numerical inverse laplace transformation by …inverse laplace transformation integral...
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Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Numerical Inverse Laplace Transformation byconcentrated matrix exponential distributions
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2
1 Dept. of Networked Systems and Services, Budapest University of Technology
and Economics2 MTA�BME Information Systems Research Group
MAM10, Hobart, 2019-02-13
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Outline
Motivation:
minX∈PH(N)
SCV (X ) = 1/N
but
minX∈ME(N)
SCV (X ) < 1/N2.
How to utilize it for e�cient inverse Laplace transformation?
Outline
Inverse Laplace transformation
The Abate-Whitt framework
Integral interpretation of the Abate-Whitt framework
Concentrated matrix exponential distributions
Numerical comparisons of ILT methods
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Laplace transformation
Laplace transform is de�ned as
h∗(s) =
∫ ∞t=0
e−sth(t)dt. (1)
The inverse transform problem is to �nd an approximate value of hat point T (i.e., h(T )) based on the complex function h∗(s).
Assumptions∫∞t=0
e−sth(t)dt is �nite for Re(s) > 0,
h(t) is real → h∗(s̄) = h̄∗(s) and h∗(s̄) + h∗(s) = 2Re(h∗(s)).
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Inverse Laplace transformation
There are several approaches for numerical inverse Laplace
transformation (NILT).
The method based on matrix exponential distributions falls into the
Abate-Whitt framework.
The Abate-Whitt framework contains NILT procedures by
Euler,
Gaver-Stehfest,
. . .
J. Abate., W. Whitt, A uni�ed framework for numerically inverting
Laplace transforms. INFORMS Journal on Computing, 18(4):408�421,
2006.G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Basic de�nition
The idea is to approximate h by a �nite linear combination of the
transform values via
h(T ) ≈ hn(T ) :=n∑
k=1
ηkT
h∗(βkT
), T > 0, (2)
where the nodes βk and weights ηk are (potentially) complex
numbers, which depend on n, but not on the transform h∗(.) or the
time argument T .
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Gaver-Stehfest method
Only for even n.
For 1 ≤ k ≤ n
βk = k ln(2),
ηk = ln(2)(−1)n/2+k
min(k,n/2)∑j=b(k+1)/2c
jn/2+1
(n/2)!
(n/2
j
)(2j
j
)(j
k − j
),
where bxc is the greatest integer less than or equal to x .
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Euler method
Only for odd n.
For 1 ≤ k ≤ n
βk =(n − 1) ln(10)
6+ πi(k − 1),
ηk = 10(n−1)/6(−1)kξk ,
where
ξ1 =1
2ξk = 1, 2 ≤ k ≤ (n + 1)/2
ξn =1
2(n−1)/2
ξn−k = ξn−k+12−(n−1)/2
((n − 1)/2
k
)for 1 ≤ k < (n − 1)/2.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Location of nodes
Location of βk nodes on the complex plane for the Gaver (n = 10)
and Euler (n = 11) methods.
Euler
Gaver
-5 5 10 15Re
10
20
30
40Im
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Integral interpretation
For Re(βk) > 0, ∀k , we reformulate the Abate�Whitt framework as
hn(T ) =1
T
n∑k=1
ηkh∗(βkT
)=
1
T
n∑k=1
ηk
∫ ∞0
e−βkTth(t)dt
=
∫ ∞0
h(t)f nT (t)dt,
where
f nT (t) =1
T
n∑k=1
ηke−βk
Tt , f n1 (t) =
n∑k=1
ηke−βk t .
If f n1 (t) was the Dirac impulse function at point 1 then the Laplace
inversion would be perfect.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Properties of f nT (t)
But f n1 (t) di�ers from the Dirac impulse function depending on the
order of the approximation (n) and the applied inverse
transformation method (weights ηk , nodes βk).
Euler
Gaver
0.5 1.0 1.5 2.0
-2
2
4
6
8
Gaver (n = 10), Euler (n = 11)
Euler
Gaver
0.5 1.0 1.5 2.0
-5
5
10
15
Gaver (n = 22), Euler (n = 23)
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Scaling
f nT (t) is a scaled version of f n1 (t) =∑n
k=1 ηke−βk t :
f nT (t) =1
Tf n1
( t
T
).
f111(t)
f1011(t)
2 4 6 8 10 12
-2
2
4
6
8
Scaling T = 1→ 10: f 111 (t) and f 1110 (t) with the Euler method
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Consequence of scaling
exact
Euler
Gaver
200 400 600 800
2
4
6
8
h(t) = btc for Gaver (n = 14), Euler (n = 15)
Integration with f nT (t) averages out �x size steps for large T values
for all Abate�Whitt framework methods!
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Oscillations
The NILT of the unit step function at 1, h∗(s) = e−s
s , with Gaver
and Euler methods
exact
Euler
Gaver
50 100 150 200 250
0.2
0.4
0.6
0.8
1.0
1.2
Gaver (n = 10), Euler (n = 11)
exact
Euler
Gaver
50 100 150 200 250
0.2
0.4
0.6
0.8
1.0
1.2
Gaver (n = 54), Euler (n = 55)
Oscillations near the jump. The amplitude does not decrease for
higher n.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Concentrated matrix exponential (CME) distributions
We aim to �nd a good candidate f n1 (t) =∑n
k=1 ηke−βk t to
approximate δ1(t). If f n1 (t) ≥ 0, t ≥ 0, then it is the probability
density function of a matrix exponential distribution.
The quality of the approximation to δ1(t) is measured by the
squared coe�cient of variance:
scv =m0m2
m21
− 1,
where mi =∫∞t=0
t i f n1 (t)dt.
(We may assume normalization so that m0 = m1 = 1.)
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Concentrated matrix exponential distributions
With improved numerical methods and di�erent representations, we
have obtained high order low scv matrix exponential distributions of
the form
n∑k=1
ηke−βk t = ce−λt
(n−1)/2∏i=0
cos2(ωt − φi ) ≥ 0
for up to n = 1000 (odd n), with
scv(f n1 ) <1
n2.
See the talk by Miklós Telek.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Concentrated matrix exponential distributions
f n1 (t) for n = 4 for CME method:
CME
0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Matrix exponential distributions with low scv
f n1 (t) for n = 10 (n = 11 for Euler):
CME
Euler
Gaver
0.5 1.0 1.5 2.0
-5
5
10
15
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - step function
NILT of the step function for n = 20:
exact
CME
Euler
Gaver
50 100 150 200 250
0.2
0.4
0.6
0.8
1.0
1.2
CME method is overshoot and undershoot free: the approximations
always stay between inf(h(t)) and sup(h(t)). This property is due
to f n1 (t) ≥ 0.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - step function
NILT of the step function for n = 50 and n = 100:
exact
CME
Euler
Gaver
50 100 150 200 250
0.2
0.4
0.6
0.8
1.0
1.2
n = 50
exact
CME
Euler
Gaver
50 100 150 200 250
- 0.5
0.5
1.0
1.5
2.0
n = 100
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - exponential function
NILT of the exponential function for n = 10:
exact
CME
Euler
Gaver
100 200 300 400 500
0.2
0.4
0.6
0.8
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - shifted exponential function
h(t) = 1(t > 1)e1−t
exact
CME
Euler
Gaver
100 200 300 400 500
0.2
0.4
0.6
0.8
1.0
n = 20
exact
CME
Euler
Gaver
100 200 300 400 500
0.2
0.4
0.6
0.8
1.0
n = 50
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - square wave function
exact
CME
Euler
Gaver
200 400 600 800 1000
- 0.2
0.2
0.4
0.6
0.8
1.0
1.2
n = 20
exact
CME
Euler
Gaver
200 400 600 800 1000
0.5
1.0
1.5
n = 50
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
NILT results - square wave function
NILT of the square wave function for n = 500:
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Properties of the CME method
Improvements provided by the CME method compared to classical
methods:
no oscillations near jumps
overshoot and undershoot free
more accurate when the order is increased
machine precision is su�cient for all calculations
Issue:
no explicit expression for nodes and coe�cients (βk and ηk);they are best pre-calculated.
Available online.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Properties of the CME method
Improvements provided by the CME method compared to classical
methods:
no oscillations near jumps
overshoot and undershoot free
more accurate when the order is increased
machine precision is su�cient for all calculations
Issue:
no explicit expression for nodes and coe�cients (βk and ηk);they are best pre-calculated. Available online.
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions
Inverse Laplace transformationIntegral interpretation of the Abate�Whitt framework
Concentrated matrix exponential distributionsNumerical comparisons
Homepage
Homepage for the results:
http://inverselaplace.org/
The homepage includes:
list of features
theoretical background
citations
online javascript app
downloadable packages (currently available for Mathematica,
Matlab and IPython)
G. Horváth1, I. Horváth2, S. Almousa1, M. Telek1,2 Inverse Laplace transformation by CME distributions