Fast summation of Helmholtz monopoles in 2-D

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Fast summation of Helmholtz monopoles in 2-D

• Term Project of AMSC 698R, Fall 2003
• Weigang Zhong
• Department of Mathematics
• University of Maryland

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Outline

• Helmholtz Equation
• Solution of Helmholtz Equation
• Properties of Hankel function and Bessel function
• Implementation of MLFMM
• Data Structure
• R S Expansion
• SS, SR, RR Reexpansion
• Results of MLFMM

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Problem Description

• Consider the wave equation with
  speed c.
• with angular frequency
• We find that satisfies the Helmholtz
  equation
• where is the wave number.
• The solutions of the Helmholtz equation represent
  solutions of the
• wave equation.
•  

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2-D Helmholtz Equation
I will consider the Helmholtz equation that is
the simplest eigenvalue equation in two
dimensions. It can be solved relative to many
coordinate systems. I will discuss it in the
polar coordinate system.
of a unit charge
We can define the field
Located at the point
by the formula
Where denotes the Hankel function of order
zero.
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2-D Helmholtz Equation cond

• Fundamental solution
• (charge, monopole, source, free field Greens
  function)

Which satisfies
For the 2D Helmholtz Equation, the field is
generated by a set of N monopoles.
Where we can apply MLFMM
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Objective

• In case we want to generate a field with more
  than 10,000
• Nodes. The direct summation requires O(N2)
  operations,Where
• N is the total number of nodes in the domain. It
  is technically
• Impractical.
• By using MLFMM, we can dramatically reduce the
  cost, roughly
• Saying, O(NlogN).

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Translation operators for 2-D Helmholtz

• Understanding the translational addition theorem
  is
• imperative for understanding the fast multipole
  method.
• Addition theorem for cylinder harmonics

• Translations represented by cylinder harmonics

Here,
are the angles that
make with the
x-axis, respectively.
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Plot Hankel function
Hankel function is singular at (0,0). H0 (0)NaN
NaNi
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Plot Bessel function of the first order
Bessel function is regular in this domain. J0
(0)1
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S-Expansion
For the summation in my problem
S-expansion of mother function is
where
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SS Reexpansion
The SS translation operator is
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SR Reexpansion
The SR translation operator is
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RR Reexpansion
The RR translation operator is
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Implementation of MLFMM

• 1000 source points
• 999 target points

• Set up data structure
• Determine the domain
• Map the domain to the unit square
• Index all the boxes, build box hierarchies for
  the source target points separately
• S Expansion
• SS, SR, RR Reexpansion

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Implementation of MLFMM cond

• Upward Pass
• Use the source hierarchy to compute the
  coefficients of S and SS expansions in the
  upward pass.
• Downward Pass
• Use the source hierarchy and the target hierarchy
  to compute the coefficients of SR expansions.
  Use the target hierarchy to compute the
  coefficients of RR expansions in the downward
  pass.
• Final summation

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Truncation Number vs. Error (N1000, Cluster
number s73)
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Cluster number vs. error (N1000, truncation num
P10)
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Cluster number vs. Cputime (N1000, truncation
num P10)
Cpu time of the straightforward method is
135.74102.13, optimal cluster number is 72
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CPU time Comparison (N1000, truncation num P10,
cluster number s73)
Magnified Figure for N100800
Max absolute errors lt 10-6
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Future work

• Adaptive MLFMM
• Irregular domain
• 3-D case

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Reference

• 1. Rapid Solution of Integral Equations of
  Scattering Theory in Two Dimensions, V. Rokhlin,
  Department of Computer Science, Yale University,
  New Haven, Connecticut, March 22 1989
• 2. LINEAR MATHEMATICS IN INFINITE DIMENSIONS
  Signals Boundary Value problems and Special
  Functions, U. H. Gerlach , Department of
  Mathematics, Ohio State University, 08/15/2003
• 3. Part III Partial differential equations,
  Nils Andersson, Department of Mathematics,
  University of Southampton, 10/02/2001
• 4. Plane Waves Solution for the 2D-Helmholtz
  Short Wave Equation, O. Laghrouche and P.
  Bettess, School of Engineering, University of
  Durham, Durham DH1 3LE, England
• 5. The multipole expansion, http//farside.ph.ut
  exas.edu/teaching/jk1/lectures/, Richard
  Fitzpatrick, University of Texas at Austin, Fall
  1996
• 6. Computation of Scattering from N spheres
  using multipole reexpansion, Nail A. Gumerov and
  Ramani Duraiswami, Institute for Advanced
  Computer Studies, University of Maryland at
  College Park, April 2002
• 7. Acoustical Scattering from N Spheres Using a
  Multilevel Fast Multipole Method, Nail A. Gumerov
  and Ramani Duraiswami, Institute for Advanced
  Computer Studies, University of Maryland at
  College Park

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• Thank You !