Page 18 - Laplace transform numerical inversion
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Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 16
The number of calls to the function
ˆ
F
(
p
)
for each algorithm is therefore:
Chebyshev:
18 =
N
K
Legendre:
18 =
N
K
Laguerre:
36 =
N
K
Q
Fourier:
2048 =
N
DFT
Gaver-Stehfest:
4608 =
N
K
M
Den Iseger:
16392 =
N
L
(
N
DFT
+ 1)
5.2 Tests results
None of the tested algorithm has given fully suitable results. Algorithms based on a real Laplace
argument could not properly rebuild an always positive declining exponential function such as
e
−
µt
. Every inversion attempt ended up at some point with negative values. Some kind of tweak-
ing can contribute in improving the inversion performance, but no any universal solution could be
found.
When going beyond the main constraint by assuming that the Laplace function to be inverted is
defined in the complex plane, one could only find one single algorithm that could invert the func-
tion with enough accuracy. The problem is now only shifted, since one has to find a way to build
some kind of analytic continuation to set the Laplace functions to be inverted in the complex plane.
Since one can not exhibit an algorithm that straightforwardly solve the issue, one can only give a
few hints to mainly correct a posteriori inversion results.
5.2.1 Pre-correction
The Laplace transform is a linear operator. To ensure a more accurate inversion one can add a
"correction function", say
ˆ
F
corr
(
p
)
, to the function
ˆ
F
model
(
p
)
to be inverted to act on the result. A
strategy would be to add a "security" function whose analytical solution is known and that either
corrects or minimises known artefacts. In short, instead of calculating :
L
−
1
h
ˆ
F
model
(
p
)
i
=
f
model
(
t
)
(20)
one would rather perform :
L
−
1
h
ˆ
F
model
(
p
) + ˆ
F
corr
(
p
)
i
=
f
model
(
t
) +
f
corr
(
t
)
(21)
What we call pre-correction can indeed contribute in improving the inversion result. Unfortunately,
it may also only delay the artefact or worse, potentially bring instability into a system that was
originally stable. Furthermore, there is no general rule to be followed to build up a suitable tailor
made correction function.
5.2.2 Post-correction
Another alternative would be to calculate the behaviour of the inverted function towards the
infinity and correspondingly perform a corrective post-processing. The Laplace transform offers
a powerful theorem that allows one to know the value of the inverted function
f
(
t
)
towards the
infinity when knowing the function
ˆ
F
(
p
)
in the Laplace domain for
p
= 0
:
final value theorem
⇐⇒
f
(+
∞
) = lim
p
→
0
p
ˆ
F
(
p
)
(22)
After the final value is known, the inversion algorithm can detect the vicinity of this value and
adapt its mode to avoid late time inversion artefacts. One can either choose to stop the inversion,
or to remove wrong values, or even to extend the reconstructed function with another processing
type by asymptotic analysis or algorithm simplification.
