1. Fast RBF Algorithms

2. Main tasks performed Solving the linear system
Evaluating the summation

3. Evaluating the coefficients
Possible to obtain 2 sets of linear systems in λ and γ.
Cost of solving : O(n3)
Need for fast algorithm

4. Algo 1: The Krylov Subspace based algorithm (Beatson-Faul-Goodsell-Powell algorithm)
S : space of functions satisfying interpolation conditions
S(k) : Krylov subspace of S formed on kth iteration
Inner product : where s and t belong to S and t is given by

5. Krylov Subspace properties
Norm defined accordingly
Property of norm : Norm of error decreases with each iteration -> Algo converges.

6. Algorithm Initial guess s0 employed
Method of progress analogous to conjugate gradient method using linear operator A
where βk is defined to make dk’s orthogonal
αk chosen to minimize the square of the norm of the error.

7. The A operator

8. Preliminaries of implementation, Results
Choose Lj , the set of q indices xl which minimizes ||xj - xl ||2
Choose q, calculate λjl.
Computational complexity : O(n2) or O(n log(n)) (for multipole summation)

9. Algo 2: Multipole methods
(Beatson and Greengard)
Objective : To evaluate the summation s(x) assuming known coefficients

10. Multiquadric RBF in 1D
Considering simple multiquadric RBF in 1D given by
Can be written as Laurent series expansion about panel center t.

11. Laurent series expansion

12. Far-field expansion
Truncation of Laurent series expansion for large values of |x - t|, leading to quick evaluation.
Far-field expansion chosen according to panels

13. Breaking domain into panels
Domain can be broken into binary panels
s(x) can be evaluated in the panel [0,1/8] (for example as

14. Discrete Gauss Transform
Can also be used instead of Laurent series for Far-field expansions

15. Bottomline
Paneling and far-field expansions used for coarsening the effect of faraway points
Computational complexity goes from O(N) to O(mp), where m is the bottom panel number and p is the summation truncation.

16. Algo 3: Tree-based algorithms
(Deng & Driscoll)
Advanced variant of multipole methods
Aim : Evaluate the summation s(x)

17. Tree Consider a domain in Rd. Nodes of tree : cell in Rd.
Branch of tree : made by dividing the domain in each dimension by 2
Number of children of each cell : 2d

18. Taylor expansion Taylor expansion of RBF (and hence s(x)) in cell C :
Ti : Set of cells C which interact with cell containing xi

19. Tree algorithm s(xi) is evaluated as
where H(xi) is the hierarchy of cells along the tree containing xi.
sC can be evaluated using a Taylor expansion as in previous slide
Multipole Acceptance Criterion : used to test cell for far-field expansion

20. Bottomline
Computational complexity of crude method : O(MN)*order of RBF evaluation
Computational complexity with tree algo : O((M+N)log(M))

21. THANK YOU

22. References
R. K. Beatson and L. Greengard, A short course on fast multipole methods, inWavelets,Multilevel Methods and Elliptic PDEs (M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, eds.), Oxford University Press, Oxford, UK, 1997, pp. 1–37.
Q.Deng and T. A. Driscoll, A fast treecode for multiquadric interpolation with varying shape parameters, SIAM J. Sci. Comput. 34 (2012), A1126–A1140.
G. Roussos and B. J. C. Baxter, Rapid evaluation of radial basis functions, J. Comput. Appl. Math. 180 (2005), 51–70
A. C. Faul andM. J.D. Powell, Krylov subspacemethods for radial basis function interpolation, in Proceedings of the International Conference on Numerical Analysis (Dundee), August 1999.
A. C. Faul, G. Goodsell, and M. J. D. Powell, A Krylov subspace alogrithm for multiquadric interpolation in many dimensions, IMAJ.Numer.Anal. 25 (2005), 1–24.