A case study D.S. Karachalios, I.V. Gosea, A.C. Antoulas · Outline Introduction and motivation....

Approximation of the Bessel functionA case study

D.S. Karachalios, I.V. Gosea, A.C. Antoulas

February 28, 2018

Affiliations:MPI Magdeburg, Rice University Houston

12th Elgersburg Workshop

Partners:

Outline

Introduction and motivation.

Overview of the approximation methods we apply:

1 The Loewner Framework.

2 The AAA algorithm.

3 The Vector Fitting (VF) method.

Numerical results and comparison among the methods.

Further examples (Physical & Artificial).

Conclusion and further developments.

Dimitris, [email protected] Irrational Approximation 2/26

mailto:[email protected]

Introduction

1. In general, model order reduction (MOR) is used to transform large,complex models (n) of time dependent processes into smaller(k n), simpler models that are still capable or representingaccurately the behavior of the original process under a variety ofconditions.

2. In particular, interpolatory model reduction methods constructreduced models whose (rational) transfer function matches that of theoriginal system at selected interpolation points.

3. Irrational transfer function correspond to infinite-dimensionaldynamical system. Is it possible to approximate such a function(without performing any spatial discretization)?

Main tool: The Loewner framework → data driven MOR method.Computes an independent linear realization (E,A,B,C).



Introduction







Introduction







Introduction




Main tool: The Loewner framework → data driven MOR method.

Computes an independent linear realization (E,A,B,C).



Introduction







Motivation

Irrational examples: [Curtain/...’09], [Filip/...’17], [Beattie/...’12], [Nakatsukasa/...’16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Euler - Bernoulli beam

frequency (Hz)101 102 103 104 105

Mag

nitu

de

10-6

10-4

10-2

100

102Frequency response of the original beam model

Hyperbolic sinus function

x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

H(x

)

0

0.2

0.4

0.6

0.8

1Function with two sharp peaks

Exponential function

x0 1 2 3 4 5 6 7 8 9 10

H(x

)

0

0.2

0.4

0.6

0.8

1Exponential function

Inverse of the Bessel fct



Motivation



frequency (Hz)101 102 103 104 105

Mag

nitu

de

10-6

10-4

10-2

100



x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

H(x

)

0

0.2

0.4

0.6

0.8



x0 1 2 3 4 5 6 7 8 9 10

H(x

)

0

0.2

0.4

0.6

0.8





Motivation



frequency (Hz)101 102 103 104 105

Mag

nitu

de

10-6

10-4

10-2

100



x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

H(x

)

0

0.2

0.4

0.6

0.8



x0 1 2 3 4 5 6 7 8 9 10

H(x

)

0

0.2

0.4

0.6

0.8





Motivation



frequency (Hz)101 102 103 104 105

Mag

nitu

de

10-6

10-4

10-2

100



x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

H(x

)

0

0.2

0.4

0.6

0.8



x0 1 2 3 4 5 6 7 8 9 10

H(x

)

0

0.2

0.4

0.6

0.8





Motivation



frequency (Hz)101 102 103 104 105

Mag

nitu

de

10-6

10-4

10-2

100



x-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

H(x

)

0

0.2

0.4

0.6

0.8



x0 1 2 3 4 5 6 7 8 9 10

H(x

)

0

0.2

0.4

0.6

0.8





Motivation and Methods

What if we don’t have access to the matrix realization or to theexplicit form of the transfer function? (Only data provided)

Answer

→ Data Driven approach!

Take measurements and use:

1 The Loewner framework - [Mayo/Antoulas ’07]

2 The AAA algorithm - [Nakatsukasa/Sete/Trefethen ’16]

3 The Vector Fitting method - [Gustavsen/Semlyen ’99]





Answer → Data Driven approach!











































Overview of the methods

A simple SISO example - (spring - mass - damper). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Spring-mass-damper equation

mx(t) + dx(t) + kx(t) = F (t)

State variable: x1 = x,x2 = x, output y = x.x1 = x2mx2 = −kx1 − dx2 + Fu = F , y = x1

Compact form - System

x(t) = Ax(t) + Bu(t),y(t) = Cx(t)

Where x =

[xx

],

A =

[0 1

− km − d

m

],B =

[01m

], C =[1 0]

Transfer Function

H(s) = C(sI−A)−1B =1

ms2 + ds+ k

We assume that: m=1, d=1 and k=1.






mx(t) + dx(t) + kx(t) = F (t)



x(t) = Ax(t) + Bu(t),y(t) = Cx(t)

Where x =

[xx

],

A =

[0 1

− km − d

m

],B =

[01m

], C =[1 0]

Transfer Function

H(s) = C(sI−A)−1B =1

ms2 + ds+ k







mx(t) + dx(t) + kx(t) = F (t)



x(t) = Ax(t) + Bu(t),y(t) = Cx(t)

Where x =

[xx

],

A =

[0 1

− km − d

m

],B =

[01m

], C =[1 0]

Transfer Function

H(s) = C(sI−A)−1B =1

ms2 + ds+ k







mx(t) + dx(t) + kx(t) = F (t)



x(t) = Ax(t) + Bu(t),y(t) = Cx(t)

Where x =

[xx

],

A =

[0 1

− km − d

m

],B =

[01m

], C =[1 0]

Transfer Function

H(s) = C(sI−A)−1B =1

ms2 + ds+ k





Method 1: The Loewner Framework - [Mayo/Antoulas ’07]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theory

Given: a row array of pairs ofcomplex numbers:

(ωk, Sk) : k = 1, ..., N

with ωk ∈ C, Sk ∈ C. We canpartition the data in two sets:

left data: (µj , νj), j = 1, ..., p

right data: (λi, wi), i = 1, ...,m

The objective is to find H(s) ∈ Csuch that:

H(λi) = wi and H (µj) = νj

Example

Sample the transfer function of thespring-mass-damper:ω =

[1 2 3 4 5 6 7 8

]S =

[13

17

113

121

131

143

157

173

]• left data:

µ =[

1 3 5 7]

V =[

13

113

131

157

]• right data:

λ =[

2 4 6 8]

W =[

17

121

143

173

]





Theory

Given: a row array of pairs ofcomplex numbers:

(ωk, Sk) : k = 1, ..., N

with ωk ∈ C, Sk ∈ C. We canpartition the data in two sets:

left data: (µj , νj), j = 1, ..., p

right data: (λi, wi), i = 1, ...,m

The objective is to find H(s) ∈ Csuch that:

H(λi) = wi and H (µj) = νj

Example

Sample the transfer function of thespring-mass-damper:ω =

[1 2 3 4 5 6 7 8

]S =

[13

17

113

121

131

143

157

173

]• left data:

µ =[

1 3 5 7]

V =[

13

113

131

157

]• right data:

λ =[

2 4 6 8]

W =[

17

121

143

173

]Dimitris, [email protected] Irrational Approximation 7/26




Theory

The Loewner matrix L ∈ Cp×m, isdefined as:

L =

v1−w1

µ1−λ1· · · v1−wm

µ1−λm

.... . .

...vp−w1

µp−λ1· · · vp−wm

µp−λm

The shifted Loewner matrixLs ∈ Cp×m, is defined as:

Ls =

µ1v1−w1λ1

µ1−λ1· · · µ1v1−wmλm

µ1−λm

.... . .

...µpvp−w1λ1

µp−λ1· · · µpvp−wmλm

µp−λm

Example

The Loewner matrix

L =

− 4

21 − 221 − 8

129 − 10219

− 691 − 8

273 − 10559 − 12

949

− 8217 − 10

651 − 121333 − 14

2263

− 10399 − 4

399 − 142451 − 16

4161

The shifted Loewner matrix

Ls =

− 1

21 − 121 − 5

129 − 7219

− 591 − 11

273 − 17559 − 23

949

− 9217 − 19

651 − 291333 − 39

2263

− 13399 − 3

133 − 412451 − 55

4161





Theory

The Loewner matrix L ∈ Cp×m, isdefined as:

L =

v1−w1

µ1−λ1· · · v1−wm

µ1−λm

.... . .

...vp−w1

µp−λ1· · · vp−wm

µp−λm

The shifted Loewner matrixLs ∈ Cp×m, is defined as:

Ls =

µ1v1−w1λ1

µ1−λ1· · · µ1v1−wmλm

µ1−λm

.... . .

...µpvp−w1λ1

µp−λ1· · · µpvp−wmλm

µp−λm

Example

The Loewner matrix

L =

− 4

21 − 221 − 8

129 − 10219

− 691 − 8

273 − 10559 − 12

949

− 8217 − 10

651 − 121333 − 14

2263

− 10399 − 4

399 − 142451 − 16

4161

The shifted Loewner matrix

Ls =

− 1

21 − 121 − 5

129 − 7219

− 591 − 11

273 − 17559 − 23

949

− 9217 − 19

651 − 291333 − 39

2263

− 13399 − 3

133 − 412451 − 55

4161





The following results allow us to construct reduced order models.

Theorem

If (L,Ls) is regular, then E = −L, A = −Ls, B = V, C = W is arealization of the data. Hence, H(z) = W(Ls − zL)−1V is the requiredinterpolant.

In the case of redundant data we perform a rank revealing SVD of:[L,Ls

]or

[LLs

]then

[L,Ls

]= YΣ`X

∗ and

[LLs

]= YΣrX

∗.

Theorem

The quadruple E = −Y∗LX, A = −Y∗LsX, B = Y∗V, C = WX, isthe realization of an approximate data interpolant.

Remark: Above is the SISO case. Moreover, the Loewner Framework can beapplied also for MIMO case via tangential interpolation.






Theorem



]or

[LLs

]then

[L,Ls

]= YΣ`X

∗ and

[LLs

]= YΣrX

∗.

Theorem








Theorem



]or

[LLs

]then

[L,Ls

]= YΣ`X

∗ and

[LLs

]= YΣrX

∗.

Theorem








Theorem



]or

[LLs

]then

[L,Ls

]= YΣ`X

∗ and

[LLs

]= YΣrX

∗.

Theorem








Theorem



]or

[LLs

]then

[L,Ls

]= YΣ`X

∗ and

[LLs

]= YΣrX

∗.

Theorem






Method 1: The Loewner Framework - spring-mass-damper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Example

We compute the singular values for the augmented matrix [L Ls]:

σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.

Reduce the dimension of the Loewner model from 4 to dimension 2;

The reduced model (C, E, A, B) is obtained by projecting the raw model

(W,L,Ls,V): C = WX, E = −Y∗LX, A = −Y∗LsX, B = Y∗V.Ez = Az + Bu

y = Cz→ Hr(s) = C(sE− A)−1B

(1

s2 + s+ 1

)Remark: Poles= eig(A, E) =

(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1





Example


σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.





(1

s2 + s+ 1


(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1





Example


σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.





(1

s2 + s+ 1


(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1





Example


σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.



(W,L,Ls,V): C = WX, E = −Y∗LX, A = −Y∗LsX, B = Y∗V.

Ez = Az + Bu


(1

s2 + s+ 1


(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1





Example


σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.





(1

s2 + s+ 1

)

Remark: Poles= eig(A, E) =

(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1





Example


σ([L Ls]) =

0.27197

0.0638123.3768 · 10−17

4.522 · 10−18

, → rank = 2.





(1

s2 + s+ 1


(−0.5 + 0.86603 i−0.5 − 0.86603 i

), Zeros = eig

([A B

C 0

],

[E 00 0

])≈ inf3,1




Method 2: The AAA algorithm - [Nakatsukasa/Sete/Trefethen ’16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The algorithm uses Barycentric representation of interpolants.

Z = [s1, ..., sn]T ,F = [f1, ..., fn]T

⇒r(s) = n(s)d(s) =

∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.

R = mean(F), e = [1, ..., 1], J = [1, ..., j, ..., n]for m = 1, ..., r n

1 [v, j] = max|F−ReT | & J(J == j) =[]

2 z = [z, Z(j)], f = [f , F (j)] → update supports points and data

3 C = [C, 1Z−Z(j) ] → next column vector of the Cauchy matrix

4 A = SF ∗C−C ∗ Sf → Loewner matrix via scaling matrices SF, Sf

5 [v,v, V ] = svd(A(J, :)), w = V (:,m) → weight vector - min sv

6 N = C ∗ (w. ∗ f), D = C ∗w, R = F, R(J) = N(J)./D(J)

7 Error = ||F−R||inf → tolerance (reached?)

end






Z = [s1, ..., sn]T ,F = [f1, ..., fn]T ⇒r(s) = n(s)d(s) =

∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.

R = mean(F), e = [1, ..., 1], J = [1, ..., j, ..., n]

for m = 1, ..., r n1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end







∑rk=1

wkfks−sk∑r

k=1wk

s−sk

.


1 [v, j] = max|F−ReT | & J(J == j) =[]







end




Method 3: Vector Fitting (VF) - [Gustavsen/Semlyen ’99]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VF aims at finding an approximant expressed in pole-residue form, as

f(s) =

r∑n=1

cns− an

+ d + sh.

VF solves the above problem as a linear problem in two stages.1. Stage: Pole identification: Specify the starting poles

an, n = 1..., r.Then multiply with an unknown function σ(s).

σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh

Overdetermined Ax = b with unknowns: cn, d, h, cn.

2. Stage: Residue identification: We can solve the original problem withthe zeros of σ(s) as a new poles an for f(s).

Overdetermined Ax = b with unknowns: cn, d, h.

repeat until converge.






f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.

VF solves the above problem as a linear problem in two stages.

1. Stage: Pole identification: Specify the starting polesan, n = 1..., r.Then multiply with an unknown function σ(s).

σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.


an, n = 1..., r.

Then multiply with an unknown function σ(s).

σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1

∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh










f(s) =

r∑n=1

cns− an

+ d + sh.



σ(s)f(s) =∑r

n=1cn

s−an+ d + sh , σ(s) =

∑rn=1

cns−an

+ 1∑rn=1

(cn

s−an+ 1

)f(s) =

∑rn=1

cns−an

+ d + sh







Bessel ApproximationDefinitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Bessel function of the first kind:

Jn(s) = 12πi

∮e(

s2)(t− 1

t)t−n−1dt, s ∈ C

The aim is to approximate 1J0(s)

over Ω = [0, 10]× [−1, 1].





Jn(s) = 12πi

∮e(

s2)(t− 1



over Ω = [0, 10]× [−1, 1].





Jn(s) = 12πi

∮e(

s2)(t− 1



over Ω = [0, 10]× [−1, 1].



Sampling BesselChoose interpolation points in two different ways.

1. Structured grid with 2121 conjugate points.

Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5

1Structured grid with 2121 points

2. Random uniformly distributed points as 2000 conjugate points.

Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5

1Sampling points as conjugate pairs 2M=2000

Remark: In both cases, conjugates pairs are under consideration inorder to built a real model approximant.





Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5



Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5







Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5



Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5





Bessel ApproximationMethod 1: The Loewner Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Singular values and superimposed graphs - H(s) , Hr(s).

100 101 102 103

10-15

10-10

10-5

100Singular values of the [L Ls] and [L;Ls]

[L Ls][L,Ls]

Error O(10−11) and the 2× 11 compressed points from the initial2121.

real part0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5

1Projected interpolation points

RightLeft



Bessel ApproximationMethod 1: The Loewner Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Poles and Zeros diagram

real part-5 0 5 10 15

imag

inar

y pa

rt

-10

-5

0

5

10

Poles and Zeros PolesZeros

Poles =

−8.32213293322054 − 1.4252 i−8.32213289862456 + 1.4252 i

−5.51461491999547−2.40481847965605

2.404825557695775.520078110286318.65372791291101

11.791535600890814.9135964357538

17.6548692348549 − 1.561 i17.654869354827 + 1.561 i

Bessel original roots =

2.404825557695775.520078110286318.65372791291101

11.791534439014214.930917708487718.0710639679109

.



Left & Right Projected Points

Structured grid 2121 points compressed to 22.

Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft

Random Uniformly grid 2000 points compressed to 24.

Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft

Remark: If we directly use those compressed points together with thecorresponding values as the interpolation data set (points/values), theinterpolant constructed this way will coincide with the interpolantcomputed from the initial v 2000 points. Hence, those are optimal points.





Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft


Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft






Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft


Re(s)0 1 2 3 4 5 6 7 8 9 10

Im(s

)

-1

-0.5

0

0.5


⇒real part

0 1 2 3 4 5 6 7 8 9 10

imag

inar

y pa

rt

-1

-0.5

0

0.5


RightLeft




Bessel ApproximationMethod 2: The AAA algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The AAA approximant Hr(s) and the original function.

Real Part-20 -15 -10 -5 0 5 10 15

Imag

inar

y P

art

-10

-5

0

5

10

Poles & Zeros - AAA algorithm

Absolute error over the Ω domain: O(10−11) + support points.

Re(z)0 1 2 3 4 5 6 7 8 9 10

Im(z

)

-1

-0.5

0

0.5

1Support points z with AAA algorithm Support points



Bessel ApproximationMethod 3: The VF method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The VF approximant Hr(s) and the original function.

real part-5 0 5 10 15 20

imag

inar

y pa

rt

-10

-5

0

5

10

Poles and Zeros with VF PolesZeros

Zero/Pole Cancellation

Absolute error over the Ω domain: O(10−6).

After the pole/zero cancellation,obtain an order r=11 approximant.

The largest error appears in thevicinity of the 3rd pole.



Bessel ApproximationMethods comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Case/Method Loewner AAA VF1st: 2121 structured points O(10−11) O(10−11) O(10−6)2nd: 2000 uniformly points O(10−11) O(10−13) O(10−6)

Table: Error comparison

1. The Loewner Framework:builds:(r − 1, r) real approximant.is a direct method.main complexity is due to SVD.

2. The AAA algorithm:builds: (r, r) complex approximant.is an iterative method.main complexity is due to SVDs of an incremental dimension.

3. The VF method:builds:(r + 1, r) real approximant.is an iterative method.main complexity is due to: solving 2 least squares.



























An Euler - Bernoulli Beam

Further examples treated with the Loewner Framework - [R. Curtain/K. Morris ’09]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PDE

∂2w(x, t)

∂t2+

∂2

∂x2

[EI

∂2w(x, t)

∂x2+ cdI

∂3w(x, t)

∂x2∂t

]= 0

Boundary Conditions and Input - Output

w(0, t) = 0, ∂w∂x

(0, t) = 0, EI∂2w(L,t)

∂x2 + cdI∂3w(L,t)

∂x2∂t= 0

−EI∂3w(L,t)

∂x3 − cdI∂4w(L,t)

∂x3∂t= u(t), y(t) =

∂w(L,t)∂t

H(s) =sN(s)

(EI + sCdI)m3(s)D(s)

m(s) =

[−s2

EI+cdIs

] 14, N(s) = cosh(Lm(s))sin(Lm(s)) − sinh(Lm(s))cos(Lm(s)),

D(s) = 1 + cosh(Lm(s))cos(Lm(s))



An Euler - Bernoulli Beam

Further examples treated with the Loewner Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Approximant: sample domain → [10, 104]Hz and Error curve.

Hz101 102 103 104 105

log-

Mag

nitu

de

10-4

10-2

100

Init.points n=800 - Loewner Appr. - order r=16 - Sampl.freq.domain: [1.0e+01,1.0e+04]HLoewner approximant

Imaginary axis101 102 103 104 105

abs(

erro

r)

10-10

10-5

Error curve |H-Hr|

Pole/Zero diagram.

Re(s) ×10-6-3 -2.5 -2 -1.5 -1 -0.5 0

Im(s

)

×104

-3

-2

-1

0

1

2

3Loewner Poles and Zeros for Irrational Beam Loewner Poles

Loewner Zeros

Parameter values: E = 69, GPa = 6, 9 · 1010N/m2-Young’s modulus elasticity constant,

I = (1/12) · 7 · 8.53 · 10−11m4-moment of inertia, cd = 5 · 10−4-damping constant,L = 0.7m, b = 7cm, h = 8.5mm-length,base,height of the rectangular cross section.



Hyperbolic sinus

Example from [Filip/Nakatsukasa/Trefethen/Beckermann ’17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H(x) = 100π(x2−0.36)sinh(100π(x2−0.36)) , x ∈ [−1, 1]

real axis-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Mag

nitu

de

0.2

0.4

0.6

0.8

1

1.2Init.points n=800 - Loewner Appr.- order r=38 - Sampling domain: [-1,1]

HLoewner approximant

real axis-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

abs(

erro

r)

10-14

10-13

10-12

Error curve with max|H-Hr| = 6.9747e-12

|H-Hr|

Pole/Zero diagram (”Far” - ”Zoom”)

Re(s)-2 0 2 4 6 8 10 12 14

Im(s

)

-0.5

0

0.5Loewner Poles and Zeros

Loewner PolesLoewner Zeros

Re(s)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Im(s

)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Loewner Poles and Zeros




Transfer function from 1D Heat equation

Example from [Beattie/Gugergin 12’]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H(s) = e−√s with s ∈ I = jω : ω ∈ R+.

frequency (Hz)10-1 100 101 102

log-

Mag

nitu

de

10-4

10-3

10-2

10-1

Init.points n=800 - Loewner Appr. - order r=6 - Sampling domain: [1.0e-01,1.0e+02]

HLoewner approximanterror curve

0 50 100 150 200 250 300 350 40010-20

10-15

10-10

10-5

100Singular values for [L Ls] and [L;Ls]

[Lr sLr][Lr;sLr]

X: 6Y: 0.006154

O(1e-3)

Poles/Zeros diagram and the impulse responce of the system.

Re(s)-40 -20 0 20 40 60 80

Im(s

)

-50

0

50Poles and Zeros from Loewner rational approximant


time (s)0 0.5 1 1.5 2 2.5 3

h

0

0.2

0.4

0.6

0.8

1Impulse Response

h(t)h

r(t)-Loewner



Conclusions and further developments

We investigated the practical applicability of three rationalapproximation methods for fitting irrational transfer functions.

The Loewner framework computes a full SVD. We are able tooptimize the computational cost:

By investigating the theoretical upper bounds of the singular values ofL, we can use a theoretical bound as a ”seed” for a shorter version ofSVD (or rSVD where r stands for randomized SVD). This could beaccessible with the Zolotarev bounds. [Beckermann & Townsend’16]Another approach is to substitute SVD with ”pseudoskeleton”approximation - CUR decomposition:

1. Max volume/Cross approximation [B.Kramer & A. Gorodetsky ’16]2. DEIM - CUR [D.C. Sorensen & M. Embree ’16]

Left and Right compressed projected points are special points!

1. Aim is to analyse the compressed information.2. Connection with potential theory and Zolotarev numbers.















By investigating the theoretical upper bounds of the singular values ofL, we can use a theoretical bound as a ”seed” for a shorter version ofSVD (or rSVD where r stands for randomized SVD). This could beaccessible with the Zolotarev bounds. [Beckermann & Townsend’16]

Another approach is to substitute SVD with ”pseudoskeleton”approximation - CUR decomposition:



















1. Max volume/Cross approximation [B.Kramer & A. Gorodetsky ’16]

2. DEIM - CUR [D.C. Sorensen & M. Embree ’16]





























1. Aim is to analyse the compressed information.

2. Connection with potential theory and Zolotarev numbers.












Bibliography

A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control,https://doi.org/10.1137/1.9780898718713,SIAM, Philadelphia, 2005,

A.C. Antoulas, S. Lefteriu and A.C. Ionita, A tutorial introduction to the Loewner Framework for ModelReduction, in Model Reduction and Approximation: Theory and Algorithms, edited by: P.Benner, M. Ohlberger, A.Cohen & K. Willcox, Series: Computational Science Engineering, pages 335-376, SIAM, Philadelphia, 2017,

Y. Nakatsukasa, O. Sete and L.N. Trefethen, The AAA Algorithm for rational approximation, arXiv: 1612.00337[math.NA], 29 pages, 2016

B. Gustavsen and A. Semlyen, Rational Approximation of frequency domain responses by vector fitting , IEEE Trans.Power Delivery, vol. 14, no. 3, pp. 1052-1061, July 1999

A.C. Ionita, Lagrange Rational Interpolation and its Applications to Approximation of Large-Scale DynamicalSystems,PhD thesis, Rice University, 2013

R. Curtain and K. Morris, Transfer Functions of Distributed Parameter Systems: A Tutorial, Automatica, 45(5),1101-1116

D.S. Karachalios, I.V Gosea & A.C. Antoulas, Case study: Approximations of the Bessel Function,arXiv:1801.03390v1 [math.NA], 18 pages, 2017,

THANK YOU VERY MUCH FOR YOUR ATTENTION!ANY QUESTIONS...?


https://doi.org/10.1137/1.9780898718713


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A case study D.S. Karachalios, I.V. Gosea, A.C. Antoulas · Outline Introduction and motivation....

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