Fast radial basis function interpolation via preconditioned Krylov iteration

SIAM Journal on Scientific Computing

Volumn 29, Issue 5, 2007, Pages 1876-1899

  Gumerov, Nail A ^a   Duraiswami, Ramani ^b  

^a UNIVERSITY OF MARYLAND   (United States)

^b NONE   (United States)

Author keywords

Cardinal function preconditioner; Computational geometry; Fast multipole method; Preconditioned conjugate gradient; Radial basis function interpolation

Indexed keywords

COMPUTATIONAL GEOMETRY; CONJUGATE GRADIENT METHOD; FUNCTIONS; MATRIX ALGEBRA; RADIAL BASIS FUNCTION NETWORKS;

BASE FUNCTION; CARDINAL FUNCTION PRECONDITIONER; FAST-MULTIPOLE METHODS; ITERATIVE ALGORITHM; KRYLOV SUBSPACE; MATRIX-VECTOR PRODUCTS; PRECONDITIONED CONJUGATE GRADIENT; PRECONDITIONERS; RADIAL BASIS; RADIAL BASIS FUNCTION INTERPOLATION;

INTERPOLATION;

EID: 52649096416     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060662083     Document Type: Conference Paper

References (30)

1. R. K. BEATSON AND M. J. D. POWELL, An iterative method for thin plate spline interpolation that employs approximations to Lagrange functions, in Numerical Analysis 1993, D. F. Griffiths and G. A. Watson eds., Longman Sci. Tech., Harlow, UK, 1994, pp. 17-39.
2. R. K. BEATSON AND L. GREENGARD, A short course on fast multipole methods, in Wavelets, Multilevel Methods and Elliptic PDEs, M. Ainsworth, J. Levesley, W. A. Light, and M. Marietta, eds., Oxford University Press, New York, 1997, pp. 1-37.
3. R. K. BEATSON, J. B. CHERRIE, AND C. T. MOUAT, Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration, Adv. Comput. Math., 11 (1999), pp. 253-270.
4. R. K. BEATSON, W. A. LIGHT, AND S. BILLINGS, Fast solution of the radial basis function interpolation equations: Domain decomposition methods, SIAM J. Sci. Comput., 22 (2000), pp. 1717-1740.
5. d: tools for fast evaluation, in Curve and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L. Schumaker, eds., Vanderbilt University Press, Nashville, TN, 2000, pp. 47-56.
6. R. K. BEATSON, J. B. CHERRIE, AND D. L. RAGOZIN, Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines, SIAM J. Math. Anal., 32 (2001), pp. 1272-1310.
7. J. BLOOMENTHAL, C. BAJAJ, J. BLINN, M. P. CANI-GASCUEL, A. ROCKWOOD, B. WYVILL, AND G. WYVILL, Introduction to Implicit Surfaces, Morgan Kaufmann Publishers, San Francisco, CA, 1997.
8. J. C. CARR, R. K. BEATSON, J. B. CHERRIE, T. J. MITCHELL, W. R. FRIGHT, B. C. MCCALLUM, AND T. R. EVANS, Reconstruction and representation of 3D objects with radial basis functions, ACM SIGGRAPH 2001, Los Angeles, CA, 2001, pp. 67-76.
9. H. CHENG, L. GREENGARD, AND V. ROKHLIN, A fast adaptive Multipole algorithm in three dimensions, J. Comput. Phys., 155 (1999), pp. 468-498.
10. T. H. CORMEN, C. E. LEISERSON, R. L. RIVEST, AND C. STEIN, Introduction to Algorithms, MIT Press, Cambridge, MA, 2004.
11. n, SIAM J. Sci. Comput., 23 (2002), pp. 1549-1571.
12. W. D. ELLIOTT AND J. A. BOARD, Fast Fourier transform accelerated fast multipole algorithm, SIAM J. Sci. Comput., 17 (1996), pp. 398-415.
13. A. C. FAUL, G. GOODSELL, AND M. J. D. POWELL, A Krylov subspace algorithm for multiquadric interpolation in many dimensions, IMA J. Numer. Anal., 25 (2005), pp. 1-24.
14. L. GREENGARD AND V. ROKHLIN, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325-348.
15. L. GREENGARD, The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, MA, 1988.
16. L. GREENGARD AND V. ROKHLIN, On the Efficient Implementation of the Fast Multipole Algorithm, Technical report RR-602, Department of Computer Science, Yale University, New Haven, CT, 1988.
17. L. GREENGARD AND J. STRAIN, The fast Gauss transform, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 79-94.
18. G. GOODSELL, On finding p-th nearest neighbours of scattered points in two dimensions for small p, Comput. Aided Geom. Design, 17 (2000), pp. 387-392.
19. N. A. GUMEROV AND R. DURAISWAMI, Fast multipole method for the biharmonic equation in three dimensions, J. Comput. Phys., 215 (2005), pp. 363-383.
20. N. A. GUMEROV AND R. DURAISWAMI, Comparison of the Efficiency of Translation Operators used in the Fast Multipole Method for the 3D Laplace Equation, Technical report CSTR-4701, Department of Computer Science, University of Maryland, College Park, MD, 2005.
21. N. A. GUMEROV AND R. DURAISWAMI, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, Oxford, UK, 2005.
22. N. A. GUMEROV, R. DURAISWAMI, AND E. A. BOROVIKOV, Data Structures, Optimal Choice of Parameters, and Complexity Results for Generalized Multilevel Fast Multipole Methods in d Dimensions, Technical report CS-TR-4458, Department of Computer Science, University of Maryland, College Park, MD, 2003.
23. R. L. HARDY, Theory and applications of the multiquadric-biharmonic method, 20 years of discovery 1968-1988, Comput. Math. Appl., 19 (1990), pp. 163-208.
24. C. A. MICCHELLI, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), pp. 11-22.
25. S. MANEEWONGVATANA AND D. M. MOUNT, Analysis of approximate nearest neighbor searching with clustered point sets, in Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implementation Challenges, M. H. Goldwasser, D. S. Johnson, C. C. McGeoch, eds., DIMACS Ser. Discrete Math. Theoret. Comp. Sci. 59, AMS, Providence, RI, 2002, pp. 105-123.
26. V. C. RAYKAR, C. YANG. R. DURAISWAMI, AND N. A. GUMEROV, Fast Computation of Sums of Gaussians in High Dimensions, Technical report CS-TR-4767, Department of Computer Science, University of Maryland, College Park, MD, 2005.
27. H. SAMET, The Design and Analysis of Spatial Data Structures, Addison-Wesley, New York, 1989.
28. V. SIMONCINI AND D. B. SZYLD, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., 25 (2003), pp. 454-477.
29. G. TURK AND J. F. O'BRIEN, Modelling with implicit surfaces that interpolate, ACM Trans. Graph., 21 (2002), pp. 855-873.
30. E. W. WEISSTEIN, Kissing number, in MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/KissingNumber.html.