1. Fast Multipole: It's All About Adding Functions in Finite Precision
Alan Edelman
Per-Olof Persson
MIT: Dept of Mathematics,
Computer Science Artificial Intelligence Laboratory
University of Maryland, College Park
April 27, 2004
11/10/2018

2. Exclusive Sum Problem Given vector x compute “exclusive sum”:
yi = ∑j≠i xj
Example: x=[ ]
y=[ ]
Students quickly think of “sum(x)-x”
Now impose NO SUBTRACTION rule
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3. Tree algorithm for exclusive sum10 26 3 7 11 15
Tree algorithm for exclusive sum 1 2 36 3 4 5 6
Up the tree:
Pairwise Adds
Down the tree:
Sibling Swap + Parent 7 8
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4. Parallel Prefix Like Algorithm10 26 3 7 11 15
Parallel Prefix Like Algorithm 1 2 36 3 4 5 6
Up the tree:
Pairwise Adds
Down the tree:
Sibling Swap + Parent 7 8
Adds may be done in parallel
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5. Parallel Exclusive Prefix Algorithm10 26 3 7 11 15
Parallel Exclusive Prefix Algorithm 1 2 36 3 4 5 6
Down the tree:
if right : add left
if left : add 0
+ Parent
Up the tree:
Pairwise Adds 7 8
Adds may be done in parallel
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6. The prefix operator yk=∑i=1xi
“Parallel Prefix” goes back to 1800’s: Babbage, carry, look-ahead addition
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7. Simple algorithm for exclusive sum
Compute directly:
exclusive_prefix_sum=[ ]
exclusive_suffix_sum=[ ]
Vector sum: y=[ ]
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8. Comparison Tree Direct Complexity 3n 2n Simplicity involved simple
Typical Venue
Parallel
Serial
FMM Connection
FMM like
Why not???
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9. The Algorithm Menu Direction of Summation Data Structure Prefix 
Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
Topology
Line (1d)
Plane (2d)
Space (3d)
Circle, sphere, manifolds, …
Data Structure
Direct
Hierarchical
Function Representation
Exact
Multipole
Interpolation
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10. The Algorithm Menu Direction of Summation Data Structure Prefix 
Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
Topology
Line (1d)
Plane (2d)
Space (3d)
Circle, sphere, manifolds, …
Data Structure
Direct
Hierarchical
Function Representation
Exact
Multipole
Interpolation
2d FMM: 1c+2c+3b+4b+5b
ParPrefix: 1a+2a+3a+4b+5a
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11. The Algorithm Menu Direction of Summation Prefix  Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
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12. The Algorithm Menu Direction of Summation Prefix  Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
Pseudocode:
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13. The Algorithm Menu Direction of Summation Prefix  Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
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14. The Algorithm Menu Direction of Summation Prefix  Suffix 
Bidirectional 
Exclusions
Inclusive
Exclusive
Neighborhood Exclusive
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15. Algorithms as Pictures
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16. The Algorithm Menu Direction of Summation Data Structure Exclusions
Topology
Data Structure
Function Representation
Discrete algorithms cover 1,2,3 and 4
Item 5 needs
Functions
Discrete Representation of Functions
Addition on this representation
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17. Exclude (Numbers or functions)
MATLAB: Tree algorithm expressed recursively
>> exclude([ ])
ans =
>> exclude([1/(t-1) 1/(t-2) 1/(t-3) 1/(t-4)])
ans = [1/(t-2)+1/(t-3)+1/(t-4),
1/(t-1)+1/(t-3)+1/(t-4),
1/(t-4)+1/(t-1)+1/(t-2),
1/(t-3)+1/(t-1)+1/(t-2)]
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18. Reals:Floats::Functions:Discretized math vs. computation
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19. Function Representation: The analog of rounding
Polynomials:
f(x)=∑0 ≤ k <p ak(x-c)k
Truncated Multipole Series:
f(x)=∑1 ≤ k ≤ p ak/(x-c)k
Virtual Charges:
f(x)=∑1 ≤ k ≤ p ak/(x-xk)
Interpolate
Number of points
Locations of points
Approximate
L2 (least squares)
L∞ (“Remez”)
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20. Function Representation: The analog of rounding
f(x)=1/(x-0.75)
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21. Function Approx:Multipole vs taylor
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22. Virtual Charges f(x)=∑1 ≤ k ≤ p ak/(x-xk)
Prechoose xk: Then interpolate, or least squares, or maybe Remez
Compute xk: Many choices
New idea(?): Lanczos
Eigenvalues not linear, but
Tridiagonal eigenvalue problem is solid as a rock
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23. 11/10/2018

24. Lanczos appears to be more accurate than multipole closer to singularities
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25. Lanczos Virtual Charges
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26. Non-tree algorithms …
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27. Non-tree in 2d: cumulate sums
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28. Non-tree in 3d: cumulate sums
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29. Non-tree in 3d: cumulate sums
Any number of dimensions ok
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30. Direct FMM 1d Experiments
Our Experiments are preliminary:
We would be delighted if others would proceed more extensively.
Comments:
Need accuracy near the boundary
Virtual charges worked better
One representation not two (multipole + taylor)
More charges needed as we proceeded to maintain accuracy (other choices?)
No power of two restriction
Easy to implement (only func rep needed)
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31. Pascal’s Triangle Flip Shift Taylor Shift Multipole
Representations of Mobius functions but with a catch!
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32. General Orthogonal Polynomials
Pascal’s Triangle
Flip
Shift Taylor
Shift Multipole
Representations of Mobius functions but with a catch!
General Orthogonal Polynomials
mi(x) = ∑j=0,∞ mi+jxj  H
πi(x) = ∑j=0,i < πi,xj> xj  LT
fi(x) = ∑j=i, ∞ <xi, πj>x-j+1  L
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33. Conclusion Formulation of FMM Direct (non-tree) algorithm
Higher dimensions
Lanczos Virtual Charges
Orthogonal Polynomials
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