Page 19 - Laplace transform numerical inversion

Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 - p 17/18
5.2.3 Analytic continuation & inversion in the complex plane
Den Iseger algorithm has proven to be the most successful and accurate inversion method among
the ones tested here. As already mentioned, it does not entirely satisfy our requirements in the
context of petroleum engineering since the function to be inverted needs to be known in the
complex plane, i.e. with p ∈ C .
A solution to fill the gap between the function ˆ
F (p), only known in the Laplace domain for a
real Laplace argument p ∈ R , and this particular inversion algorithm, would be to implement an
analytic continuation process. It should be applied on the function ˆ
F (p) and would yield a function
˚
F (p) defined in the complex plane for p ∈ C. This latter function ˚
F (p) could eventually be inverted
by using a less constrained inversion algorithm such as Den Iseger. The full process would be as
follows:
ˆ
F (p), p ∈ R
↓
Analytic
continuation
↓
˚
F (p), p ∈ C
↓
Re L−1 [ . ]
↓
f (t)
Now, it should be pointed out that analytic continuation is not a trivial task to perform. In short,
its principle is to extend definition domain of functions by mapping a real line onto a complex
plane. Among the difficulties, one can mention for instance the fact that certain functions with
a unique definition in the real space can have several definitions in the complex plane that do
not preserve all the properties of the original function, e.g. ln(t), t ∈ R ←→ ln(p), p ∈ C . Another
difficulty is to define strategies to avoid potential singularities.
Today, nothing warranties that new issues will not be brought to the numerical inversion initial
issue by adding this new processing. Furthermore, Den Iseger algorithm has proven to be a good
candidate in the current study after assessing performances on only a single test function. A new
study should be lead to generalise these first results.
Acknowledgements
We would like to thank Phillip Fair from Shell, Bob Hite from Blue Ridge PTA and Tony Fitzpatrick from
Schlumberger for their constructive remarks regarding Laplace transform numerical inversion issues and
analytic continuation.
References
[1] Brian Davies and Brian Martin. Numerical inversion of the Laplace transform: a survey and
comparison of methods. Journal of Computational Physics, 33(1):1 32, October 1979.
[2] Harald Stehfest. Algorithm 368: Numerical inversion of laplace transforms [d5]. Commun.
ACM, 13:47 49, January 1970.
[3] Harald Stehfest. Remark on algorithm 368: Numerical inversion of laplace transforms. Com-
mun. ACM, 13:624 , October 1970.
[4] Harvey Dubner and Joseph Abate. Numerical inversion of laplace transforms by relating them
to the finite fourier cosine transform. J. ACM, 15(1):115 123, 1968.