrepresented groups.
Boundary value problems find application in numerous areas of science and engineering. Examples include fluid mechanics, solid mechanics, electromagnetism, quantum mechanics, geophysics, and imaging. For this reason several research groups worldwide work on software libraries for such solvers. We focus on a set of problems for which integral equation formulations provide an optimal framework, while maximizing robustness, accuracy, and scalability using features specific to this type of equations. The set of tools that we envision follow the example of black-box solvers provided in some major numerical libraries. For example, FISHPACK and Intel’s Math Kernel Libraries provide routines for scalar boundary value problems on simple domains (rectangular or spherical) using uniform discretizations. The goal of this project is to develop and deploy a library of black-box solvers for linear non-oscillatory boundary value problems in three-dimensional complex geometries. The algorithms in the library will be based on integral equation formulations, and support scalar and vector problems and highly non-uniform discretizations. Its main features will be high accuracy, adaptivity and high performance on manycore and heterogeneous architectures. Upon completion of the project we would like to report the following outcomes. - We have designed, implemented and released a volume fast multipole method for continuous functions and particles that supports several different boundary value problems and scales on distributed and shared memory architectures. - We have designed and released parallel sort software that scales to thousands of cores. - We have published 14 research papers on surface representations, linear solvers, preconditioners, and time-stepping methods that can be used in conjunction with our scheme. - Trained four graduate students and three postdoctoral researchers. - Provided summer internships for undergraduate students - Organized several minisymposia on solvers for integral equations - Further disseminated this work through presentations and invited talks. The codes described above are available at: http://padas.ices.utexas.edu/software
