model reduction of large-scale dynamical systemsaca/aca_monopoli_iii.pdf · model reduction of...

Model reduction of large-scale dynamical systemsLecture III: Krylov approximation and rational interpolation

Thanos Antoulas

Rice University and Jacobs University

email: [email protected]: www.ece.rice.edu/ aca

International School, Monopoli, 7 - 12 September 2008

Thanos Antoulas (Rice U. & Jacobs U.) Reduction of large-scale systems 1 / 38

Outline

1 Krylov approximation methods

2 The Arnoldi and the Lanczos proceduresThe Arnoldi procedureThe Lanczos procedureAn example

3 Krylov methods and moment matchingRemarks

4 Rational interpolation by Krylov projectionRealization by projectionInterpolation by projection

5 Choice of Krylov projection points: Optimal H2 model reduction

6 Summary: Lectures II and III


Krylov approximation methods

Outline










Given Σ =

(A BC D

), expand the transfer function around s0:

H(s) = η0 + η1(s − s0) + η2(s − s0)2 + η3(s − s0)

3 + · · ·

Moments at s0: ηj , j ≥ 0. Find Σ =

(A BC D

), with

H(s) = η0 + η1(s − s0) + η2(s − s0)2 + η3(s − s0)

3 + · · ·

such that for appropriate k :

ηj = ηj , j = 1, 2, · · · , k

Moment matching methods can be implemented in a numerically stableand efficient way.



Krylov approximation methods: Special cases

• s0 =∞ Moments: Markov parametersProblem: (partial) realizationSolution computed through: Lanczos and Arnoldi procedures• s0 = 0Problem: Pade approximationSolution computed through: Lanczos and Arnoldi procedures• In general: arbirtary s0 ∈ CProblem: Rational interpolationSolution computed through: Rational Lanczos

• Computation of moments: numerically problematic

• Key fact for numerical reliability: If (A, B, C, D) given

• moment matching without moment computation⇒ iterative implementation.


The Arnoldi and the Lanczos procedures

Outline








The Arnoldi and the Lanczos procedures The Arnoldi procedure

The Arnoldi procedure

Given is A ∈ Rn×n, and b ∈ Rn. Let Rk (A, b) ∈ Rn×k be the reachability orKrylov matrix. It is assumed that Rk has full column rank equal to k .

Devise a process which is iterative and at the k th step we have

AVk = Vk Hk + Rk , Vk , Rk ∈ Rn×k , Hk ∈ Rk×k , k = 1, 2, · · · , n

=V VA R+

H

These quantities have to satisfy the following conditions at each step.

The columns of Vk are orthonormal: V∗k Vk = Ik , k = 1, 2, · · · , n.

span col Vk = span colRk (A, b), k = 1, 2, · · · , n

The residual Rk satisfies the Galerkin condition: V∗k Rk = 0,k = 1, 2, · · · , n.

This problem can be solved by the Arnoldi procedure.



Arnoldi: recursive implementation

Given: A ∈ Rn×n, b ∈ Rn

Find: V ∈ Rn×k , f ∈ Rn, and H ∈ Rk×k , such that

AV = VH + fe∗k where H = V∗AV, V∗V = Ik , V∗f = 0,

with H in upper Hessenberg form.

1 v1 = b‖b‖ , w = Av1; α1 = v∗1w

f1 = w− v1α1; V1 = (v1); H1 = (α1)

2 For j = 1, 2, · · · , k − 1

1 βj =‖ fj ‖, vj+1 =fjβj

2 Vj+1 =(Vj vj+1

), Hj =

(Hj

βje∗j

)3 w = Avj+1, h = V∗j+1w, fj+1 = w− Vj+1h

4 Hj+1 =(

Hj h)



Properties of Arnoldi

Hk is obtained by projecting A onto the span of the columns of Vk : Hk = V∗k AVk .

The remainder Rk has rank one and can be written as Rk = rk e∗k , where ek is the k th unit vector; thus rk ⊥ Rk .

This further implies that vk+1 =rk‖rk‖

, where vk+1 is the (k + 1)st column of V. Consequently, Hk is an upper

Hessenberg matrix.

Hk =

h1,1 h1,2 h1,3 · · · h1,k−1 h1,kh2,1 h2,2 h2,3 · · · h2,k−1 h2,k

h3,2 h3,3 h3,k−1 h3,k

. . ....

.

.

.

hk−1,k−1 hk−1,khk,k−1 hk,k

Let pk (λ) = det(λIk − Hk ), be the characteristic polynomial of Hk . This monic polynomial is the solution of thefollowing minimization problem

pk = arg min ‖p(A)b‖2

where the minimum is taken over all monic polynomials p of degree k . Since pk (A)b = Ak b +Rk · p, where pi+1

is

the coefficient of λi of the polynomial pk , it also follows that the coefficients of pk provide the least squares fit betweenAk b and the columns ofRk .

There holds

rk =1

‖pk−1(A)b‖pk (A)b, Hk,k−1 =

‖pk (A)b‖‖pk−1(A)b‖



An alternative way of looking at Arnoldi

Consider a matrix A ∈ Rn×n, a starting vector b ∈ Rn, and the correspondingreachability matrix Rn = [b Ab · · · An−1b]. The following relationship holdstrue:

ARn = RnF where F =

0 0 · · · 0 −α01 0 · · · 0 −α10 1 · · · 0 −α2

...0 0 · · · 1 −αn−1

and χA(s) = sn + αn−1sn−1 + · · ·+ α1s + α0, is the characteristic polynomialof A. Compute the QR factorization of Rn:

Rn = VU, V∗V = In, U upper triangular



It follows that

AVU = VUF ⇒ AV = V UFU−1︸︷︷︸A

⇒ AV = VA

Since U is upper triangular, so is U−1; furthermore F is upper Hessenberg.Therefore A being the product of an upper triangular times an upperHessenberg times an upper triangular matrix is upper Hessenberg.

The k-step Arnoldi factorization can now be obtained by considering the firstk columns of the above relationship, to wit:

[AV]k =[VA]

k ⇒ A[V]k = [V]k Akk + fe∗k

where f is a multiple of the (k +1)-st column of V. Notice that Akk is still upperHessenberg, while the columns of [V]k provide an orthonormal basis for thespace spanned by the first k columns of the reachability matrix Rn.


The Arnoldi and the Lanczos procedures The Lanczos procedure

The symmetric Lanczos procedure

If A = A∗ then the Arnoldi procedure is the same as the symmetric Lanczosprocedure. In this case Hk is tridiagonal:

Hk =

α1 β2β2 α2 β3

β3 α3

. . .. . .

. . .

αk−1 βkβk αk

This matrix shows that the vectors in the Lanczos procedure satisfy a threeterm recurrence relationship

Avi = βi+1vi+1 + αivi + βivi−1, i = 1, 2, · · · , k − 1

Remark. If the remainder rk = 0, the procedure has terminated, in whichcase if (λ, x) is an eigenpair of Hk , (λ, Vk x) is an eigenpair of A (since Hk x =λx implies AVk x = Vk Hk x = λVk x).



Two-sided Lanczos

The two-sided Lanczos procedure. Given A ∈ Rn×n which is notsymmetric, and two vectors b, c∗ ∈ Rn, devise a process which is iterativeand the k th step there holds:

AVk = Vk Hk + Rk , A∗Wk = Wk Hk + Sk , k = 1, 2, · · · , n.

Biorthogonality: W∗k Vk = Ik ,

span col Vk = span colRk (A, b), span col Wk = span colRk (A∗, c∗),

Galerkin conditions: V∗k Sk = 0, W∗k Rk = 0, k = 1, 2, · · · , n.

Remarks.

• The second condition of the second item above can also be expressed as span rows W∗k = span rowsOk (c, A), whereOk isthe observability matrix of the pair (c, A).

• The assumption for the solvability of this problem is detOk (c, A)Rk (A, b) 6= 0, k = 1, 2, · · · , n.

• The associated Lanczos polynomials are defined as pk (λ) = det(λIk − Hk ), and the induced inner product is defined as〈p(λ), q(λ)〉 = 〈p(A∗)c∗, q(A)b〉 = c p(A) · q(A) b.



Two-sided Lanczos: recursive implementation

Given: the triple A ∈ Rn×n, b, c∗ ∈ Rn

Find: V, W ∈ Rn×k , , g ∈ Rn, and H ∈ Rk×k , such that

AV = VH + fe∗k , A∗W = WH∗ + ge∗k where

H = V∗AW, V∗W = Ik , W∗f = 0, V∗g = 0. The projections πL and πU above,are given by V∗, W, respectively.

1 β1 :=√|b∗c∗|, γ1 := sgn (b∗c∗)β1

v1 = b/β1, w1 := c∗/γ1

2 For j = 1, · · · , k , set

1 αj = w∗j Avj

2 rj = Avj − αjvj − γjvj−1, qj = A∗wj − αjwj − βjwj−1

3 βj+1 =√|r∗j qj |, γj+1 = sgn (r∗j qj)βj+1

4 vj+1 = rj/βj+1, wj+1 = qj/γj+1



Properties of two-sided Lanczos

Hk is obtained by projecting A as follows: Hk = W∗k AVk .

The remainders Rk , Sk have rank one and can be written as Rk = rk e∗k ,Sk = qk e∗k .

This further implies that vk+1, wk+1 are scaled versions of rk , qkrespectively Consequently, Hk is a tridiagonal matrix.

The generalized Lanczos polynomials pk (λ) = det(λIk − Hk ),k = 0, 1, · · · , n−1, p0 = 1, are orthogonal: 〈pi , pj〉 = 0, for i 6= j .

The columns of Vk , Wk and the Lanczos polynomials satisfy thefollowing three-term recurrences

γk vk+1 = (A− αk )vk − βk−1vk−1βk wk+1 = (A∗ − αk )wk − γk−1wk−1γk pk+1(λ) = (λ− αk )pk (λ) − βk−1pk−1(λ)βk qk+1(λ) = (λ− αk )qk (λ) − γk−1qk−1(λ)


The Arnoldi and the Lanczos procedures An example

Example: symmetric Lanczos

Consider the following symmetric matrix:

A =

2 1 2 11 2 0 12 0 2 11 1 1 0

With the starting vector b = [1 0 0 0]∗, we obtain


The Arnoldi and the Lanczos procedures An example

V1 =

[ 1000

], H1 = [2], R1 =

[ 0121

]

V2 =

1 00 1√

60 2√

60 1√

6

, H2 =[

2√

6√6 8

3

], R2 =

0 00 1√

540 −1√

540 1√

54

V3 =

1 0 00 1√

61√3

0 2√6

−1√3

0 1√6

1√3

, H3 =

[2

√6 0√

6 83

1√18

0 1√18

43

], R3 =

[0 0 0

0 0√

32

0 0 0

0 0 −√

32

]

V4 =

1 0 0 00 1√

61√3

1√2

0 2√6

−1√3

0

0 1√6

1√3

−1√2

, H4 =

2√

6 0 0√6 8

31√18

0

0 1√18

43

√3√2

0 0√

3√2

0

, R4 = 04×4

where

AVk = Vk Hk + Rk , V∗k Rk = 0 ⇒ Hk = V∗k AVk , k = 1, 2, 3, 4.


Krylov methods and moment matching

Outline









Arnoldi and moment matching

The Arnoldi factorization can be used for model reduction as follows. Recallthe QR factorization of the reachability matrix Rk ∈ Rn×k ; a projection VV∗can then be attached to this factorization:

Rk = VU ⇒ V = Rk U−1

where V ∈ Rn×k , V∗V = Ik , and U is upper triangular. The reduced ordersystem is:

Σ =

(A BC

)where A = V∗AV , B = V∗B , C = CV

Theorem. Σ as defined above satisfies the equality of the Markov parametersηi = ηi , i = 1, · · · , k . Furthermore, A is in Hessenberg form, and B is amultiple of the unit vector e1.



Proof. First notice that since U is upper triangular, v1 = B‖B‖ , and since

V∗Rk = U it follows that B = u1 =‖ B ‖ e1; therefore B = V∗B.VV∗B = VB = B, hence AB = V∗AVV∗B = V∗AB; in general, since VV∗ is aprojection along the columns of Rk , we have VV∗Rk = Rk ; moreover:Rk = V∗Rk ; hence

(η1 · · · ηk ) = CRk = CVV∗Rk = CRk = (η1 · · · ηk )

Finally, the upper triangularity of U implies that A is in Hessenberg form.

Remark.Similarly, one can show that reduction by means the two-sided Lanczosprocedure preserves 2k Markov parameters.


Krylov methods and moment matching Remarks

Remarks

The number of operations is O(k2n) vs. O(n3), which implies efficiency.The requirement for memory is large if k is relatively large.

Only matrix-vector multiplications are required. No matrix factorizationsand/or inversions. There is no need to compute the transformed n-thorder model and then truncate. This eliminates ill-conditioning.

Drawbacks: • Numerical issue: Arnoldi/Lanczos methods looseorthogonality. This comes from the instability of the classicalGram-Schmidt procedure. Remedy: re-orthogonalization.• no global error bound.• Σ tends to approximate the high frequency poles of Σ. Remedy: matchexpansions around other frequencies⇒ rational Lanczos.


Rational interpolation by Krylov projection

Outline








Rational interpolation by Krylov projection Realization by projection

Partial realization by projection

Given a system Σ = (A, B, C), where A ∈ Rn×n and B, C∗ ∈ Rn, We seek alower dimensional model Σ = (A, B, C), where A ∈ Rk×k , B, C∗ ∈ Rk , k < n,such that Σ preserves some properties of the original system, throughappropriate projection methods. In other words, we seek V ∈ Rn×k andW ∈ Rn×k such that W∗V = Ik , and the reduced system is given by:A = W∗AV, B = W∗B, C = CV.

Lemma

With V = [B, AB, · · · , Ak−1B] = Rk (A, B) and W any left inverse of V , Σ isa partial realization of Σ and matches k Markov parameters.

From a numerical point of view, one would not use V as defined above since usually the columns of V are almost linearlydependent. As it turns out any matrix whose column span is the same as that of V can be used.

Proof.

We have CB = CVW∗B = CRk (A, B)e1 = CB; furthermore

CAj B = CRk (A, B)W∗AjRk (A, B)e1 = CRk (A, B)W∗Aj B = CRk (A, B)ej+1 = CAj B, j = 1, · · · , k − 1.


Rational interpolation by Krylov projection Interpolation by projection

Rational interpolation by projection

Suppose now that we are given k distinct points sj ∈ C. V is defined as thegeneralized reachability matrix

V =[(s1In − A)−1B, · · · , (sk In − A)−1B

],

and as before, let W be any left inverse of V. Then

Lemma

Σ defined above, interpolates the transfer function of Σ at the sj , that is

H(sj) = C(sj In − A)−1B = C(sj Ik − A)−1B = H(sj), j = 1, · · · , k .

Proof.

The following string of equalities leads to the desired result:

C(sj Ik − A)−1B = CV(sj Ik − W∗AV)−1W∗B

= C[(s1In − A)−1B, · · · , (sk In − A)−1B

] (W∗(sj In − A)V

)−1W∗B

=[C(s1In − A)−1B, · · · , C(sk In − A)−1B

] ([· · · W∗B · · · ]

)−1 W∗B

=[C(s1In − A)−1B, · · · , C(sk In − A)−1B

]ej

= C(sj In − A)−1B.



Matching points with multiplicity

We now wish to match the value of the transfer function at a given points0 ∈ C, together with k − 1 derivatives. For this we define the generalizedreachability matrix

V =[(s0In − A)−1B, (s0In − A)−2B, · · · , (s0In − A)−k B

],

together with any left inverse W thereof.

Lemma

Σ interpolates the transfer function of Σ at s0, together with k − 1 derivativesat the same point, j = 0, 1, · · · , k − 1:

(−1)j

j!

d j

dsjH(s)

∣∣∣∣∣s=s0

= C(s0In − A)−(j+1)B = C(s0Ik − A)−(j+1)B =(−1)j

j!

d j

dsjH(s)

∣∣∣∣∣s=s0

Proof.

Let V be as defined as above, and W be such that W∗V = Ik . It readily follows that the projected matrix s0Ir − A is in companionform (expression on the left) and therefore its powers are obtained by shifting its columns to the right:

s0Ik − A = W∗(s0In − A)V = [W∗B, e1, · · · , ek−1] ⇒ (s0Ik − A)` = [ ∗ · · · ∗︸︷︷︸`−1

, W∗B, e1, · · · , ek−`].

Consequently [W∗(s0In − A)V]−`W∗B = e` , which finally impliesC(s0Ik − A)−`B = CV [W∗(s0I− A)V]−` W∗B = CVe` = C(s0In − A)−`B, ` = 1, 2, · · · , k.



General result: rational Krylov

A projector which is composed of any combination of the above three casesachieves matching of an appropriate number of Markov parameters andmoments. Let the partial reachability matrix be

Rk (A, B) =[B AB · · · Ak−1B

],

and partial generalized reachability matrix be:Rk (A, B;σ) =

[(σIn − A)−1B (σIn − A)−2B · · · (σIn − A)−k B

].

Rational Krylov

(a) If V as defined in the above three cases is replaced by V = VR, R ∈ Rk×k ,det R 6= 0, and W by W = R−1W, the same matching results hold true.(b) Let V be such that

span col V = span col [Rk (A, B) Rm1(A, B;σ1) · · · Rm`(A, B;σ`)] ,

and W any left inverse of V. The reduced system matches k Markovparameters and mi moments at σi ∈ C, i = 1, · · · , `.



Two-sided projections: the choice of W

Let Ok (C, A) ∈ Rk×n, be the partial observability matrix consisting of the firstk rows of On(C, A) ∈ Rn×n. The first case is

V = Rk (A, B), W = (Ok (C, A)Rk (A, B)︸︷︷︸Hk

)−1Ok (C, A).

Lemma

Assuming that detHk 6= 0, Σ is a partial realization of Σ and matches 2kMarkov parameters.

Given 2k distinct points s1, · · · , s2k , we will make use of the followinggeneralized reachability and observability matrices:

V =[(s1In − A)−1B · · · (sk In − A)−1B

], W =

[(sk+1In − A∗)−1C∗ · · · (s2k In − A∗)−1C∗

].

Lemma

Assuming that det W∗V 6= 0, the projected system Σ where V = V andW = W(V∗W)−1 interpolates the transfer function of Σ at the 2k points si .



Remarks(a)The same procedure as above can be used to approximate implicitsystems, i.e., systems that are given in a generalized formEx(t) = Ax(t) + Bu(t), y(t) = Cx(t), where E may be singular. The reducedsystem is given by

E = W∗EV, A = W∗AV, B = W∗B, C = CV,

where

W∗ =

C(sk+1E− A)−1

...C(s2k E− A)−1

, V =[(s1E− A)−1B · · · (sk E− A)−1B

](b) Sylvester equations and projectors. The solution of an appropriateSylvester equation AX + XH + BG = 0, provides a projector that interpolatesthe original system C, A, B at minus the eigenvalues of H. Therefore theprojectors above can be obtained by solving Sylvester equations.


Choice of Krylov projection points: OptimalH2 model reduction

Outline









Choice of Krylov projection points:Optimal H2 model reduction

Recall: the H2 norm of a stable system is:

‖Σ‖H2 =

(∫ +∞

−∞h2(t)dt

)1/2

where h(t) = CeAtB, t ≥ 0, is the impulse response of Σ.

Goal: construct a Krylov projector such that

Σk = arg mindeg(Σ) = rΣ : stable

∥∥∥Σ− Σ∥∥∥H2

=

(∫ +∞

−∞(h− h)2(t)dt

)1/2



First-order necessary optimality conditions

Let (A, B, C) solve the optimal H2 problem and let λi denote the eigenvaluesof A. The necessary conditions are

H(−λ∗i ) = H(−λ∗i ) anddds

H(s)

∣∣∣∣s=−λ∗i

=dds

H(s)

∣∣∣∣s=−λ∗i

Thus the reduced system has to match the first two moments of the originalsystem at the mirror images of the eigenvalues of A.

The H2 norm: if H(s) =∑n

k=1φk

s−λk⇒ ‖H‖2

H2=

n∑k=1

ck H(−λ∗i )

Corollary. With H(s) =∑r

k=1φk

s−λk, the H2 norm of the error system, is

J =∥∥∥H− H

∥∥∥2

H2=

n∑i=1

φi

[H(−λi )− H(−λi )

]+

r∑j=1

φj

[H(−λj )− H(−λj )

]

Conclusion. The H2 error is due to the mismatch of the transfer functionsH− H at the mirror images of the full-order and reduced system poles λi , λi .



An iterative algorithm

Let the system obtained after the (j − 1)st step be (Cj−1, Aj−1, Bj−1), whereAj−1 ∈ Rk×k , Bj−1, C∗j−1 ∈ Rk . At the j th step the system is obtained as follows

Aj = (W∗j Vj)

−1W∗j AVj , Bj = (W∗

j Vj)−1W∗

j B, Cj = CVj ,

whereVj =

[(λ1I− A)−1B, · · · , (λk I− A)−1B

],

W∗j =

[C(λ1I− A)−1, · · · , C(λk I− A)−1

],

and: −λ1, · · · ,−λk ∈ σ(Aj−1),i.e., −λi are the eigenvalues of the (j − 1)st iterate Aj−1.

The Newton step: can be computed explicitly

λ(k)1

λ(k)2...

←−

λ(k)1

λ(k)2...

− J−1

λ

(k−1)1

λ(k−1)2

.

.

.

⇒ local convergence guaranteed.



An iterative rational Krylov algorithm (IRKA)

The proposed algorithm produces a reduced order model H(s) that satisfiesthe interpolation-based conditions, i.e.

H(−λ∗i ) = H(−λ

∗i ) and

d

dsH(s)

∣∣∣∣s=−λ∗i

=d

dsH(s)

∣∣∣∣s=−λ∗i

1 Make an initial selection of σi , for i = 1, . . . , k

2 W = [(σ1I− A∗)−1C∗, · · · , (σk I− A∗)−1C∗]

3 V = [(σ1I− A)−1B, · · · , (σk I− A)−1B]

4 while (not converged)

A = (W∗V)−1W∗AV,σi ←− −λi(A) + Newton correction, i = 1, . . . , kW = [(σ1I− A∗)−1C∗, · · · , (σk I− A∗)−1C∗]V = [(σ1I− A)−1B, · · · , (σk I− A)−1B]

5 A = (W∗V)−1W∗AV, B = (W∗V)−1W∗B, C = CV



Moderate-dimensional example

total system variables n = 902, independent variables dim = 599, reduceddimension k = 21reduced model captures dominant modes

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−40

−38

−36

−34

−32

−30

−28

−26

−24

−22

−20

Frequency x 108(rad/s)

Sin

gu

lar

valu

es (

db

)

Frequency responseSpectral zero method with SADPA

n=902 dim=599 k=21

Original

Reduced(SZM)

−0.0136 −0.0135 −0.0135 −0.0134 −0.0134−3

−2

−1

0

1

2

3

Dominant spectral zerosTheoretical and found with SADPA

Real

Imag

Spz: original

Spz: dominant

Spz: SADPA computed

R

C RC

L

C RC

RL

RCC

RLL-

?

- -

??? ?

-

?

. . .

?

u

y

1



H∞ and H2 error norms

Relative norms of the error systems

Reduction Methodn = 902, dim = 599, k = 21 H∞ H2

PRIMA 1.4775 -Spectral Zero Method with SADPA 0.9628 0.841

Optimal H2 0.5943 0.4621Balanced truncation (BT) 0.9393 0.6466

Riccati Balanced Truncation (PRBT) 0.9617 0.8164


Summary: Lectures II and III

Outline









Approximation methods: SummaryPPPPPPPPq

��

Krylov

• Realization• Interpolation• Lanczos• Arnoldi

SVD

@@@R

��

�

Nonlinear systems Linear systems• POD methods • Balanced truncation• Empirical Gramians • Hankel approximation@

@@

@R�

��

Krylov/SVD Methods

��

r@

@R

rProperties

• numerical efficiency

• n� 103

• choice of matching moments

Properties

• Stability

• Error bound

• n ≈ 103



Complexity considerations

• Dense problemsMajor cost Balanced Truncation:Compute gramians≈ 30N3 (eigenvalue decomp.)

Perform balancing≈ 25N3 (sing. value decomp.)

Rational Krylov approximation:

Decompose (A− σi E) for k points≈ 23 kN3

Remark : Iterations (Sign, Smith) can accelerate the computation of gramians (esp. on parallel machines)

• Approximate and/or sparse decompositionsMajor cost Balanced Truncation:Compute gramians≈ c1αkN

Perform balancing O(n3)

Rational Krylov approximation:

Iterative solves for (A− σi E)x = b≈ c2kαN, where k = number of expansion points; α = average number of non-zero

elements per row in A, E.


model reduction of large-scale dynamical systemsaca/aca_monopoli_iii.pdf · model reduction of...

Documents