A connection between time domain model order reduction and moment - - PowerPoint PPT Presentation

▶
a connection between time domain model order reduction
SMART_READER_LITE
LIVE PREVIEW

A connection between time domain model order reduction and moment - - PowerPoint PPT Presentation

Oct 05, 2022 588 likes •1.14k views

A connection between time domain model order reduction and moment matching Manuela Hund joint with Jens Saak September 2, 2016 Partners: Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

slide-1
SLIDE 1

A connection between time domain model

  • rder reduction and moment matching

Manuela Hund

joint with Jens Saak

September 2, 2016

Partners:

slide-2
SLIDE 2

Introduction

Model order reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Linear time-invariant (LTI) system E ˙ x(t) = A x(t) + B u(t) y(t) = C x(t)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 2/24

slide-3
SLIDE 3

Introduction

Model order reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Linear time-invariant (LTI) system E ˙ x(t) = A x(t) + B u(t) y(t) = C x(t)

Er = V T EV ∈ Rm×m Ar = V T AV ∈ Rm×m Br = V T B ∈ Rm×p Cr = CV ∈ Rq×m

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 2/24

slide-4
SLIDE 4

Introduction

Model order reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Linear time-invariant (LTI) system E ˙ x(t) = A x(t) + B u(t) y(t) = C x(t)

Er = V T EV ∈ Rm×m Ar = V T AV ∈ Rm×m Br = V T B ∈ Rm×p Cr = CV ∈ Rq×m

Reduced LTI system: Er ˙ xr(t) = Ar xr(t)+ Br u(t) yr(t) = Cr xr(t)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 2/24

slide-5
SLIDE 5

Time domain MOR based on

  • rthogonal polynomials

Basic idea

[JIANG/CHEN 2012]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Single-Input Single-Output (SISO) system (p = q = 1): E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t).

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 3/24

slide-6
SLIDE 6

Time domain MOR based on

  • rthogonal polynomials

Basic idea

[JIANG/CHEN 2012]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Single-Input Single-Output (SISO) system (p = q = 1): E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Approximation of state and input:

(∀i : vi ∈ Rn, wi ∈ R, gi : [t0, tf ] → R)

x(t) ≈ xm (t) =

m−1

  • i=0

vigi(t), u(t) ≈ um (t) =

m−1

  • i=1

wi ˙ gi(t). Artificial initial condition: x0 = x(t0) ≈ xm(t0) =

m−1

  • i=0

vigi(t0).

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 3/24

slide-7
SLIDE 7

Time domain MOR based on

  • rthogonal polynomials

Restriction

[HUND 2015]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Single-Input Single-Output (SISO) system (p = q = 1): E ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t). Approximation of state and input:

(∀i : vi ∈ Rn, wi ∈ R, gi : [t0, tf ] → R)

x(t) ≈ xm+1(t) =

m

  • i=1

vigi(t), u(t) ≈ um+1(t) =

m

  • i=1

wi ˙ gi(t). Fixed initial condition: x0 = x(t0) =    . . .    ∈ Rn.

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 3/24

slide-8
SLIDE 8

Time domain MOR based on

  • rthogonal polynomials

Procedure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 1 (Differential recurrence formula)

[HUND 2015]

For three sequenced orthogonal polynomials gi(t), where i ∈ N0, it holds: gn(t) = αn ˙ gn+1(t) + βn ˙ gn(t) + γn ˙ gn−1(t), n = 1, 2, . . . , where αn, βn, γn are diffential recurrence coefficients.

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 4/24

slide-9
SLIDE 9

Time domain MOR based on

  • rthogonal polynomials

Procedure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 1 (Differential recurrence formula)

[HUND 2015]

For three sequenced orthogonal polynomials gi(t), where i ∈ N0, it holds: gn(t) = αn ˙ gn+1(t) + βn ˙ gn(t) + γn ˙ gn−1(t), n = 1, 2, . . . , where αn, βn, γn are diffential recurrence coefficients. application of differential recurrence formula in xm+1(t) ⇒ application to state equation leads to expressions depending on ˙ gi(t)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 4/24

slide-10
SLIDE 10

Time domain MOR based on

  • rthogonal polynomials

Procedure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 1 (Differential recurrence formula)

[HUND 2015]

For three sequenced orthogonal polynomials gi(t), where i ∈ N0, it holds: gn(t) = αn ˙ gn+1(t) + βn ˙ gn(t) + γn ˙ gn−1(t), n = 1, 2, . . . , where αn, βn, γn are diffential recurrence coefficients. application of differential recurrence formula in xm+1(t) ⇒ application to state equation leads to expressions depending on ˙ gi(t) coefficient comparison leads to a linear system of equations Hv = f , where H ∈ Rmn×mn, v ∈ Rmn×1, f ∈ Rmn×1

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 4/24

slide-11
SLIDE 11

Time domain MOR based on

  • rthogonal polynomials

Procedure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 1 (Differential recurrence formula)

[HUND 2015]

For three sequenced orthogonal polynomials gi(t), where i ∈ N0, it holds: gn(t) = αn ˙ gn+1(t) + βn ˙ gn(t) + γn ˙ gn−1(t), n = 1, 2, . . . , where αn, βn, γn are diffential recurrence coefficients. application of differential recurrence formula in xm+1(t) ⇒ application to state equation leads to expressions depending on ˙ gi(t) coefficient comparison leads to a linear system of equations Hv = f , where H ∈ Rmn×mn, v ∈ Rmn×1, f ∈ Rmn×1 determine projection matrix V by orthogonalization of span {v1, . . . , vm}

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 4/24

slide-12
SLIDE 12

Time domain MOR based on

  • rthogonal polynomials

Structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H =                 E − β1A −γ2A · · · · · · · · · −α1A E − β2A −γ3A ... . . . −α2A E − β3A −γ4A ... . . . . . . ... ... ... ... ... . . . . . . ... ... ... ... . . . ... ... ... −γmA · · · · · · · · · −αm−1A E − βmA                 , v =    v1 . . . vm    , f =    Bw1 . . . Bwm    .

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 5/24

slide-13
SLIDE 13

Time domain MOR based on

  • rthogonal polynomials

Structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H =                 E − β1A −γ2A · · · · · · · · · −α1A E − β2A −γ3A ... . . . −α2A E − β3A −γ4A ... . . . . . . ... ... ... ... ... . . . . . . ... ... ... ... . . . ... ... ... −γmA · · · · · · · · · −αm−1A E − βmA                 , v =    v1 . . . vm    , f =    Bw1 . . . Bwm    .

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 5/24

slide-14
SLIDE 14

Contributions

Structure exploitation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H = −                 β1A γ2A · · · · · · · · · α1A β2A γ3A ... . . . α2A β3A γ4A ... . . . . . . ... ... ... ... ... . . . . . . ... ... ... ... . . . ... ... ... γm−1A · · · · · · · · · αm−2A βm−1A                 +           E · · · · · · ... ... . . . . . . ... ... ... . . . . . . ... ... · · · · · · E          

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 6/24

slide-15
SLIDE 15

Contributions

Transformation to a Sylvester equation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds: e.g. [HORN/JOHNSON 1991]

(BT ⊗ A)vec(X) = vec(F) ⇔ AXB = F

Sylvester equation:

Hv = f ⇔ AVO + EVP = F, where O = −tridiag(γ, β, α) ∈ Rm×m, P = Im ∈ Rm×m and α =

  • α1 . . . αm−1
  • , β =
  • β1

· · · βm

  • , γ =
  • γ2

· · · γm

  • .
  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 7/24

slide-16
SLIDE 16

Contributions

Equivalence to moment matching

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 2 (Duality theorem) e.g. [VANDENDORPE 2004]

The columns of the solution of the Sylvester equation AV − EVS = BL form a basis of a rational Krylov subspace KΛ(S)(A, E, B) if and only if (S, L) is observable.

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 8/24

slide-17
SLIDE 17

Contributions

Equivalence to moment matching

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 2 (Duality theorem) e.g. [VANDENDORPE 2004]

The columns of the solution of the Sylvester equation AV − EVS = BL form a basis of a rational Krylov subspace KΛ(S)(A, E, B) if and only if (S, L) is observable.

Theorem 3

[HUND 2015]

time domain MOR ⇔ moment matching for Hermite, Legendre or Chebychev polynomials for Laguerre (see also [EID 2009]) for Jacobi polynomials assuming distinct eigenvalues of (Im, O)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 8/24

slide-18
SLIDE 18

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AVO + EV = F O =            −1 1 · · · · · · −1 1 ... . . . . . . ... ... ... ... . . . . . . ... ... ... . . . ... ... 1 · · · · · · · · · −1            , F =B

  • w1

. . . wm

  • .
  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 9/24

slide-19
SLIDE 19

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AVOO−1 + EVO−1= FO−1 OO−1 =−            −1 1 · · · · · · −1 1 ... . . . . . . ... ... ... ... . . . . . . ... ... ... . . . ... ... 1 · · · · · · · · · −1                       1 · · · · · · · · · · · · 1 ... . . . . . . ... ... . . . . . . ... ... . . . . . . ... ... . . . · · · · · · · · · 1            , F =B

  • w1

. . . wm

  • .
  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 9/24

slide-20
SLIDE 20

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AV + EVO−1= FO−1 OO−1 =−            −1 1 · · · · · · −1 1 ... . . . . . . ... ... ... ... . . . . . . ... ... ... . . . ... ... 1 · · · · · · · · · −1                       1 · · · · · · · · · · · · 1 ... . . . . . . ... ... . . . . . . ... ... . . . . . . ... ... . . . · · · · · · · · · 1            , F =B

  • 1

. . .

  • .
  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 9/24

slide-21
SLIDE 21

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sylvester equation using Laguerre polynomials

AV − EVS = BL, where S =       1 · · · · · · 1 ... . . . . . . ... ... . . . · · · 1       , L = −

  • 1

. . . 1

  • .
  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 10/24

slide-22
SLIDE 22

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sylvester equation using Laguerre polynomials

AV − EVS = BL, where S =       1 · · · · · · 1 ... . . . . . . ... ... . . . · · · 1       , L = −

  • 1

. . . 1

  • .

Necessary: Full rank of observability matrix

Ob(S, L) =

  • LT

(LS)T . . .

  • (LS)m−1TT

!

= m

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 10/24

slide-23
SLIDE 23

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplification caused by the structure of L and S: L = − S(1,:)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 11/24

slide-24
SLIDE 24

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplification caused by the structure of L and S: L = − S(1,:) LS = − S(1,:)S = −S2

(1,:)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 11/24

slide-25
SLIDE 25

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplification caused by the structure of L and S: L = − S(1,:) LS = − S(1,:)S = −S2

(1,:)

. . . LSm−1 = − S(1,:)Sm−1 = −Sm

(1,:)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 11/24

slide-26
SLIDE 26

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplification caused by the structure of L and S: L = − S(1,:) LS = − S(1,:)S = −S2

(1,:)

. . . LSm−1 = − S(1,:)Sm−1 = −Sm

(1,:)

Aim

Computing S(1,:) for different powers

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 11/24

slide-27
SLIDE 27

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rewriting S in terms of binomial coefficients: S =          1 1 1 · · · 1 1 1 · · · 1 . . . ... ... ... . . . . . . ... 1 1 · · · · · · 1          =         

  • 1
  • 2
  • · · ·

m−1

  • 1
  • · · ·

m−2

  • .

. . ... ... ... . . . . . . ...

  • 1
  • · · ·

· · ·

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-28
SLIDE 28

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

n

  • k=0

k l

  • =

n + 1 l + 1

  • ,

for integer l, n ≥ 0 Potentize S : S2 =         

  • 1
  • 2
  • · · ·

m−1

  • 1
  • · · ·

m−2

  • .

. . ... ... ... . . . . . . ...

  • 1
  • · · ·

· · ·

       

2

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-29
SLIDE 29

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

n

  • k=0

k l

  • =

n + 1 l + 1

  • ,

for integer l, n ≥ 0 Potentize S : S2 =         

  • 1
  • 2
  • · · ·

m−1

  • 1
  • · · ·

m−2

  • .

. . ... ... ... . . . . . . ...

  • 1
  • · · ·

· · ·

       

2

=          1

  • 2

1

  • 3

1

  • · · ·

m

1

  • 1
  • 2

1

  • · · ·

m−1

1

  • .

. . ... ... ... . . . . . . ... 1

  • 2

1

  • · · ·

· · · 1

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-30
SLIDE 30

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

  • n
  • =
  • n

n

  • = 1,
  • n

1

  • =
  • n

n − 1

  • = n,
  • n

k

  • =
  • n − 1

k − 1

  • +
  • n − 1

k

  • for integers n, k ≥ 1

Potentize S : S3 =         

  • 1
  • 2
  • · · ·

m−1

  • 1
  • · · ·

m−2

  • .

. . ... ... ... . . . . . . ...

  • 1
  • · · ·

· · ·

       

3

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-31
SLIDE 31

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

  • n
  • =
  • n

n

  • = 1,
  • n

1

  • =
  • n

n − 1

  • = n,
  • n

k

  • =
  • n − 1

k − 1

  • +
  • n − 1

k

  • for integers n, k ≥ 1

Potentize S : S3 =          1 1 1 · · · 1 1 1 · · · 1 . . . ... ... ... . . . . . . ... 1 1 · · · · · · 1                   1

1

  • 2

1

  • 3

1

  • · · ·

m

1

  • 1

1

  • 2

1

  • · · ·

m−1

1

  • .

. . ... ... ... . . . . . . ... 1

1

  • 2

1

  • · · ·

· · · 1

1

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-32
SLIDE 32

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

  • n
  • =
  • n

n

  • = 1,
  • n

1

  • =
  • n

n − 1

  • = n,
  • n

k

  • =
  • n − 1

k − 1

  • +
  • n − 1

k

  • for integers n, k ≥ 1

Potentize S : S3 =          2

  • 3

1

  • 4

2

  • · · ·

m+1

m−1

  • 2
  • 3

1

  • · · ·

m

m−2

  • .

. . ... ... ... . . . . . . ... 2

  • 3

1

  • · · ·

· · · 2

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-33
SLIDE 33

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It holds e.g [KNUTH 1969]

  • n
  • =
  • n

n

  • = 1,
  • n

1

  • =
  • n

n − 1

  • = n,
  • n

k

  • =
  • n − 1

k − 1

  • +
  • n − 1

k

  • for integers n, k ≥ 1

Potentize S : Sm =          m−1

  • m

1

  • m+1

2

  • · · ·

2m−2

m−1

  • m−1
  • m

1

  • · · ·

2m−3

m−2

  • .

. . ... ... ... . . . . . . ... m−1

  • m

1

  • · · ·

· · · m−1

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 12/24

slide-34
SLIDE 34

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Observability matrix: Ob(S, L) = −          S(1,:) S2

(1,:)

S3

(1,:)

. . . Sm

(1,:)

         = −         

  • 1
  • 2
  • · · ·

m−1

  • 1
  • 2

1

  • 3

2

  • · · ·

m

m−1

  • 2
  • 3

1

  • 4

2

  • · · ·

m+1

m−1

  • .

. . . . . . . . ... . . . m−1

  • m

1

  • m+1

2

  • · · ·

2m−2

m−1

       

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 13/24

slide-35
SLIDE 35

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Observability matrix: Ob(S, L) = −           1 1 1 1 · · · 1 1 2 3 4 · · · m 1 3 6 10 · · ·

m(m+1) 2

1 4 10 20 · · ·

m(m+1)(m+2) 6

. . . . . . . . . . . . ... . . . 1 m

m(m+1) 2 m(m+1)(m+2) 6

· · · 2m−2

m−1

        

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 13/24

slide-36
SLIDE 36

Contributions

Proofing equivalence for Laguerre polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Observability matrix: Ob(S, L) = −           1 1 1 1 · · · 1 1 2 3 4 · · · m 1 3 6 10 · · ·

m(m+1) 2

1 4 10 20 · · ·

m(m+1)(m+2) 6

. . . . . . . . . . . . ... . . . 1 m

m(m+1) 2 m(m+1)(m+2) 6

· · · 2m−2

m−1

         The observability matrix Ob(S, L) is the Pascal matrix and has thus determinant 1 (e.g [BRAWER/PIROVINO 1992]). ⇒ Ob(S, L) has full rank

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 13/24

slide-37
SLIDE 37

Experiments

Perform step response with

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

intial state x0 =    . . .    ∈ Rn, starting time t0 = 0, time intervall t ∈ [0, 1], time step τ = 0.001 (in implicit Euler), input u(t) =      , t ∈ [0, 0.1)

1 2 sin

  • π
  • 10t − 3

2

  • + 1

2

, t ∈ [0.1, 0.2) 1 , t ∈ [0.2, 1] .

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 14/24

slide-38
SLIDE 38

Experiments

Example: Triple Chain (n = 1202)

[SAAK 2009]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure: visualized triple chain example

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 15/24

slide-39
SLIDE 39

Experiments

Example: Triple Chain (n = 1202)

[SAAK 2009]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 16/24

slide-40
SLIDE 40

Experiments

Example: Triple Chain (n = 1202)

[SAAK 2009]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2 IRKA-OO

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 16/24

slide-41
SLIDE 41

Experiments

Example: Triple Chain (n = 1202)

[SAAK 2009]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2 IRKA-OO IRKA BT

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 16/24

slide-42
SLIDE 42

Experiments

Example: Triple Chain (n = 1202)

[SAAK 2009]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  • riginal system

Hermite (m = 24) Laguerre Legendre,Chebychev 10−2 10−1 100 101 102 total time [s]

Comparison of total time spent (m = 40)

simulation time reduction time H\f Syl∗ H\f Syl∗ H\f Syl∗

0.44 0.02 0.74 0.13 39.44 0.17 * [BENNER/K¨ OHLER/SAAK 2011]

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 17/24

slide-43
SLIDE 43

Experiments

Example: Penzl’s FOM (n = 1006)

[PENZL 1999]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A =     A1 A2 A3 A4     ∈ R1006×1006

10 20 30 40 Magnitude (dB) 10 1 10 2 10 3

  • 90
  • 45

45 90 Phase (deg) Bode Diagram Frequency (rad/s)

where A1 = −1 100 −100 −1

  • ,

A2 = −1 200 −200 −1

  • ,

A3 = −1 400 −400 −1

  • ,

A4 = diag(−1, . . . , −1000) ∈ R1000×1000, BT = C = [6 . . . 6

6

1 . . . 1

1000

] ∈ R1×1006.

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 18/24

slide-44
SLIDE 44

Experiments

Example: Penzl’s FOM (n = 1006)

[PENZL 1999]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−12 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 19/24

slide-45
SLIDE 45

Experiments

Example: Penzl’s FOM (n = 1006)

[PENZL 1999]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−12 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2 IRKA-OO

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 19/24

slide-46
SLIDE 46

Experiments

Example: Penzl’s FOM (n = 1006)

[PENZL 1999]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 10 15 20 25 30 35 40 10−12 10−9 10−6 10−3 100 103

reduced order m

  • y − yr

y

  • L2

Time domain relative error

Hermite Laguerre Legendre Chebychev1 Chebychev2 IRKA-OO IRKA BT

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 19/24

slide-47
SLIDE 47

Experiments

Example: Penzl’s FOM (n = 1006)

[PENZL 1999]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  • riginal system

Hermite (m = 24) Laguerre Legendre,Chebychev 10−2 10−1 100 101 total time [s]

Comparison of total time spent (m = 40)

10−2 10−1 100 101 simulation time reduction time H\f Syl∗ H\f Syl∗ H\f Syl∗

0.1 0.01 0.1 0.03 0.19 0.05 * [BENNER/K¨ OHLER/SAAK 2011]

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 20/24

slide-48
SLIDE 48

Experiments

Numerical Problems and Remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Problems

Hermite (original approach): H numerically not invertible for m ≥ 25

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 21/24

slide-49
SLIDE 49

Experiments

Numerical Problems and Remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Problems

Hermite (original approach): H numerically not invertible for m ≥ 25

Remark

time domain MOR: expansion points fixed moment matching: expansion points selected per model

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 21/24

slide-50
SLIDE 50

Experiments

Numerical Problems and Remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Problems

Hermite (original approach): H numerically not invertible for m ≥ 25

Remark

time domain MOR: expansion points fixed moment matching: expansion points selected per model

−50 50 −50 50

Re(λ) Im(λ)

Spectrum m = 40 (original) Laguerre Legendre Chebychev1 Chebychev2 −2 2 −500 500

Re(λ) Im(λ)

Spectrum m = 40 (variation)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 21/24

slide-51
SLIDE 51

Conclusions and Outlook

Conclusions

connection between time domain MOR and moment matching identified for classes of polynomials (extending result by [EID 2009] for Laguerre) disadvantage: unknown remainder term estimation

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 22/24

slide-52
SLIDE 52

Conclusions and Outlook

Conclusions

connection between time domain MOR and moment matching identified for classes of polynomials (extending result by [EID 2009] for Laguerre) disadvantage: unknown remainder term estimation

Outlook

second order systems M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t) y(t) = Cx(t) proper error estimations (Fourier basis)

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 22/24

slide-53
SLIDE 53

References I

  • P. Benner, M. K¨
  • hler, and J. Saak, Sparse-dense Sylvester equations in

H2-model order reduction, Preprint MPIMD/11-11, Max Planck Institute Magdeburg, Dec. 2011.

  • R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear

Algebra and its Applications, 174 (1992), pp. 13–23.

  • R. Eid, Time domain model reduction by moment matching, PhD thesis,

Technische Universit¨ at M¨ unchen, 2009.

  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge

University Press, Cambridge, 1991.

  • M. Hund, Zeitbereichs-Modellreduktion und Sylvestergleichungen, Master’s thesis,

Otto-von-Guericke Universit¨ at Magdeburg, 2015. Y.-L. Jiang and H.-B. Chen, Time domain model order reduction of general

  • rthogonal polynomials for linear input-output systems, IEEE Trans. Automat.

Control, 57 (2012), pp. 330–343.

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 23/24

slide-54
SLIDE 54

References II

  • D. E. Knuth, The art of computer programming. Vol. 1: Fundamental

algorithms, Second printing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969.

  • T. Penzl, Algorithms for model reduction of large dynamical systems, Technical

report SFB393/99-40, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, FRG, 1999.

  • J. Saak, Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in

PDE Control and Model Order Reduction, Dissertation, TU Chemnitz, July 2009. available from http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642.

  • A. Vandendorpe, Model reduction of linear systems, an interpolation point of

view, PhD thesis, Universit´ e Catholique De Louvain, 2004.

Thank you for your attention!

  • M. Hund, hund@mpi-magdeburg.mpg.de

A connection between time domain MOR and moment matching 24/24