1. The Fast Multipole Method: It’s All about Adding Functions
Alan Edelman
MIT: Dept of Mathematics,
Lab for Computer Science
FOCM 2002
Saturday, August 10
11/15/2018

2. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

3. The matrix-vector viewpoint:
f =V(y, x) q
potentials at y1 , ..., yN
charges at x1 , ..., xN
Vij = r(yi - xj) qj
||yi-xj||
fi=S
Example: j
Direct evaluation: O(N2) work
Greengard, Rokhlin FMM (1987): (N)
11/15/2018

4. Some applications: N-body Problems Potential Evaluation
Future Fast Fourier Transform
Divide and Conquer Eigenvalue Solvers
Polynomial Roots
More waiting to be found ….
11/15/2018

5. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

6. It’s all about adding functions:
fi=S qj
||yi-xj|| j
f(y)=S fj(y) j
fj(y)= qj
||y-xj||
11/15/2018

7. What might these functions look like?
f(y)=S fj(y) j
f1(y)
f2(y)
11/15/2018

8. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing Functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

9. The Exclusive Add Why not sum and subtract each element? sum(x)-x
y = x
yi = ji xj
Why not sum and subtract each element?
sum(x)-x
What if NaN’s or numbers on different scales are present?
11/15/2018

10. The Exclusive Add A Good Algorithm is worth seeing five ways!
y = x
yi = ji xj
A Good Algorithm is worth seeing five ways!
1. Pseudocode
2. MATLAB
3. On a tree
4. Matrix Notation
5. Kronecker Product Notation
11/15/2018

11. Exclusive Add (pseudocode)
yi = ji xj
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
11/15/2018

12. Exclusive Add in MATLAB
yi = ji xj
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
function y=exclude(v)
n=length(v);
if n==1, y=0*v; else
w=v(1:2:n)+v(2:2:n); % Pairwise adds
w=exclude(w); % Recur
y(1:2:n)=w+v(2:2:n); y(2:2:n)=w+v(1:2:n); % Update Adds
end
11/15/2018

13. Exclusive Add on a Tree
yi = ji xj 36
Pairwise sums “up the tree”
Modify evens/odds down
11/15/2018

14. With Matrices T + =
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15. Kronecker product Approach + =
A =(I4( )) A4 (I4( ))T + I4  ( ) 11 11
0 1
1 0
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
11/15/2018

16. Kronecker product Approach + =
A =(I4( )) A4 (I4( ))T + I4  ( ) 11 11
0 1
1 0
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
11/15/2018

17. Kronecker product Approach + =
A =(I4( )) A4 (I4( ))T + I4  ( ) 11 11
0 1
1 0
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
11/15/2018

18. Kronecker product Approach + =
A =(I4( )) A4 (I4( ))T + I4  ( ) 11 11
0 1
1 0
1)Pairwise Sums
2)Recursive Excl Add
3)Update “evens & odds”
11/15/2018

19. The Family Directions Left Right Left/Right Inclusive Exc=0 Prefix
All Algorithms
1)Pairwise Sums
2)Recursive “foo”
3)Update
Directions
Left
Right
Left/Right
Inclusive
Exc=0
Prefix
Suffix
Reduce
Exclusive
Exc=1
Exc Prefix
Exc Suffix
Exc Add
Neighbor Exc
Exc=2
Left Multipole
Right " " "
Multipole
11/15/2018

20. Multipole is a doubly exclusive add
We add functions with poles not numbers
We exclude ourselves and our nearest neighbors
For reasons analogous to the floating point example:
The function can not be represented accurately near the singularities.
This is 1d, higher dimensions analogous
Algorithm: Pairwise Add, Recursion, Update Missing Pieces
11/15/2018

21. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing Functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

22. “Rounding” functions to p coefficients
Best you can do:
1)Pick an interval
2)Pick a function space inside or outside the interval.
3)Find the best approximation on that function space.
11/15/2018

23. “Rounding” functions to p coefficients
Ideal polynomial Approximation
Truncated Taylor Series
f(x) =  ak(x-c)k
Theory: Converges in a disk from c to nearest singularity
Practical: Accurate where smooth, far from the singularity
Alternative Polynomial: Interpolate rather than Taylor
11/15/2018

24. “Rounding” functions to p coefficients
Ideal Multipole Approximation
Truncated Multipole Series
f(x) =  ak/(x-c)k+1
Theory: Converges outside a disk from c to nearest singularity
Practical: Accurate where smooth, far from the singularity
Alternative Multipole: Interpolate
11/15/2018

25. “Rounding” functions to p coefficients
Examples
R multipole/Taylor Gu
R Chebyshev interpolation Dutt, Gu, Rokhlin
S1 Chebyshev interpolation Dutt, Rokhlin
R singular funs of int ops Yarvin, Rokhlin
C multipole/Taylor Greengard, Rokhlin
C discretized Poisson form Anderson
C singular funs of int ops Hrycak, Rokhlin
R3 multipole/Taylor Greengard
R3 discretized Poisson Anderson
R3 singular funs of int ops Greengard, Rokhlin
Any Virtual Charges E, McCorquodale
11/15/2018

26. Finite Precision Arithmetic
Idea adding real numbers
We all know what this means
x=1+1
x=e+
On a computer:
Storage: must round to d-bit precision
Arithmetic: need a finite precision algorithm
11/15/2018

27. Finite Precision Arithmetic
Idea adding functions
We all know what this means
f(x)=sin2(x)+cos2(x) = 1
f(x)=log(x)+log(x) = log(x2)
On a computer:
Storage: must round to d-bit precision
Arithmetic: need a finite precision algorithm
11/15/2018

28. Functions to add Slowly growing singularity f(x)=q/(x-c)
f(x)=q*log(x-c)
f(x)=q*cot(x-c) but not f(x)=exp(-(x-c)2)
11/15/2018

29. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing Functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

30. Adding Functions Issues with adding polynomials common center accuracy
Same issues appear with multipole
11/15/2018

31. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
Gil Strang’s latest favorite matrix
i+j i
11/15/2018

32. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

33. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

34. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

35. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

36. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

37. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

38. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

39. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

40. The Pascal Matrix P=pascal(5) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15
Pij=( ) i,j = 0,…,n-1
i+j i
11/15/2018

41. Pascal and Cholesky(Pascal)
P=pascal(5)
Pij=( ) i,j = 0,…,n-1
L=chol(pascal(5))
Lij=( ) i,j = 0,…,n-1 i j
i+j i
P=LL’ ( )=  ( )( )
i+j j
i k
j j-k
11/15/2018

42. Binomial Identities (1-x)-(j+1) = i=0 ( ) xi  P
i+j i 
(1+x) j = i=0 ( ) xi  L’ j
j i
(x-1)-(j+1) = i=j ( ) x-(i+1)  L 
i j
11/15/2018

43. “Flipping” (multipoletaylor)
w = F v
 wi (x-d)i =  vj / (x-c) j+1
i=0
j=0
Fij = (c-d)-i Pij (d-c)-(j+1)
Proof: Differentiate i times then evaluate at x=d
or cleverly use (1-x)-(j+1) = ( )xi
i+j i
11/15/2018

44. “Shifting” (multipolemultipole)
w = S v
 wi /(x-d) i+1 =  vj / (x-c) j+1
i=0
j=0
Sij = (c-d) (i+1) Lij (c-d)-(j+1)
11/15/2018

45. “Shifting” (taylortaylor)
w = S v
 wi (x-d)i =  vj(x-c)j
i=0
j=0 T
Sij = (d-c)-i Lij (d-c) j
11/15/2018

46. Outline Multipole: What are we adding?
The exclusive add without the multipole
Representing Functions on a computer
“Adding” functions and the Pascal Matrix
Research Opportunities
11/15/2018

47. Group Representations
Let V be a vector space of functions of x
e.g. polynomials, rational functions, etc.
Let G be a group acting on {x},
e.g. rotations, non-singular matrices.
Clearly the map  from f(x) to h(x)=f(g-1x) is linear.
Clearly (gh)= (g) (h).
We say that  is a representation of G.
11/15/2018

48. Group Representations Research
Generalize Fast Multipole to Arbitrary Representations!
Algebraic Multipole Theory???
11/15/2018