Approximating Inverse Laplace

Transforms Chris Pardi

Introduction

In this talk I want to look at a current problem I have in my research.

This is finding a suitable method for gaining a good approximation to

an inverse Laplace transform.

I am going to look at:

What I am looking at that requires the calculation of a inverse

Laplace transform.

The function I need to invert.

An initial attempt at the inversion.

My planned next step.

While doing time dependant Hartree-Fock calculations we

are required to solve a system of coupled second order

partial differential equations.

In the spherically symmetric case where we let the full

Hartree-Fock wavefunction be .

These equations look like:

Where:

).,(),(),(

trQtrHt

trQi iiii

ii

),(),(

),,,( Yr

trQtr

).,(2

)1(

2

1),(

22

2

trVr

ll

rtrH ii

i

ii

i

ii

Time Dependant Hartree-Fock

Calculations

Boundary Conditions

The previous equations are all partial differential equations

that are first order in time and second order in space. Hence

we require an initial and two boundary conditions.

The initial condition is specified through stationary Hartree-

Fock and the boundary at the origin is known via symmetry

arguments.

The second boundary condition is the outgoing wave

condition and tells us .

When calculating the solution computationally we truncate

the boundary at some finite point, R. So we require a

boundary condition at R that satisfies the outgoing wave

condition.

0)(lim rQr

Boundary Conditions

Absorbing Boundary Condition

(ABC) The equation that gives us our second boundary condition is

the following:

Where is the Green’s function of the problem

given by:

Before this can be implemented an expression

for is required in order for the ABC to be efficient.

.),(

),,(2

1),(

0

dr

tRQRRGtRQ

t

),',( rrG

),()'(),',(),(),',( )(

rrrrGrHrrG

i i

ext

ii

,0),',(0

rrG

.0),',(r

rrG,0),',(

Rrr

rrG

M. Heinen et al. Phys. Rev. E, 79(5) 056709 (2009).

),',( rrG

The Hamiltonian, H

The Hamiltonian in the exterior region, r>R, only has three

components the kinetic, the centrifugal term and the

Coulomb interaction.

For the Coulomb potential we can make the assumption that

the charge is totally contained within the computed region.

This allows us to use the relation, , constant.

This leaves the Hamiltonian, for r>R as:

.2

)1(

2

1),(

22

2)(

r

ll

rrtrH

ext

r

Solving for the Green’s Function

This gives the following differential equation for the Greens

function:

Quite often the previous equations are solved using the

Laplace transform method:

),()'(2

)1(

2

122

2

rrGr

ll

rr

GGi i

ii

,0),',(0

rrG

.0),',(r

rrG

,0),',(

Rrr

rrG

.),',(),',(0

derrGrrg

Solving for the Green’s Function

This gives the ordinary differential equation:

This can be solved to give:

Where are Whitaker functions.

The above needs to be inverted to get back to our Green’s

function used in the ABC.

However the inversion complicated and so an approximation

to this process is needed.

).'()1(

2

122

2

rrgr

ll

ri

r

g

).1(4

1,

2,

)2(

)2(

2

1),,(

2

2

,

,ll

i

i

p

ipW

ipW

iRRg

iRp

,W

An Initial Attempt

Firstly the asymptotic series for the Whitaker functions:

where: is substituted into .

Just taking the leading term of the above approximation

gives:

This can be expanded in partial fractions of , then inverted

to give:

Fad is the Faddeeva function.

,1)(),1)...(1()( 0anaaaa n ),',( rrg

.)1()1(2

4),,(

2

1i i

i

ii

RRRg

,2

1Fad

22

2),,(

2

1i

iii ii

i

iRRG

An Initial Attempt: Results

This Green’s function was then tested as a boundary condition to

the differential equation:

.5.3

2

12

2

t

QiQ

rr

Q .)(ionNormalisat0

2R

drrQ

An Initial Attempt: Results

This is a plot of along imaginary shows how the

approximations are not accurate near the origin.

),,( RRg

An Updated Procedure

So a better way of approximating is needed.

It has been shown in various papers that approximating the special

functions in an Laplace transform by Padé approximations can

provide a accurate estimation of its inverse.

A Padé approximant is a quotient of two polynomials that match a

Taylor series up to a certain order.

They often provide a much better approximation of a function than

a Taylor series, despite being calculated from its Taylor Series.

),,( RRg

...1

.. 1

01

10 iML

i

iM

M

L

L zczbzb

zazaa

Padé Approximants

The graph approximations of the inverse Laplace transform below:

Where k are modified

Bessel functions of the

second kind.

William B. Bickford, Zeitschrift für Angewandte Mathematik und Physik, Volume 17, Number 2, 362-365

Conclusion

Shown that just using a asymptotic series for Whittaker

functions is not accurate enough to give a useful inverse

Laplace transform.

Shown that using a Padé approximant for the Whittaker

functions may provide a good way of getting reasonable

results.