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1
Numerical Laplace Transform
Inversion and Selected
Applications
Patrick O. Kano, Ph.D.
March 5, 2010
2
Outline
� The talk is organized as follows:
1. Basic definitions and analytic inversion
2. Issues in numerical Laplace transform inversion
3. Introduce three of the most commonly known
numerical inversion procedures
1. Talbot’s Method
2. Weeks’ Method
3. Post’s Formula
4. Illustrate through applications
1. Pulse propagation in dispersive materials
2. Calculation of the matrix exponential
5. Future directions and conclusions
Overview
Contributions
3
Laplace Transform Definition
� Laplace transform solution methods are a standard of mathematics, physics, and engineering undergraduate education.
� Textbook examples however utilize known Laplace transform pairs.
Difficult Time
Dependant
Problem
Solve Simpler
Laplace Space
Problem
Time
Dependant
Solution
[ ] ( ) )(0)(1
tfsFtfELL →→=
−
?
A Laplace transform is a tool to
make a difficult problem into a
simpler one.
4
� A sufficient existence condition is that f(t) be
� piecewise continuous for nonnegative values of t
� of exponential order
� Intuitively, the Laplace transform can
be viewed as the continuous analog to a power series.
tMetf
M
t ≤≤ 0 allfor )(such that
, and enonnegativexist There
σ
σ
Laplace Transform Definitions
( ) (s)sdttfestfLsF st Re where allfor )()()()(0
<== ∫∞
− σ
2
)(
Transform No
tetf =
( ) dtetfxa st
n
ex
tfa
tn
n
n
s
n
−∞∞
=
=
→
→
→ ∫∑ →∑ ∫
−
00
)(
5
Laplace Transform Inversion� How does one return from the Laplace space representation to the time
domain?
We can alleviate some of the suspense at the very beginning by cheerfully
confessing that there is no single answer to this question.
Instead, there are many particular methods geared to appropriate situations.
This is the usual situation in mathematics and science and, hardly necessary to
add, a very fortunate situation for the brotherhood.
Richard Bellman
6
Analytic Inversion� The analytic inversion of the Laplace transform is a well-known application
of the theory of complex variables.
� For isolated singularities, the Bromwich contour is the standard approach.
∫∞+
∞−
=i
i
stdsesF
itf
σ
σπ
)(2
1)(
Realσ
Abscissa of convergence
Isolated singularities
Imaginary
� Laplace transform inversion is not a unique operation. � In practice, one can assume that the analytic inverse is well-defined.
Lerch’s theorem
( )( ) ( )
( ) ( ) 0 ,0 allfor then,
)(
transformLaplace same thehave and )(function twoif
2
0
1
21
21
=−>
=≡
∫ dttftf
fLfLsF
tftf
τ
τ
7
Numerical Inversion Issues� The numerical inversion of the Laplace transform is an inherently ill-
posed problem.
� To combat these numerical issues one may use 2 tactics
1. Fixed-point high precision variables.
2. Use of multiple algorithms, each with efficacy for certain classes of functions.
∫Λ
= dsesFi
tfst)(
2
1)(
π
Inherent sensitivity due to the multiplication by a exponential function of time.
Algorithmic and finite precision errors can lead to exponential divergence of numerical solutions.
8
� Mathematica
� ARPREC
� An Arbitrary Precision Computation Package
� Lawrence Berkley National Laboratory
� D. Bailey, Y. Hida, X. Li, B. Thompson
Fixed-Point High Precision Variables� High precision variables are required for most inversion methods.
� This requirement is important consequences:� Numerical LT methods are typically slower than other time-
propagation methods.� Implementation requires either
� An environment where high precision variables are innate.� Additional high precision variable software packages.
Double Precision: 10-308-10308
� GMP
� GNU Multiple Precision Arithmetic Library
� MP
� Matlab Based Toolbox
� Ben Barrows
� Matlab file exchange
10308~e709σ Large for even
modest times.
9
Numerical Inversion
� Post’s Formula
� Alternative to integration; arises from Laplace’s method
� Post (1930), Gaver (1966), Valko-Abate (2004)
� The Weeks method
� Laguerre polynomial expansion
� Ward (1954), Weeks (1966), Weideman (1999)
� Fourier series expansion
� Fourier related method
� Koizumi (1935), Dubner-Abate (1968), DeHoog-Knight-Stokes (1982), D’Amore (1999)
� Talbot’s method
� Deformed contour method
� Talbot (1979), Weideman & Trefethen (2007)
Euler
1707-1783
Laplace
1749-1827
Heaviside
1850-1925
Time�
Post
1930
Weeks
1966
Fourier
Series
1968
Talbot
1979
10
Talbot’s Method (1979)� Talbot’s method is based on a deformation of the
Bromwich contour.
� The idea is to replace the contour with one which opens towards the negative real axis.
� Talbot’s method requires that
( )jj sKs
ssF
iessingularit allfor Im
as 0)(
<
∞→→
( ) ( )
πθπ
σνλ
θνλθσθ
<<
++=
-
real are ,,
cotis
11
Talbot’s Method� The method is easily implemented in Mathematica.
0.39140
0.04710
0.14120
1.62580
Run TimePrecision
3
/12 )()(
t
etfesF
ts
π
−− =↔=
1 , ,021 === νλσ
( ) ( )
( )[ ]
θθπ
θθθθλλθ
πθπνσ
θνλθσθ
π
π
dd
dssFe
itf
id
ds
is
st
∫−
=
−+−=
<<==
++=
)(2
1)(
cot1cot
-,1,0
cot
The Talbot method
answers are accurate
up-to the computation
precision for time t=1.
Timeval = 1;
Rval = 1/2;
Flap[s_]=Exp[-2*Sqrt[s]];
Tfunexact[t_] =Exp[-1/t]/Sqrt[Pi*t*t*t];
Valexact = N[Tfunexact[Timeval],1000]
STalbot[r_,a_]=r*a*Cot[a]+I*r*a;
dsda[r_,a_]=I*r*(1+I*(a+Cot[a]*(a*Cot[a]-1)));
TimeDfun[r_,t_]:=1/(2*Pi*I)*NIntegrate[Exp[STalbot[r,a]*t]*Flap[STalbot[r,a]]*dsda[r,a],{a,-Pi,Pi},WorkingPrecision→→→→20];
{Timeval,Approxval}=Timing[TimeDfun[Rval,Timeval]]
RelError = Abs[Approxval-Valexact]/Valexact
12
Talbot’s Method
� Attempts have been made to automate the selection:� “Algorithm 682: Talbot’s method of the Laplace inversion problems”,
Murli & Rizzardi, 1990. [FORTRAN]
� This is an active area of research. � Optimizing Talbot’s contours for the inversion of the Laplace
transform, A. Weideman, 2006� Parabolic and Hyperbolic contours for computing the Bromwich
integral, A. Weideman & L.N. Trefethen, 2007
The primarily difficulty lies in the selection of appropriate values for the contours parameters.
( )2cos)(2
)(2
ttfs
ssF =↔
+=
1 , ,021 === νλσ
Mathematica’s adaptive
integration fails for the
same parameter values.
13
Post’s Formula (1930)
� There are two features of Post’s formula which are particularly attractive
� It contains no parameters, save the order of the derivative and the precision of the computations.
� The inversion is performed using
� Only real values for s
� Without priority knowledge of poles
� Post’s formula manifests the same inherent ill-posedness from which all numerical inversion procedures suffer.
� Errors are amplified � multiplicative factor grows quickly with the order of the derivative q
� The method converges slowly
� One needs an expression or approximation for the higher order derivatives of F(s)
( )
tqs
q
qqq
qsF
ds
d
t
q
qtf
/
1
)(!
1lim)(
=
+
∞→
−=
�Emil Post’s inversion procedure provides an alternative to Bromwich contour integration
14
Derivation� Post’s formula can be
derived using Laplace’s method
( ) ( )
( ) ( )0
2
20
0
2
2
0
0
00
k
2~)I(k,
behaviour asymptotic thehas
I(k)
integral the,k as
then,0 if and0 if
τπ
τ
ττ
τ
τ
τ
ττ
he
dτ
gd
dhe
dτ
gd
dτ
dg
kg
kg
−
∞−
∫=
∞→
≠=
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
+++
−
+++=
−
−
=−===
=−
=
−=
=
+
−
−
+
−−
∞−
∞−
∞−
∫
∫
∫
K
K
2
1
2
1
ln12
22
2
0
0
/ln
0
0
288
1
12
11)(
!
1~)(
288
1
12
112k!
formula sStirling'With
)(k2~)(1
)(2
~)(1
1
d
gd ,ln/ , ,Assign
)(1
at evaluate and Rearrange
1)(
respect to with derivative theTake
)(
0
kktkF
t
k
ktf
kkkek
tfet
kFt
k
tfek
t
tkF
ttgfht
dfet
kF
tks
dfesF
sk
dfesF
k
kk
kk
kkk
k
tkkk
tkkk
skkk
th
s
π
π
π
τττττττ
ττ
τττ
ττ
τ
ττ
τ
τ
Approximate
15
Derivatives� Finite differences an obvious method by which to approximate the
derivatives of a reasonably behaved function.
� The Gaver functionals can be computed by a recursive algorithm:
( )
( ) ( )( ) ( )
( ) ( )
)(lim)(
)2ln(2)2ln()(
1
)2ln(2)2ln(1)(
0
1
tftf
tjqF
j
q
q
q
tqtf
nxFxnFnxF
tqF
q
q
tqtf
q
j
j
q
q
∞→
=
=
+
=
−+=∆
∆
−=
∑ −
Gaver-Post Formula
1966
( )
( ) ( ) ( ) np p,1 1G
n1 2ln2ln
1
11
n
p
0
≤≤
−
+=
≤
=
+−−n
p
n
p
n
Gp
nG
p
n
tnF
tnG
q
qGtf =)(q
16
Derivatives� Post’s formula does not require a finite difference
approximation.
� For a particular function form, e.g. composition of two functions, a tailored method may be more robust.
( )
( )q
m
m
q
q
pmqm
mpq
m
pq
pq
pq
q,p
q
pp
p
q
q
xgB
dx
gdB
qB
B
Bdx
gd
m
qB
dx
gd,,
dx
gd,
dx
dgBg(x)
dx
fdf(g(x))
dx
d
)(
1for 0
1
1
1
,
1,
0,
0,0
1,
1
1
,
1
1
2
2
0
=
=
≤=
=
−
−=
=
−−
+−
=
+−
+−
=
∑
∑ K
Bell Polynomials of the Second Kindwww.mathworks.com/matlabcentral/fileexchange/14483
Faa di Bruno’s formula
17
Acceleration� Sequence acceleration methods be used to greatly increase accuracy
� The proper application of an acceleration convergence method requires some additional knowledge about the series.
� Post’s formula is logarithmically convergent
Slow
Sequence
{fq(t)}
Linear or NonlinearTransform
Fast
Sequence
{hq(t)}
<<=
−
−=
∞→
+
eConvergencLinear 10
eConvergenc cLogarithmi1lim
)()(
)()( 1
ca
tftf
tftfa
q
q
q
11
lim)()(
)()(lim
)()()(~)(
1
2
11
=+
=−
−
+++
∞→
+
∞→ q
q
tftf
tftf
q
tc
q
tctftf
q
q
q L
Gaver (1966)
18
Acceleration� Wynn-rho algorithm is well suited to logarithmically convergent
sequences.
� Studies have shown that it is useful for the Post formula:
� NSum in Mathematica implements these acceleration methods.
Post Inversion Formula and Sequence Acceleration
UA VIGRE Project 2009
J. Cain & B. Berman
1
1
0
1
1
2
1
3
2
2
2
2
12
2
0
2
1
1
2-Q
1
2
1
11
1
0
1
1
0
2-Q
0
2
0
10
0
0
0
1
0
2-Q
0
0
0
0
by Q,even for ,lim)(
function for theion approximatan yields algorithm The
−−−
−
−
−
−
−
−
∞→
==
==
==
==
=
Q
Q
Q
f
f
f
f
ftf
ρρ
ρ
ρ
ρρρρ
ρρρρρ
ρρρρρ
ρ
MM
MMM
MMMM
L
L
kQq
Qk
k q
kq
k
q
k
q
k
−−=
−=
+−
+= +
−++
1,,0
3,,0
1 1
111
K
K
ρρρ
ρ
ä "AlternatingSigns" method for summands with alternating signs
"EulerMaclaurin" Euler|Maclaurin summation method
"WynnEpsilon" Wynn epsilon extrapolation method
19
Application of Post’s Formula
� Rapid computation of the
� distribution of an initial optical pulse
� in a fixed dielectric medium
� with a nontrivial material dispersion relation.
NSF Grant ITR-0325097
An Integrated Simulation Environment for High-Resolution Computational Methods in
Electromagnetics with Biomedical Applications
Moysey Brio, et. al.
( )srε
Material
Biological materials often have a dielectric constant
which is a complex function of wavelength.
Input Pulse of Light Out Pulse of a Different Shape
Create databases of pre-computed tables which can be used by devices which
must operate in real-time.
20
Cole-Type Dispersion Relation� Many real world materials can be described by a Cole-type dispersion model.
( )[ ]{ }{ }{ }
( ){ }( )
( )∞∈
∞∈
∞∈
∞∈
∈
∈
++
+=
∞
−∞ ∑
,0
,0
,0
),0[
]1,0(
)1,0[
1)(
01
σ
ε
τ
δε
ε
σ
τ
δεεε
n
n
n
n
nba
n
nr
b
a
sss
nn
02.0
958.7)(
105.3
30.0
05.53)(
100.4
10.0
96.7)(
0.100
10.0
96.7)(
0.32
4
4
7
4
3
3
4
3
2
2
2
1
1
1
=
=
⋅=
=
=
⋅=
=
=
=
=
=
=
a
ms
a
ns
a
ns
a
ps
τ
δε
τ
δε
τ
δε
τ
δεBrain
White
Matter
� A standard method used in computational optics is to incorporate the dispersion
relation by means of an associated difference equation.
� For fractional coefficients, it is not clear how to translate into an associated
equation.
Fractional a
Coefficients
{ } 1
02.0
0.4
=
=
=∞
b
σ
ε
21
� Maxwell’s equations are the starting point for this analysis.
� In the Laplace space, the convolution and derivatives become multiplications.
( )( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) τττεε
µ
dxEttxEtxD
DE
txt
HtxE
txt
DtxH
txH
txD
t
,,,
nt displaceme theand strength field electricbetween assumption General
,,
,,
0,
0,
0
00
0
vvvvvv
vv
vv
vv
vv
vv
vv
vv
∫ −Φ+=
∂
∂−=×∇
∂
∂=×∇
=⋅∇
=⋅∇
Maxwell’s Equations
Temporal Convolution
( ) )()()(0
sGsFdtgfL
t
=
−∫ τττ ( ) ),()(),(
0
sxEsdtxEL
tvvvv
φτττ =
−Φ∫
22
� Maxwell’s equations now have a simpler form.
� Eliminating the magnetic field H from the problem,
� One obtains the wave equation in Laplace space
Maxwell’s Equations
( ) ( ) ( )
( )
( )( ) ( ) ( ) ( )
( ) ( ) ( )( )0,,,
0,,)(1,
0,
0,
,)(1,
0
00
0
=−−=×∇
=−+=×∇
=⋅∇
=⋅∇
+=
txHsxHssxE
txEsxEsssxH
sxH
sxE
sxEssxD
vvvvvv
vvvvvv
vv
vv
vvvv
µ
εφε
φε
( ) )0,()0,(),()(1),( 000000
22 =∂
∂−=−=+−∇ tx
t
EtxEssxEsssxE
vv
vvvvvvεµεµφεµ
( ) ( )
)0,()0,( 0
2
=∂
∂==×∇
∇−⋅∇∇=×∇×∇
txt
EtxH
HHH
vv
vv
vvv
ε
23
� One can more succinctly state this last equation as
� Applying a Fourier transform yields the desired solution in the joint space
002
22
22
1 )(1)(
)0,()0,(),(
),(1
),()(),(
µεφε
ε
=+=
=∂
∂+==
−=−∇
css
txt
EtxEssxV
sxVc
sxEsc
ssxE
r
r
vv
vvvv
vvvvvv
Maxwell’s Equations
222 )(
),(),(
kcss
skVskE
r
v
vvvv
+=
ε
24
Database Coefficients in the Joint Space� The solution in a dielectric medium can be characterized by one
coefficient α and its time derivative.
� Compute high order derivatives of α(k,s) and β(k,s)
� beta derivatives are trivially obtain from the alpha derivatives.
( ) ( )
( ) ( ) ( )sksskkcss
sk
tkEt
sktkEskskE
r
,, )(
1,
)0,(,)0,(,),(
222
vv
v
v
vvvvvvvv
αβε
α
αβ
=+
=
=∂
∂+==
For a given dispersion relation εr(s), the coefficients are pre-computed and
stored in a matrix of k vs time.
( ) ( ) ( )skqDsksDskD qqq ,,, 1ααβ −+=
The crux of the problem is the arbitrary precision calculation of the q-th derivative of α .
25
Derivative Approaches1.1. Standard GaverStandard Gaver--WynnWynn--RhoRho
� Finite Differences + Wynn-Rho Acceleration
� A brute force application entails a computation for each k and s.
2.2. GaverGaver--PostPost� Finite Differences + Wynn-Rho Acceleration
� The arbitrary precision computation of the dispersion relation εr(s) is time consuming.
� Dispersion relation is independent of k
� More efficient to store εr(s) and call for each k evaluation of α.
3.3. BellBell--PostPost� Analytic Derivatives + Wynn-Rho Acceleration� Store εr(s) and its derivatives.
� Use Faa di Bruno’s formula for the qth derivative of the computation of two functions.
( ) ( )
+=
= ∑ −
= tjqsk
j
n
q
q
tqtk
q
j
j
q
)2ln(,
2)2ln(),(
0
1 αα
26
Bell-Post Method� The problem of determining the time dependence of α(k,t) and thus
the electric field is reduced to evaluating the susceptibility function and its arbitrary order derivatives.
( )
( )
( )
( )( )[ ]
( )
( ) ( ) ( )sDnnssnDsDssgD
sgDsgDsDgBkcss
pskD
kcsssg
ssf
dx
gd,,
dx
gd,
dx
dgBg(x)
dx
fdf(g(x))
dx
d
kcsssk
r
n
r
n
r
nn
q
p
pq
pqp
r
p
q
r
pq
pq
q,p
q
pp
p
q
q
r
εεε
εα
ε
εα
212
0
12
,1222
222
1
1
2
2
0
222
)1(2)(
Rule Leibniz
)(,),(),(1!
),(
)(
1)(
)(
1,
−−
=
+−
+
+−
+−
=
−++=
+
−=
+=
=
=
+=
∑
∑
K
K
27
Cole-Type Dispersion Relation� For white brain matter the derivatives of εr(s) can be found by using the
Faa di Bruno formula.
( )[ ]{ }{ }{ }
( ){ }( )
( )∞∈
∞∈
∞∈
∞∈
∈
∈
++
+=
∞
−∞ ∑
,0
,0
,0
),0[
]1,0(
)1,0[
1)(
01
σ
ε
τ
δε
ε
σ
τ
δεεε
n
n
n
n
nba
n
nr
b
a
sss
nn
( ) ( )
( )
( )
( ) ( )
( )
( ) ( ) ( )kbk
j
kk
pap
j
ap
kq
kq
q
k
k
q
q
b
a
n
nq
q
nq
q
q
r
q
gjbgf
sjasg
sgggBsgfds
sgfd
ssf
ssg
sgfds
d
s
q
ds
sdn
+−−
=
−−−
=
−
+−
=
−
−
+
+
+−=
−−=
=
+=
=
+−
=
∏
∏
∑
∑
1)1()(
1)(
)(,,,)())((
1)(
)(
)(!1)(
1
0
11
0
1
121
,
0
1
1
0
τ
τ
δεε
σε
K
28
Mathematic Implementation Flow DiagramInputs
1. Choose [qmin,qmax]
2. Inversion time t
3. Take an explicit expression for εr(s) and its derivatives
4. A set of wavenumber k
Evaluate εr(s) and its derivatives at s=q/t
Compute s2εr(s) and its derivatives via Leibniz’s rule
Compute the Bell polynomials from the recursion relation
For k, compute s2εr(s) + c2k2
Compute the qth and (q-1)th derivatives of α(k,s)
Compute the qth derivative of β(k,s)
Approximate the inversion coefficients via Post’s formula
Apply Wynn-rho acceleration
Repeat for each q
29
Brain White Matter: Run Time
� The Bell-Post and Gaver-Post methods are faster than a standard Gaver.
� The acceleration dominates over the sequence computation times.
� The time follows a polynomial growth with q-max.
Time=(16/3)t0100 Digits Precision
pAqT =
1.6550.148Brute
Gaver
1.438-0.189Gaver-
Post
1.670-0.592Bell-
Post
pLog ACase
30
Brain White Matter: Accuracy
Time=(16/3)t0100 Digits Precision
� The Bell-Post and Gaver-Post methods have comparable accuracy� At higher precision
� and Post formula derivative orders.
31
The Weeks Method (1966)� The Weeks’ method is one of the most well known algorithms for the
numerical inversion of a scalar Laplace space function.� It popularity is due, in part, to the fact that it returns an explicit
expression for the time domain function.� The Weeks method assumes that
� a smooth time domain function of bounded exponential growth
� can be expressed as the limit of an expansion in scalar Laguerrepolynomials.
( )
( )
( ) ( )nx
n
nx
n
N
n
n
bt
n
t
N
NN
xedx
d
n
exL
btLeaetf
tftf
−
−
=
−
∞→
=
=
=
∑
!
2)(
)(lim
1
0
σ
The coefficients {an}
1. contain the information particular to
the Laplace space function
2. may be complex scalar, vectors, or
matrices
3. time independent
32
The Weeks Method� Two free scaling parameters σ and b, must be
selected according to the constraints that
� b>0 � ensures that the Laguerre polynomials are well behaved for large t
� σ>σ0-abscissa of convergence
0 1 2 3 4 5 6 7 8 9 10
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
e−
t Ln(2
t)
0123Laguerre Polynomials
33
The Weeks Method� The computation of the coefficients begins with a
Bromwich integration in the complex plane.
� Assume the expansion
� Equate the two expressions
( )∫∞
∞−
+=
∞<<∞>+=
dyiyFee
tf
iys
iytt
σπ
σσσσ
2)(
y- , , 0
( )∑∞
=
−=0
2)(n
n
bt
n
tbtLeaetf
σ
( ) ( )∫∑∞
∞−
∞
=
− += dyiyFebtLeaiyt
n
n
bt
n σπ2
12
0
34
Key Weeks Method Facts� It is known that the weighted Laguerre coefficients have the
Fourier representation.
� Performing
� the appropriate substitution,
� assuming it is possible to interchange the sum and integral
� equating integrands leaves
( ) ( )( )∫
∞
∞−
+
−
+
−= dy
biy
biyebtLe
n
n
iyt
n
bt
12
12
π
( )( )
( )∑∞
=+
+=+
−
01
nn
n
n iyFbiy
biya σ
35
Moebius Transformation� One may apply a transformation from complex
variable s to a new complex variable w
biy
biywiyσs
bs
bsw
+
−=+=
+−
−−= , or with
σ
σ
Isolated singularities of F(s) in the s-half-plane are mapped to the exterior of the
unit circle in the w-plane.
36
W-plane Representation� With the change of variables, one obtains
� a power series in w
� whose radius of convergence is greater than 1.
� The function is analytic on the unit circle.
� Numerically, the evaluation of the integral can be computed very accurately using the midpoint rule
−
+−
−=∑
∞
= 1
1
1
2
0 w
wbF
w
bwa
n
n
n σ
θσπ
π
πθ
θ
θ
θd
e
ebF
e
bea
i
i
i
in
n ∫−
−
−
+−
−=
1
1
1
2
2
1
M
m
e
ebF
e
be
M
ea m
M
Mmi
i
i
inM
in
nm
m
m
mπ
θσθ
θ
θ
θ
π
=
−
+−
−≈ ∑
−=
−
−
+
+
+
1
1
1
2
2 2/1
2/1
2/1
2
37
Matrix Exponential Application� An application of the Weeks method is to the
calculation of the matrix exponential.
00 )(0 , xetxx)(tx xAdt
xd At vvvvvv
=→===What does it mean “the
exponential of a matrix”?
L++++=!3!2
32AA
AIeA
Why don’t we just calculate this?
“Nineteen Dubious Ways to Compute the Exponential of a Matrix”,
SIAM Review 20, C. B. Moler & C. F. Van Loan, 1978.
“Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five
Years Later”, SIAM Review 45, C. B. Moler & C. F. Van Loan, 2003.
� Inverse Laplace Transform (#12)
[ ]∫−
−=Bromwich
AtAsI
ie dse
2
1 st1
π Resolvent Matrix of A
Apply the Weeks method
38
Matrix Exponential Application� Matlab: Pade’ approximation with scaling and squaring (#3)
� Matlab demos� expmdemo1: Pade’ + Scaling + Squaring in an m-file
� expmdemo2: Taylor Series
� expmdemo3: Similarity Transformation
( )( )
( )( )
( ) jq
j
pq
jp
j
pq
pq
pqB
n
nA
A
Bjqjqp
qjqpBD
Bjpjqp
pjqpBN
BD
BNe
ee
−−+
−+=
−+
−+=
≈
=
∑
∑
=
=
0
0
2
2
)!(!!
!!)(
)!(!!
!!)(
)(
)( expm
39
Beam Propagation Equation� Nonparaxial scalar beam propagation equation
� Discretisation in space yields a set of ODEs
� The Laplace transform in z yields
uzyxnkx
iuiz
u
+
∂
∂−=
∂
∂),,(22
02
2
β
( )uzyxnkDiuiz
u),,(22
0+−=∂
∂β
( )[ ] ( )
2
222
0
1
0)(ˆ
β
β
β
InkDA
uAIIisIsu
−+=
+−−=− v
u = a component of the
electric field
40
Beam Propagation Equation� The Laplace space function is of a matrix exponential
� The issue is how to pick the Laguerre polynomial parameters σ and b.
� Weeks’ original suggestions
� Error-Estimate Motivated Approach
� Weideman Method
� minimization of the error estimate as a function of σ and b
� Min-Max Method
� Maximum the radius of convergence as a function of σ and b
[ ] ( ) ( )
[ ]∫−
−
−=
+−=−=
Bromwich
MtMsI
ie
AIIiMuMsIsu
dse2
1
0)(ˆ
st1
1
π
βv
tcoefficienexpansion #21
0max 0 ==
+= N
t
N b
t,σσ
Expensive
Matrix
Inversions
41
Error Estimates� A straight forward error estimate yields three contributions
� Discretisation error: Finite integral sampling
� Truncations error: Finite number of Laguerre polynomials
� Round-off error: Finite computation precision
+≤ ∑∑
−
=
∞
=
1
0
22N
nFn
NnFn
t
total aaeE εσ
Truncation Round-Off
Weideman Method
Min-Max Method
Midpoint
rule on
circular
contour
( )1
)(
)()(
2
1
1
1
2
−≤
=≤
−
=
+∫
rr
rKT
r
rKdw
w
rKa
NE
n
rw
nFnπ
( )radiuscontour circular plane-wr
matrixresolvent theof norm
=
=rK
42
Beam Propagation Equation Example� Multi-mode interference coupler
Maximum
Solution
Absolute
Error
bσ
Weideman
Min-Max
Weeks
0.00042516.7911.84
0.001191020
12.28321
By proper selection of the parameters, it is possible to perform the calculations in double precision.
43
Pathological Matrices� An application is the exponential of special matrices.
6x6 Pei Matrix
Maximum Element Relative Error
32 Coefficients
2 1 1 1 1 1
1 2 1 1 1 1
1 1 2 1 1 1
1 1 1 2 1 1
1 1 1 1 2 1
1 1 1 1 1 2
Weeks’
MinMax
(2 Search
Methods)
Weideman Eigenvalues
1
N+1 (7)
gallery(‘pei’,6)
www.math.arizona.edu/~brio/WEEKS_METHOD_PAGE/pkanoWeeksMethod.html
44
Future Directions
NLAP-CL: Robust Parallel Numerical Laplace Transform Inversion via a C-CUDA Library and Application to Optical Pulse Propagation•Mosey Brio – University of Arizona
•Patrick Kano – Applied Energetics, Inc.
•Paul Dostert – Coker College
Extend NSF supported Post’s Formula Work to 2D and 3D
Mathematica
is too slow.(CUDA)
NVIDIA’s Compute Unified Device Architecture
� December 2009 – NSF proposal submitted
Tables of
coefficients for
multiple dispersive
materials
NLAP requires multiple simple arithmetic computations in high precision.
MATLAB C-MEX Files MATLAB
NLAP
Front End
45
Summary & Conclusions� The purpose the of the presentation was to provide some insight and
illustrate applications of numerical Laplace transform inversion.
� Standard methods� Talbot’s Method� Post’s Formula� The Weeks method
� Illustrated two examples� Calculation of the matrix exponential� Optical pulse propagation in dispersive media
� Numerical Laplace transform inversion is a topic � multitude of nuances to provide avenues for further research� popularity increase as computing power improves� potential for practical application in diverse fields
� Great utility and intellectual merit to the development of a general numerical Laplace inversion toolbox.
46
Sources� Numerical Inversion of the Laplace Transform, Bellman, Kalaba, Lockett, 1966.
� Peter Valko’s NLAP website: www.pe.tamu.edu/valko/public%5Fhtml/NIL/
� “Numerical inversion of Laplace transforms using Laguerre functions”, W. Weeks, Journal of the ACM, vol. 13, no. 3, pp.419-429, July 1966.
� “The accurate numerical inversion of Laplace transforms”, J. Inst. Math. Appl., vol. 23, 1979.
� “Application of Weeks method for the numerical inversion of the Laplace transform the matrix exponential”, P. Kano, M. Brio, J. Moloney, Comm. Math. Sci., 2005.
� “Application of Post’s formula to optical pulse propagation in dispersive media”, P. Kano, M. Brio, Computers and Mathematics with Applications, 2009.
47
BACKUPS
� BACKUP
48
Laguerre Polynomials� An unstable approach to obtain the time domain
function is to generate the Laguerre polynomials is to use the recurrence relation
� A backward Clenshaw algorithm is a stable method.
( ) ( )
( ) ( )∑∞
=
−
−+
=
−=
=
−−+=+
0
1
0
11
2)(
1)(
1)(
)()(12)(1
n
nn
tb
nnn
btLaetf
xxL
xL
xnLxLxnxLn
σ
49
Analytic Inversion
( )
[ ]2order 0
1
1)(
2
2
=
=
=
s
es
sg
ssF
st
( )( ) ( )[ ]
ttees
sds
dr
sgssds
d
!mr
πirdses
t
s
st
ss
m
m
m
iσ
iσ
st
==
=
−−
=
=
=
=−
−
∞+
∞−
∫
0
0
2
2
01
1
2
1
1
1
21
0
ts
Ltf
tdsesi
i
i
st
=
=
=
−
∞+
∞−
∫
2
1
2
1)(
1
2
1σ
σπ
50
Application of Post’s Formula
� Post’s formula was implemented to
� invert the Fourier-Laplace space coefficients
� which arise from the solution of the optical dispersive wave equation.
� We considered three implementations
� Standard Gaver-Wynn-Rho
� Gaver-Post
� Bell-Post
51
Brain White MatterRelative Error, Bell-Post, k=kmax Relative Error, Gaver-Post, k=kmax
The Bell-Post method performs
well at modest values for the
precision and order of the Post
formula approximation.
At higher precision and Post formula
approximation order, the Gaver-Post has
an accuracy/unit run time comparable or
better than the Bell-Post method.
