Fast Multipole Methods represent an area of vigorous research at UMIACS. The effort is lead by Nail Gumerov and Ramani Duraiswami with several students, visitors and collaborators. In this page an overview of this research is provided.
NEW: See our book Fast Multipole Methods for the Helmholtz Equation in Three Dimensions (The Elsevier Electromagnetism Series)
Background
The fast multipole method has been called one of the ten most significant algorithms in scientific computation discovered in the 20th century (along with algorithms such as the FFT), and won its inventors, Vladimir Rokhlin & Leslie Greengard, the 2001 Steele prize, in addition to getting Greengard the ACM 1987 best dissertation award.
The algorithm allows the product of particular dense matrices with a vector to be evaluated approximately (to a specified precision) in O(Nlog N) operations, when direct multiplication requires O(N2) operations. These matrices are those whose elements arise from distributions of points {x} and {y} in Rd, i.e. each element of the matrix can be represented as
Aij=j(xi,yj).
Coupled with advances in iterative methods for the solution of linear systems, they provide O(N log N) time and memory complexity solutions of problems that hitherto required O(N3) or O(N2) time complexity and O(N2) memory complexity. For extremely large problems, the gain in efficiency and memory can be very significant.
FMM work
Our work on the FMM has focused on both generalizing the method so that it can be applied as a general matrix-vector acceleration algorithm and applying it to many different problem areas. We have also focused on extending the method to new problem areas, especially in statistics and vision. We have also developed what perhaps the first course offered on the FMM anywhere. Our class notes for the most current course are here and links to older course notes are also provided.
Links to some papers are presented below.
Nail A. Gumerov, Ramani Duraiswami, “A scalar potential formulation and translation theory for the time-harmonic Maxwell equations,” Department of Computer Science Technical Reports CS-TR-4785, Institute for Advanced Computer Studies, UMIACS-2006-09. pdf (accepted Journal of Computational Physics)
Vikas C.
Raykar, Ramani Duraiswami, “Very
fast optimal bandwidth selection for univariate
kernel density estimation,” Department of Computer Science Technical
Report CS-TR-4774, UMIACS Technical Report UMIACS-TR-2005-73.
pdf
(a shortened and edited version appears in the Proceedings Sixth SIAM
International Conference on Data Mining, April 2006 )
Nail A. Gumerov and Ramani Duraiswami. Fast radial basis function interpolation via preconditioned Krylov iteration. accepted SIAM J. Scientific Computing, 2006. pdf matlab
Nail A. Gumerov and Ramani Duraiswami. FMM accelerated BEM for 3D Laplace & Helmholtz equations. Proceedings International Conference on Boundary Element Techniques, Paris, 2006. pdf
Ramani Duraiswami and Nail A. Gumerov, Iterative methods for use with the fast multipole method. In Abstracts of the Ninth Copper Mountain Conference on Iterative Methods, Apr 2006. Talk
Ramani Duraiswami,
Dmitry N. Zotkin, and Nail A. Gumerov. Fast evaluation of the room transfer
function using multipole expansion. IEEE Transactions on Speech and Audio
Processing, Accepted,
to appear, Jan 2007.
pdf
Nail A Gumerov and Ramani Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions (The Elsevier Electromagnetism Series) Hardcover: 426 pages, Publisher: Elsevier Science (March 15, 2005), ISBN: 0080443710
Nail A. Gumerov and Ramani Duraiswami, Computation of scattering from clusters of spheres using the fast multipole method, J. Acoust. Soc. Am.. 117 (4), Pt. 1, April 2005, pp. 1744-1761 pdf
Nail A. Gumerov and Ramani Duraiswami, "Comparison of the efficiency of translation operators used in the fast multipole method for the 3D Laplace equation," University of Maryland, Department of Computer Science Technical Report CS-TR-4701 (Institute for Advanced Computer Studies Technical Report UMIACS 2005-09) pdf
Changjiang Yang, Ramani Duraiswami and Larry S. Davis A Robust and Efficient Kernel-Based Similarity Measure with Application to Object Tracking, IEEE Conference on Computer Vision and Pattern Recognition, San Diego, 2005 pdf
Vikas C. Raykar and Ramani Duraiswami, “The improved fast Gauss
transform with applications to machine learning,”
NIPS 2005 Workshop on Large Scale Kernel Machines,
Whister, BC. Organized by
Léon Bottou,
Olivier Chapelle,
Dennis Decoste,
Jason Weston,
Slides
Fast Multipole Method Based Filtering of Nonuniformly Sampled Data. Nail A. Gumerov, Ramani Duraiswami, unpublished, pdf
Meeting:
A highly topical meeting
on the fast multipole method and related algorithms was held at the Center for
Scientific Computation and Applied Mathematical Modeling at the
Current Students:
Former Students
In addition the students in our class have done interesting projects on aspects of the FMM.
Other FMM collaborators
Courses
Ph.D. Thesis
Zhihui Tang, Fast Transforms Based on Structured Matrices with Applications to the Fast Multipole Method, 2003. pdf (Several ideas were explored for speeding up translation operators. the methods that resulted were not too promising. The main results were variants of fast algorithms for Pascal matrices)
Talks
· Fast Multipole Methods for Incompressible flow simulation, Ramani Duraiswami & Nail Gumerov, Presented at the CSCAMM Workshop on Perspectives on Incompressible Flows, Abstract, Slides
·
Acoustical scattering from N spheres using
a multilevel fast multipole method, Nail Gumerov & Ramani Duraiswami, Presented
at the Acoustical Society of America meeting in
Open Source Software
· The Improved Fast Gauss Transform. available under the LGPL from Vikas Raykar's site
Commercial Software
Some of our software is being further developed (under license from the University of Maryland) by a small business (Fantalgo, LLC).
Meetings
Support:
Work supported by NSF Awards 0086075, 0219681, and internal UMIACS funds.
We welcome collaborations and joint projects/proposals with domain computational scientists seeking to employ the FMM to speed up calculations in their fields. We are also open to enquiries for consultation on any of this work, or on the FMM in general.