Fast Methods for Integral Equations - MIT OpenCourseWare ... · Fast Methods for Integral Equations...
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Introduction to Simulation - Lecture 23 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Fast Methods for Integral Equations Jacob White
Transcript of Fast Methods for Integral Equations - MIT OpenCourseWare ... · Fast Methods for Integral Equations...
Introduction to Simulation - Lecture 23
Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Fast Methods for Integral Equations
Jacob White
Outline
Solving Discretized Integral EquationsUsing Krylov Subspace MethodsFast Matrix-Vector Products
Multipole Algorithms Multipole Representation.Basic Hierarchy
Algorithmic ImprovementsLocal ExpansionsAdaptive Algorithms
Computational Results
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v+-2 0 Outside∇ Ψ =
is given on SurfaceΨ
potential
Exterior Problem in Electrostatics
First Kind Integral Equation For Charge:
“Dirichelet Problem”
potential
( ) ( ){
arg
1
'surface
xx xGreen
xCh
sFunction
eD
d
y
S
ensit
σ ′−
′′
Ψ = ∫14243
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Resonator Discretized Structure
Computed ForcesBottom View
Computed ForcesTop View
Drag Force in a Microresonator
Courtesy of Werner Hemmert, Ph.D. Used with permission.
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Basis Function Approach
Piecewise Constant Basis
3-D Laplace’sEquation
Integral Equation:
( ) 1 if is on panel jj x xϕ =( ) 0 otherwisej xϕ =
Discretize Surface into Panels
Panel j
( ) ( )1
surface
x dx x
x Sσ′
′ ′−
Ψ = ∫
( ) ( ){1 Basis Functions
Represent n
i ii
x xσ α ϕ=
≈∑
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3-D Laplace’sEquation
Basis Function Approach
Centroid Collocation
( ) ( )1
,
,i i
n
c j cj
i j
panel jx G x x dS
A
α=
′ ′Ψ =∑ ∫144424443
( )
( )
11,1 1, 1
,1 ,n
cn
n n n n c
xA A
A A x
α
α
Ψ = Ψ
L L
M O M MM
M O M MM
L L
Put collocation points atpanel centroids
icx Collocation
point
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3-D Laplace’sEquation
Basis Function Approach
Calculating Matrix Elements
Panel j
icx Collocation
point
,1
i
i jpa cnel j x x
A dS ′′−
= ∫
,
i jc centrj
idi
o
Panel Areax x
A−
≈One point quadrature
Approximation
xy
z
t
4
,1 in
0.25*
i jc oi j
j p
Ar ax x
A e= −
≈∑Four point quadrature
Approximation
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3-D Laplace’sEquation
Basis Function Approach
Calculating “Self-Term”
Panel i
icx Collocation
point
,1
i
i ipa cnel i x x
A dS ′′−
= ∫
,
0i i
i ic c
Panel AreaAx x−
≈
14243
One point quadrature
Approximation
xy
z
, is an integrable singularity1
i
i ipanel i cx x
A dS′−
′= ∫
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3-D Laplace’sEquation
Basis Function Approach Calculating “Self-Term”
Tricks of the trade
Panel i
icx Collocation
point
,1
i
i ipa cnel i x x
A dS ′′−
= ∫xy
z
Disk of radius R surrounding
collocation point
,1 1
i ic ci i
disk rest of panel
A dS dSx x x x′ ′− −
′ ′= +∫ ∫
Disk Integral has singularity but has analytic formula
Integrate in two pieces
2
0 0
1 21
i
R
d k cis
dS rdrd Rrx x
π
θ π′
′ = =−∫ ∫ ∫
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3-D Laplace’sEquation
Basis Function Approach Calculating “Self-Term”Other Tricks of the trade
Panel i
icx Collocation
point
Integrand is singular
,1
i
i ipanel i cx x
A dS′
′=−∫
14243xy
z
2) Curve panels can be handled with projection
1) If panel is a flat polygon, analytical formulas exist
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3-D Laplace’sEquation
Basis Function Approach
Galerkin (test=basis)
1,1 1, 1 1
,1 ,
n
n n n n n
A A b
A A b
α
α
=
L L
M O M M M
M O M M M
L L
( )1
,
1n
j panel i panel jj
ii j
x dS dS dSx x
b A
α=
′ ′Ψ =′−∑∫ ∫ ∫14243
14444244443
For piecewise constant Basis
( ) ( ) ( ) ( ) ( )1
,
,n
i j i jj
i ji
x x dS x G x x x dS dS
Ab
ϕ α ϕ ϕ=
′ ′ ′Ψ = ∑∫ ∫ ∫1442443 1444442444443
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3-D Laplace’sEquation
Basis Function Approach
Problem with dense matrix
( )
( )
11,1 1, 1
,1 ,n
cn
n n n n c
xA A
A A x
α
α
Ψ = Ψ
L L
M O M MM
M O M MM
L L
Integral Equation Method Generate Huge Dense Matrices
Gaussian Elimination Much Too Slow!
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( ) ( )( ) ( )
11
10
Tkkjk
k jTj j j
Ar App r p
Ap Ap
++
+=
= −∑
( ) ( )( ) ( )
Tkk
k Tk k
r Ap
Ap Apα =
1k kk kx x pα+ = +
1k kk kr r Apα+ = −
Determine optimal stepsize in kth search direction
Update the solution and the residual
Compute the new orthogonalizedsearch direction
compute kAp For discretized Integral equations, A is dense
The kth step of GCR
The Generalized Conjugate Residual AlgorithmSolving Discretized
Integral Equations
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( ) ( )( ) ( )
11
10
Tkkjk
k jTj j j
Ar App r p
Ap Ap
++
+=
= −∑
( ) ( )( ) ( )
Tkk
k Tk k
r Ap
Ap Apα =
1k kk kx x pα+ = +
1k kk kr r Apα+ = −
Vector inner products, O(n)
Vector Adds, O(n)
O(k) inner products, total cost O(nk)
Algorithm is O(n2) for Integral Equations even though # iters (k) is small!
compute kAp Dense Matrix-vector product costs O(n2)
Complexity of GCR
The Generalized Conjugate Residual AlgorithmSolving Discretized
Integral Equations
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exactly compute kApDense Matrix-vector product costs O(n2)
Fast Matrix Vector Products
The Generalized Conjugate Residual AlgorithmSolving Discretized
Integral Equations
approximately compute kApReduces Matrix-vector product costs to
O(n) or O(nlogn)
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SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
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SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
Summary
Solving Discretized Integral EquationsGCR plus Fast Matrix-Vector Products
Multipole Algorithms Multipole Representation.Basic Hierarchy
Algorithmic ImprovementsLocal ExpansionsAdaptive Algorithms
Computational ResultsPrecorrected-FFT Algorithms
