Page 9 - Laplace transform numerical inversion

Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 - p 7/18
2 3 4 5 6
0
0 1
0.5
1
1.5
2
Fp_inv(t)
f(t)
0 1 2 3 4 5 6
-1e-014
-5e-015
0
5e-015
1e-014
1.5e-014
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0
Den_Iseger inverse Laplace transform
Figure 3: Analytical solution f (t) = Γe−µt and numerical inversion result.
4 Inversion algorithms testing
In this section one presents the results obtained by each tested algorithm introduced in section
(3 ). Algorithm comparative tests were only performed on a single function f (t) given in(14) and
represented in Figure 3.For clarity sake, results are only presented here in terms of reconstruc-
tion errors and shown in a single figure for each algorithm configuration.
In ach figure, the reconstruction error, which is equal to the known analytical result minus the
inverted Laplace function, appears in blue and is therefore tagged: Error: f(t) - Fp_inv(t). Since,
with the chosen test function f (t), one particularly focuses on the sign of the Laplace inverted
fonction, one can also see in green, superimposed on the error, the result of a "sign test" tagged:
Test Fp_inv(t) < 0. This often crenellated function allows one to easily localise instants for which
exponential reconstructed function becomes negative. The test function is positive when the in-
verted function is positive and negative otherwise. This test function is normalised in magnitude
to always keep the same proportion than the reconstruction error.
Algorithms can often be tweaked to increase inversion accuracy. This was not the aim of the
study. Therefore, the error magnitude that appears on the graphes should mainly be considered
as an indication and not as a method accuracy measurement.
Ideally, with the retained test function, a perfect inversion algorithm would yield a very accurate
inverted function with an always positive sign. In this case, the sign test function would be a
continuous horizontal line above the time axis. At least, a minor function preconditioning, should
allow one to reject at late time the instant when the sign indicator becomes negative. These are
the retained criteria to assess the different algorithms.
4.1 Algorithms with real Laplace argument: p ∈ R
4.1.1 Gaver-Stehfest algorithm
One of the most popular inversion algorithm, especially in reservoir engineering, where one needs
to have the Laplace argument p to be real (p ∈ R), is the Gaver-Stehfest method described in[2]
and[3] . This algorithm is fast and usually gives good results, especially for smooth functions.
Gaver-Stehfest algorithm is based on the following approximation:
∀t ∈ R∗
+, f (t) ≈ ln(2)
t
N X
n=1
K n ˆ
F
n ln(2)
t
(18)
Parameter N , referred to as the "Stehfest number" should be even. The weighting coefficients
Kn can be calculated once and for all during a pre-processing, because they only depend on the
Stehfest number N .