Page 13 - Laplace transform numerical inversion
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Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 11
2
3
4
5
6
-2e-007
0
1
-1e-007
0
1e-007
2e-007
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0, N = 22
0
1
2
3
4
5
6
-4e-008
-3e-008
-2e-008
-1e-008
0
1e-008
2e-008
3e-008
4e-008
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0, N = 26
0
1
2
3
4
5
6
-3e-009
-2e-009
-1e-009
0
1e-009
2e-009
3e-009
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0, N = 30
0
1
2
3
4
5
6
-1.5e-011
-1e-011
-5e-012
0
5e-012
1e-011
1.5e-011
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0, N = 40
Figure 9: Inversion error of
f
(
t
) =
e
−
µt
by Stehfest
N
= 22
,
26
,
30
, and
40
algorithm.
oscillates around the true solution that dies towards zero and, in our example, occasionally be-
comes negative.
As a conclusion, big Stehfest number algorithm can contribute to minimise reconstruction error,
but it cannot be retained as an alternative for the current issue. Furthermore, as mentioned
before, big Stehfest number algorithm requires the use of an arbitrary precision floating point
arithmetic implementation. This is not costless since the inversion algorithm as well as all the
Laplace reservoir models need to be rewritten by using this high precision arithmetic to avoid
numerical approximation errors within the whole inversion process.
4.1.3 Polynomials series expansion algorithms
The principle of polynomials series expansion algorithms is to approximate the inverted function
by a weighing sum of particular polynomials. One has to calculate a given number
K
∈
N
of
weighting coefficients
α
k
that depend on the function to be inverted. In the Laplace domain, let
ˆ
F
(
p
)
be the function to be inverted and
f
(
t
)
be the result of this operation. The principle of the
series expansion method is to express
f
(
t
)
as follows:
f
(
t
) =
w
(
t
)
+
∞
X
k
=0
α
k
Φ
k
(
t
)
!
≈
w
(
t
)
K
−
1
X
k
=0
α
k
Φ
k
(
t
)
!
(19)
w
(
t
)
is a windowing function. Depending on the case,
Φ
k
(
t
)
can be either a function of Legendre
polynomials, Chebyshev polynomials of second kind or Laguerre polynomials. These three alter-
natives are illustrated in the following sections. Weighting coefficients
α
k
depend on the values
taken by
ˆ
F
(
p
)
for a limited number of
p
values. The Laplace argument
p
is here considered as
being a real number.
