A fast analysis-based discrete Hankel
transform using asymptotic expansions
Alex Townsend
MIT
IMA Leslie Fox Prize Meeting, 22nd June, 2015
Based on: T., “A fast analysis-based discrete Hankel transform using asymptotic expansions”, to appear in SINUM.
Introduction
The fast Fourier transform
The FFT computes the DFT in OpN log N q operations:
ř
Xk “ nN“´01 xn ¨ e ´2πink {N ,
0 ď k ď N ´ 1.
sinp
´2π
t
q
Imag axis
e
´2πit
Top 10
algorithm
cos
p´2
James Cooley
t
Alex Townsend @ MIT
πt q
John Tukey
xis
al a
Re
1/26
Introduction
A circularly symmetric 2D Fourier transform
Let f px, y q be a circularly symmetric function. Then, the finite Hankel transform is
"
ż1
1
´i pω1 x `ω2 y q
H0 pωq “
f px, y qe
dx dy “
f pr qJ0 prωqr dr,
ω2 “ ω21 `ω22
2π
2
2
0
x `y ď1
Discrete analogue: H0 pj0,k q «
Alex Townsend @ MIT
1
2
j0,N
`1
N
ÿ
2
f pj0,n {j0,N `1 qJ0 pj0,k j0,n {j0,N `1 q.
2
J
p
j
q
0,n
n “1 1
2/26
Introduction
A circularly symmetric 2D Fourier transform
Let f px, y q be a circularly symmetric function. Then, the finite Hankel transform is
"
ż1
1
´i pω1 x `ω2 y q
H0 pωq “
f px, y qe
dx dy “
f pr qJ0 prωqr dr,
ω2 “ ω21 `ω22
2π
2
2
0
x `y ď1
Discrete analogue: H0 pj0,k q «
1
1
2
j0,N
`1
N
ÿ
2
f pj0,n {j0,N `1 qJ0 pj0,k j0,n {j0,N `1 q.
2
J
p
j
q
0,n
n “1 1
j0,k “ Bessel roots
J0 pz q “
40
´1
Alex Townsend @ MIT
Bessel function
H. Fisk Johnson
2/26
Introduction
A circularly symmetric 2D Fourier transform
Let f px, y q be a circularly symmetric function. Then, the finite Hankel transform is
"
ż1
1
´i pω1 x `ω2 y q
H0 pωq “
f px, y qe
dx dy “
f pr qJ0 prωqr dr,
ω2 “ ω21 `ω22
2π
0
x 2 `y 2 ď1
Discrete analogue: H0 pj0,k q «
1
2
j0,N
`1
N
ÿ
2
f pj0,n {j0,N `1 qJ0 pj0,k j0,n {j0,N `1 q.
2
J
p
j
q
0,n
n “1 1
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
ř
Xk “ N
1 ď k ď N.
n“1 xn J0 p j0,k j0,n {j0,N `1 q,
H. Fisk Johnson
Alex Townsend @ MIT
2/26
Introduction
A circularly symmetric 2D Fourier transform
Let f px, y q be a circularly symmetric function. Then, the finite Hankel transform is
"
ż1
1
´i pω1 x `ω2 y q
H0 pωq “
f px, y qe
dx dy “
f pr qJ0 prωqr dr,
ω2 “ ω21 `ω22
2π
0
x 2 `y 2 ď1
Discrete analogue: H0 pj0,k q «
1
2
j0,N
`1
N
ÿ
2
f pj0,n {j0,N `1 qJ0 pj0,k j0,n {j0,N `1 q.
2
J
p
j
q
0,n
n “1 1
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
Independently rediscovered by [Yu et al., 89] and [Guizar-Sicairos & Gutiérrez-Vega, 04].
Alex Townsend @ MIT
H. Fisk Johnson
2/26
Introduction
A circularly symmetric 2D Fourier transform
Let f px, y q be a circularly symmetric function. Then, the finite Hankel transform is
"
ż1
1
´i pω1 x `ω2 y q
H0 pωq “
f px, y qe
dx dy “
f pr qJ0 prωqr dr,
ω2 “ ω21 `ω22
2π
0
x 2 `y 2 ď1
Discrete analogue: H0 pj0,k q «
1
2
j0,N
`1
N
ÿ
2
f pj0,n {j0,N `1 qJ0 pj0,k j0,n {j0,N `1 q.
2
J
p
j
q
0,n
n “1 1
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
Independently rediscovered by [Yu et al., 89] and [Guizar-Sicairos & Gutiérrez-Vega, 04].
Alex Townsend @ MIT
H. Fisk Johnson
2/26
Introduction
Many applications
Optics, electromagnetics, imaging, and the numerical solution of PDEs:
[Siegman, 77], [Oppenheim, 78], [Nachamkin & Maggiore, 80], [Candel, 81],
[Gopalan & Chen, 83], [Johnson, 86], [Johnson, 87], [Higgins & Munson, 88],
[Yu et al., 89], [Vittorio et al., 92], [Sharafeddin et al., 92], [Agnesi et al., 93],
[Zhao & Halling, 93], [Cree & Bones, 94], [Barakat, 96], [Ferrari, 99], [Secada,
99], [Wieder, 99], [Knockaert, 00], [Zhang et al., 02], [Markham & Conchello,
03], [Perciante & Ferrari, 04], [Guizar-Sicairos & Gutiérrez-Vega, 04], [Cerjan,
07], [Poularikas, 10]
Compton camera:
Hankel transform is useful for image reconstruction [Cree & Bones, 94].
Used in nuclear medical imaging
and radioactive waste detection.
Alex Townsend @ MIT
3/26
Introduction
Two existing methods
1. “Direct” approach:
Naively computes
Butterfly scheme (state-of-the-art)
t(!)
os
ec
|
in OpN 2 q operations [Amos, 86].
2. Butterfly scheme:
Online cost “ OpN log N q
Offline cost “ OpN 2 q
[O’Neil, Woolfe, & Rokhlin, 10].
Alex Townsend @ MIT
in
Offl
Execution time
X “ J0 p j 0 j 0 {j0,N `1 qx
ect
cost
Dir
e
n
i
l
On
103
105
N
Timings from [O’Neil, Woolfe, & Rokhlin, 10].
4/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1. Kernel. J0 pz q does not have constant
amplitude.
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1. Kernel. J0 pz q does not have constant
amplitude.
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1. Kernel. J0 pz q does not have constant
amplitude.
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1. Kernel. J0 pz q does not have constant
amplitude.
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1
1. Kernel. J0 pz q does not have constant
amplitude.
?
zJ0 pz q
40
´1
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1
1. Kernel. J0 pz q does not have constant
amplitude.
?
zJ0 pz q
40
´1
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
0
40
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Three challenging differences between the DFT and DHT
The discrete Hankel transform of order 0 (DHT) [Johnson, 86]:
|
X “ J0 p j 0 j 0 {j0,N `1 q x.
1
1. Kernel. J0 pz q does not have constant
amplitude.
?
zJ0 pz q
40
´1
2. Frequencies. j0,1 , . . . , j0,N are not equispaced.
0
40
0
1
3. Time samples. j0,1 {j0,N `1 , . . . , j0,N {j0,N `1 are not
equispaced.
Alex Townsend @ MIT
5/26
Introduction
Overview of talk
Increasingly nonuniform
FFT
Evaluation of
Schlömilch
expansions
Evaluation of
Fourier–Bessel
expansions
DFT
J0 p r ω| q
J0 p r j 0 q
rk “ k {N,
ωn “ nπ,
|
DHT
|
J0 p j 0 j 0 {j0,N `1 q
j0,n “ Bessel roots.
Goal: FFT-based and analysis-based fast transforms with zero offline cost.
Alex Townsend @ MIT
6/26
Introduction
Overview of talk
Increasingly nonuniform
FFT
Evaluation of
Schlömilch
expansions
Evaluation of
Fourier–Bessel
expansions
DFT
J0 p r ω| q
J0 p r j 0 q
rk “ k {N,
ωn “ nπ,
|
DHT
|
J0 p j 0 j 0 {j0,N `1 q
j0,n “ Bessel roots
Goal: FFT-based and analysis-based fast transforms with zero offline cost.
Alex Townsend @ MIT
6/26
Fast evaluation of Schlömilch expansions
Setup
Schlömilch expansion of f : r0, 1s Ñ C is [Schlömilch, 1846]:
f pr q “
N
ÿ
xn J0 pnπr q.
n “1
Oscar Schlömilch
Aim: A fast algorithm for evaluating f pr q at rk “ k {N for 1 ď k ď N.
Equivalent to the following matrix-vector product:
˛¨ ˛
¨
J0 pr1 ω1 q . . . J0 pr1 ωN q
x1
‹
˚
˚
..
..
.. ‹
...
‹
˚
‹
J0 pr ωT qx “ ˚
.
.
˝
‚˝ . ‚,
J0 prN ω1 q . . . J0 prN ωN q
xN
Alex Townsend @ MIT
rk “ k {N,
ωn “ nπ.
7/26
Fast evaluation of Schlömilch expansions
Bessel functions are trigonometric-like for large argument
For fixed M ě 1 and real z Ñ 8:
¸
c ˜
Mÿ
´1
Mÿ
´1
m
m
p´
1
q
a
p´
1
q
a
2
2m
2m
`
1
cospz ´ π4 q
J0 p z q „
´ sinpz ´ π4 q
.
2m
πz
z
z 2m`1
m“0
m “0
J0 pz q
Fraunhofer regime
Hermann Hankel
Peter A. Hansen
A 19th century hobby: [Poisson, 1823], [Kummer, 1837], [Jacobi, 1843], [Lipschitz, 1859], [Hamilton, 1907]
Alex Townsend @ MIT
8/26
Fast evaluation of Schlömilch expansions
Numerical pitfalls of an asymptotic expansionist
c
J0 pz q “
˜
¸
Mÿ
´1
Mÿ
´1
m
m
p´
1
q
a
p´
1
q
a
2
2m
2m`1
cospz ´ π4 q
´ sinpz ´ π4 q
` RM pz q
2m
2m
`
1
πz
z
z
m “0
m “0
10 0
M“2
M
M
“
10
“7
M“3
M“
4
M“
5
10 -5
|RM p10q|
|RM pz q|
M“1
10 -10
10 -15
10 -20
0
10
20
30
40
M
z
Fix M. Where is the asymptotic expansion accurate?
Alex Townsend @ MIT
9/26
Fast evaluation of Schlömilch expansions
Recap
Evaluation of a Schlömilch expansion is equivalent to the following matrix-vector
product:
¨
˛¨ ˛
J0 pr1 ω1 q . . . J0 pr1 ωN q
x1
˚
‹
˚
‹
..
..
...
‹ ˚ ... ‹ , rk “ k {N, ωn “ nπ.
J0 p r ωT q x “ ˚
.
.
˝
‚˝ ‚
J0 prN ω1 q . . . J0 prN ωN q
xN
Alex Townsend @ MIT
10/26
Fast evaluation of Schlömilch expansions
Asymptotic expansions as a matrix decomposition
The asymptotic expansion gives a matrix decomposition:
¸
c ˜
Mÿ
´1
Mÿ
´1
m
m
p´
1
q
a
p´
1
q
a
2
2m
2m`1
J0 prk ωn q “
`RM prk ωn q
cosprk ωn ´ π4 q
´ sinprk ωn ´ π4 q
1
2m
`
2m
` 32
π
2
m“0prk ωn q
m“0 prk ωn q
which is a sum of diagonally scaled DCTs and DSTs when rk “ Nk and ωn “ nπ:

˙
„
Mÿ
´1 ˆ
SN ´1 0
T
T
J0 pr ω q “
D 2 ` RM .
Dum1 P CN `1 PDum2 ` Dvm1
0
0 vm
m “0
J0
Alex Townsend @ MIT
“
JASY
0
`
RM
11/26
Fast evaluation of Schlömilch expansions
Asymptotic expansions as a matrix decomposition
The asymptotic expansion gives a matrix decomposition:
¸
c ˜
Mÿ
´1
Mÿ
´1
m
m
p´
1
q
a
p´
1
q
a
2
2m
2m`1
J0 prk ωn q “
`RM prk ωn q
cosprk ωn ´ π4 q
´ sinprk ωn ´ π4 q
1
2m
`
2m
` 32
π
2
m“0prk ωn q
m“0 prk ωn q
which is a sum of diagonally scaled DCTs and DSTs when rk “ Nk and ωn “ nπ:

˙
„
Mÿ
´1 ˆ
SN ´1 0
T
T
J0 pr ω q “
D 2 ` RM .
Dum1 P CN `1 PDum2 ` Dvm1
0
0 vm
m “0
J0
Alex Townsend @ MIT
“
JASY
0
`
RM
11/26
Fast evaluation of Schlömilch expansions
Asymptotic expansions as a matrix decomposition
The asymptotic expansion gives a matrix decomposition:
¸
c ˜
Mÿ
´1
Mÿ
´1
m
m
p´
1
q
a
p´
1
q
a
2
2m
2m`1
J0 prk ωn q “
`RM prk ωn q
cosprk ωn ´ π4 q
´ sinprk ωn ´ π4 q
1
2m
`
2m
` 32
π
2
m“0prk ωn q
m“0 prk ωn q
which is a sum of diagonally scaled DCTs and DSTs when rk “ Nk and ωn “ nπ:

˙
„
Mÿ
´1 ˆ
SN ´1 0
T
T
J0 pr ω q “
D 2 ` RM .
Dum1 P CN `1 PDum2 ` Dvm1
0
0 vm
m “0
J0
Alex Townsend @ MIT
“
JASY
0
`
RM
11/26
Fast evaluation of Schlömilch expansions
Be careful and stay safe
Error curve: |RM prk ωn q| “ Fix M. Where is the asymptotic expansion accurate?
RM p r ω| q “
nk ď
N
s0,M pq
π
ùñ
|RM prk ωn q| ă Computed by a
fixed-point iteration
Alex Townsend @ MIT
12/26
Fast evaluation of Schlömilch expansions
AL
pr
ω|
q
Partitioning and balancing competing costs
0
J EV
Theorem
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
How many partitions?
JEVAL
x
0
N
JASY
x
0
Alex Townsend @ MIT
13/26
Fast evaluation of Schlömilch expansions
AL
pr
ω|
q
Partitioning and balancing competing costs
0
J EV
Theorem
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
How many partitions?
JEVAL
x
0
N
JASY
x
0
Alex Townsend @ MIT
13/26
Fast evaluation of Schlömilch expansions
AL
pr
ω|
q
Partitioning and balancing competing costs
0
J EV
Theorem
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
How many partitions?
JEVAL
x
0
N
JASY
x
0
Alex Townsend @ MIT
13/26
Fast evaluation of Schlömilch expansions
AL
Theorem
0
J EV
1
rαN 2 s
pr
ω|
q
Partitioning and balancing competing costs
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
Too few partitions
JEVAL
x
0
N
OpN 3{2 q
JASY
x
0
OpN log N q
Alex Townsend @ MIT
13/26
Fast evaluation of Schlömilch expansions
AL
Theorem
0
J EV
1
rαN 2 s
pr
ω|
q
Partitioning and balancing competing costs
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
JASY
x
0
Too many partitions
N
OpN 2 log N q
x
JEVAL
0
OpN log N q
Alex Townsend @ MIT
13/26
Fast evaluation of Schlömilch expansions
AL
Theorem
0
J EV
1
rαN 2 s
pr
ω|
q
Partitioning and balancing competing costs
JASY
p r ω| q
0,M
The matrix-vector product
X “ J0 pr ωT qx can be computed in
OpN plog N q2 { loglog N q operations.
Oplog N { log log N q partitions
N
JASY
x
0
OpN plog N q2 { loglog N q
Alex Townsend @ MIT
JEVAL
x
0
OpN plog N q2 { loglog N q
13/26
Fast evaluation of Schlömilch expansions
Fix M. But, what M should we pick?
Pick the working tolerance: ą 0:
N “ 1,000
0.5
2
“
M
Execution time
5
M
RM pr ω| q “
“
M
“
10
0.45
0.4
0.3
0.25
“1
0 ´9
0.2
0.15
For computational efficiency:
“ 10 ´15
0.35
0.1
“ 10´3
5
10
15
20
M
M “ maxpt0.3 logp1{qu, 3q
Other algorithmic parameters are decided upon by precise error bounds.
Alex Townsend @ MIT
14/26
Fast evaluation of Schlömilch expansions
Dir
ec
Op t
N2
q
A handful of FFTs
Zero offline cost
Analysis-based Ñ guaranteed accuracy
Execution time
Analysis-based algorithmic parameters
q
w
Ne og N
l
og
l
2{
q
N
g
N
Op
plo
N
Parameter
M
sν,M pq
α
β
P
Alex Townsend @ MIT
Short description
Formula
Number of terms in asy
maxpt0.3 logp1{qu, 3q
z ě sν,M pq ñ |Rν,M pz q| ď [Table 4.1, T., 15]
Partitioning parameter
psν,M pq{πq1{2
Refining parameter
minp3{ log N, 1q
Number of partitions
rlogp30α´1 N ´1{2 q{ log βs
15/26
Fast evaluation of Fourier–Bessel expansions
Overview of talk
Increasingly nonuniform
FFT
Evaluation of
Schlömilch
expansions
Evaluation of
Fourier–Bessel
expansions
DFT
J0 p r ω| q
J0 p r j 0 q
rk “ k {N,
ωn “ nπ,
|
DHT
|
J0 p j 0 j 0 {j0,N `1 q
j0,n “ Bessel roots
Goal: FFT-based and analysis-based fast transforms with zero offline cost.
Alex Townsend @ MIT
16/26
Fast evaluation of Fourier–Bessel expansions
Setup
Fourier–Bessel expansion of f : r0, 1s Ñ C is:
f pr q “
N
ÿ
xn J0 pj0,n r q.
n “1
Aim: A fast algorithm for evaluating f pr q at rk “ k {N for 1 ď k ď N.
Equivalent to the following matrix-vector product:
¨
˛¨ ˛
J0 pr1 j0,1 q . . . J0 pr1 j0,N q
x1
˚
‹
˚
‹
|
..
..
...
‹ ˚ ... ‹ ,
J0 pr j 0 qx “ ˚
.
.
˝
‚˝ ‚
J0 prN j0,1 q . . . J0 prN j0,N q
xN
Alex Townsend @ MIT
rk “ k {N,
J0 pj0,n q “ 0.
17/26
Fast evaluation of Fourier–Bessel expansions
Bessel roots as a perturbation of an equispaced grid
Idea: Bessel roots are nearly equispaced.
Lemma (Theorem 3 in [Hethcote, 70].)
Let j0,n denote the nth positive root of J0 pz q. Then,
ˇ
ˆ
˙ ˇ
ˇ
ˇ
1
1
ˇj0,n ´ n ´
ˇď
π
.
ˇ
ˇ
4
8pn ´ 41 qπ
We can write:
j0,n “ looooomooooon
pn ´ 1{4qπ `bn ,
ω̃n
Alex Townsend @ MIT
| bn | ď
1
.
8pn ´ 14 qπ
0
40
18/26
Fast evaluation of Fourier–Bessel expansions
A matrix decomposition
Theorem
For integers K ě 1 and T ě 1 we have
|
J0 p r j 0 q “ J0 p r ω̃| ` r b | q “
Kÿ
´1
Tÿ
´1
s “´K `1 t “0
diag-weighted Schlömilch
hkkkkkkkkkkkkikkkkkkkkkkkkj
p´1qt 2´2t `s 2t ´s
Dr Js p r ω̃| qDb2t ´s `EK ,T .
t!pt ´ s q!
Proof.
Neumann addition formula and Taylor expansion of J0 pz q about z “ 0.
|
J0 p r j 0 q
Alex Townsend @ MIT
“
A
`
p1q
EK ,T
A low-rank
correction
used for this
19/26
Fast evaluation of Fourier–Bessel expansions
Numerical results
Looks like
“33
But, after rearranging sum:
Fourier-Bessel “ looooooomooooooon
p2T ` K ´ 2q ˆ Schlömilch
“10
xn J0 pj0,n k {N q,
1 ď k ď N.
n“1
10
Fourier-Bessel “ looooomooooon
p2K ´ 1qT ˆ Schlömilch
Alex Townsend @ MIT
Xk “
N2
q
T “ 3.
N
ÿ
1
Op
K “ 6,
Fourier–Bessel expansions:
Execution time
Rearrange computation:
For “ 10´15 , T and K are:
Np
Op
10 0
log
log
og
2 {l
Nq
10 -1
Direct
ǫ=10 -15
ǫ=10 -8
ǫ=10 -3
10 -2
10 2
Nq
10 3
N
10 4
10 5
20/26
The discrete Hankel transform
Overview of talk
Increasingly nonuniform
FFT
Evaluation of
Schlömilch
expansions
Evaluation of
Fourier–Bessel
expansions
DFT
J0 p r ω| q
J0 p r j 0 q
rk “ k {N,
ωn “ nπ,
|
DHT
|
J0 p j 0 j 0 {j0,N `1 q
j0,n “ Bessel roots
Goal: FFT-based and analysis-based fast transforms with zero offline cost.
Alex Townsend @ MIT
21/26
The discrete Hankel transform
Discrete Hankel transform
Let j0,k denote the nth positive root of J0 pz q. The ratios are nearly equispaced:
j0,k {j0,N `1
ˇ
ˇ
1ˇ
ˇ j
k
´
1
ˇ 0,k
4ˇ
´
.
ˇ
ˇď
0
1
3
3
ˇ j0,N `1 N ` ˇ 8pN ` qpk ´ 1 qπ2
4
4
4
In a similar manner to the Fourier–Bessel case, we have
|
J0 p j 0 j 0 {j0,N `1 q
Alex Townsend @ MIT
“
B
`
p2q
EK ,T
A low-rank
correction
used for this
22/26
The discrete Hankel transform
Numerical results
Discrete Hankel transform
For “ 10´15 , we select:
K “ 6,
T “ 3.
xn J0 pj0,n j0,k {j0,N `1 q,
1 ď k ď N.
n “1
10 2
“10
10 1
2
10 0
and
2
DHT “ loooooooomoooooooon
p2T ` K ´ 2q ˆSchlömilch
“100
Execution time
DHT “ looooooomooooooon
p2T ` K ´ 2q ˆFourier–Bessel
O
pN
2
q
We have
Alex Townsend @ MIT
Xk “
N
ÿ
N
Op
g
plo
Nq
{
log
log
Direct
ǫ=10 -15
ǫ=10 -8
ǫ=10 -3
10 -1
10 -2 2
10
Nq
10 3
N
10 4
23/26
The discrete Hankel transform
Errors in practice versus theory
c “ vector with N p0, 1q entries
}f ´ f quad }8
10 0
10
Theoretical bound
› ›
10´3 ›c ›1
-5
10
´8
Near-optimal complexity in N
and ą 0:
› ›
›c ›
1
OpN plog N q2 logp1{qp { loglog N q,
10 -10
› ›
10´15 ›c ›1
where
-15
ǫ=10
ǫ=10
ǫ=10
10 -15
1000
2000
N
Alex Townsend @ MIT
3000
-8
-3
4000
Transform
Schlömilch
Fourier–Bessel
DHT
p
1
2
3
24/26
Conclusion
FFT
Evaluation of
Schlömilch
expansions
Evaluation of
Fourier–Bessel
expansions
DFT
J0 p r ω| q
J0 p r j 0 q
|
DHT
|
J0 p j 0 j 0 {j0,N `1 q
Novel features of our algorithm for the DHT:
1. Zero offline cost.
2. Implementation adapts to individual computer architecture.
3. Guaranteed error control (no heuristics).
4. No hierarchical data structures.
Alex Townsend @ MIT
25/26
Thank you
Thank you
More information in: T., “A fast analysis-based discrete Hankel transform using
asymptotic expansions”, to appear in SINUM.
Code publicly available from:
https://github.com/ajt60gaibb/FastAsyTransforms
Alex Townsend @ MIT
26/26
References
M. J. Cree and P. J. Bones, Algorithms to numerically evaluate the Hankel transform, J. Franklin Institute, 1983.
H. F. Johnson, An improved method for computing a discrete Hankel transform, Computer Physics
Communications, 1987.
N. Hale and A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptotic
formula, SISC, 2014.
M. O’Neil, F. Woolfe, and V. Rokhlin, An algorithm for the rapid evaluation of special function transforms, Appl.
Comput. Harm. Anal., 2010.
A. Townsend, The race to compute high-order Gauss quadrature, SIAM News, 2015.
A. Townsend, A fast analysis-based discrete Hankel transform using asymptotics expansions, to appear in
SINUM, 2015.
M. Tygert, Fast algorithms for spherical harmonic expansions, II, J. Comput. Phys., 2008.
Alex Townsend @ MIT
26/26
Extra slides
Direct approach [Amos, 86]
Compute
Jν pzq :
Lots going on!
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26
Extra slides
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
26/26