Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral...

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Page 1: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 2: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 3: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 4: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 5: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 6: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 7: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 8: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 9: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 10: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 11: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 12: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 13: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 14: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 15: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 16: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 17: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 18: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 19: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 20: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 21: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 22: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 23: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 24: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 25: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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

Page 26: Numerical Inverse Laplace Transformation by …Inverse Laplace transformation Integral interpretation of the Abate Whitt framewrko Concentrated matrix exponential distributions Numerical

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