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Dealing with Hamiltonian Structure: Challenges and Successes David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Cortona, September 2008 p. 1 Alternating Pencils Cortona, September 2008 p. 2

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SLIDE 1

Dealing with Hamiltonian Structure: Challenges and Successes

David S. Watkins

watkins@math.wsu.edu

Department of Mathematics Washington State University

Cortona, September 2008 – p. 1

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SLIDE 2

Alternating Pencils

Cortona, September 2008 – p. 2

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SLIDE 3

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0

Cortona, September 2008 – p. 2

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SLIDE 4

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd

Cortona, September 2008 – p. 2

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SLIDE 5

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations

Cortona, September 2008 – p. 2

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SLIDE 6

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations (λ2M + λG + K)v = 0

Cortona, September 2008 – p. 2

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SLIDE 7

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations (λ2M + λG + K)v = 0 large, sparse matrices

Cortona, September 2008 – p. 2

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SLIDE 8

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations (λ2M + λG + K)v = 0 large, sparse matrices compute a few smallest eigenvalues

Cortona, September 2008 – p. 2

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SLIDE 9

Alternating Pencils

(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations (λ2M + λG + K)v = 0 large, sparse matrices compute a few smallest eigenvalues symmetry of spectrum

Cortona, September 2008 – p. 2

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SLIDE 10

Spectrum of an Alternating Pencil

Cortona, September 2008 – p. 3

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SLIDE 11

Reduction of Order

Cortona, September 2008 – p. 4

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SLIDE 12

Reduction of Order

(linearization)

Cortona, September 2008 – p. 4

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SLIDE 13

Reduction of Order

(linearization) same as for differential equations

Cortona, September 2008 – p. 4

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SLIDE 14

Reduction of Order

(linearization) same as for differential equations w = λv,

Cortona, September 2008 – p. 4

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SLIDE 15

Reduction of Order

(linearization) same as for differential equations w = λv, −λMv + Mw = 0

Cortona, September 2008 – p. 4

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SLIDE 16

Reduction of Order

(linearization) same as for differential equations w = λv, −λMv + Mw = 0 λ

  • G

M −M v w

  • +

K M v w

  • =
  • Cortona, September 2008 – p. 4
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SLIDE 17

Reduction of Order

(linearization) same as for differential equations w = λv, −λMv + Mw = 0 λ

  • G

M −M v w

  • +

K M v w

  • =
  • structure is preserved

Cortona, September 2008 – p. 4

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SLIDE 18

Reduction of Order

(linearization) same as for differential equations w = λv, −λMv + Mw = 0 λ

  • G

M −M v w

  • +

K M v w

  • =
  • structure is preserved

Mackey/Mackey/Mehl/Mehrmann

Cortona, September 2008 – p. 4

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SLIDE 19

Factorization

Cortona, September 2008 – p. 5

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SLIDE 20

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • Cortona, September 2008 – p. 5
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SLIDE 21

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • J =
  • 0 I

−I 0

  • Cortona, September 2008 – p. 5
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SLIDE 22

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • J =
  • 0 I

−I 0

  • N = LTJL

Cortona, September 2008 – p. 5

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SLIDE 23

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • J =
  • 0 I

−I 0

  • N = LTJL

A − λN ⇒ JTL−TAL−1 − λI

Cortona, September 2008 – p. 5

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SLIDE 24

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • J =
  • 0 I

−I 0

  • N = LTJL

A − λN ⇒ JTL−TAL−1 − λI Hamiltonian matrix: (JH)T = JH

Cortona, September 2008 – p. 5

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SLIDE 25

Factorization

  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • J =
  • 0 I

−I 0

  • N = LTJL

A − λN ⇒ JTL−TAL−1 − λI Hamiltonian matrix: (JH)T = JH Hamiltonian matrix ⇔ alternating pencil

Cortona, September 2008 – p. 5

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SLIDE 26

Special Case

Cortona, September 2008 – p. 6

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SLIDE 27

Special Case

λ

  • G

M −M v w

  • +

K M v w

  • =
  • Cortona, September 2008 – p. 6
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SLIDE 28

Special Case

λ

  • G

M −M v w

  • +

K M v w

  • =
  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • Cortona, September 2008 – p. 6
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SLIDE 29

Special Case

λ

  • G

M −M v w

  • +

K M v w

  • =
  • G

M −M

  • =

I

1 2G M

T 0 I −I 0 I

1 2G M

  • H = J

I

1 2G

I K M −1 I −1

2G I

  • Cortona, September 2008 – p. 6
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SLIDE 30

H = J I

1 2G

I K M −1 I −1

2G I

  • Cortona, September 2008 – p. 7
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SLIDE 31

H = J I

1 2G

I K M −1 I −1

2G I

  • Don’t form H explicitly.

Cortona, September 2008 – p. 7

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SLIDE 32

H = J I

1 2G

I K M −1 I −1

2G I

  • Don’t form H explicitly.

Use a Krylov subspace method.

Cortona, September 2008 – p. 7

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SLIDE 33

H = J I

1 2G

I K M −1 I −1

2G I

  • Don’t form H explicitly.

Use a Krylov subspace method. But we want the smallest eigenvalues.

Cortona, September 2008 – p. 7

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SLIDE 34

H = J I

1 2G

I K M −1 I −1

2G I

  • Don’t form H explicitly.

Use a Krylov subspace method. But we want the smallest eigenvalues. H−1 = I

1 2G I

K−1 M I −1

2G

I

  • JT

Cortona, September 2008 – p. 7

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SLIDE 35

LQG Control Problem

Cortona, September 2008 – p. 8

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SLIDE 36

LQG Control Problem

˙ x = Ax + Bu

Cortona, September 2008 – p. 8

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SLIDE 37

LQG Control Problem

˙ x = Ax + Bu I(x, u) = ∞ 1

2xTQx + xTSu + 1 2uTRu

  • dt

Cortona, September 2008 – p. 8

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SLIDE 38

LQG Control Problem

˙ x = Ax + Bu I(x, u) = ∞ 1

2xTQx + xTSu + 1 2uTRu

  • dt

L(x, u, µ) = I(x, u) + ∞

0 µT( ˙

x − Ax − Bu) dt

Cortona, September 2008 – p. 8

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SLIDE 39

LQG Control Problem

˙ x = Ax + Bu I(x, u) = ∞ 1

2xTQx + xTSu + 1 2uTRu

  • dt

L(x, u, µ) = I(x, u) + ∞

0 µT( ˙

x − Ax − Bu) dt   0 I 0 −I 0 0 0 0 0     ˙ x ˙ µ ˙ u   −   Q −AT S −A −B ST −BT R     x µ u   = 0

Cortona, September 2008 – p. 8

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SLIDE 40

LQG Control Problem

˙ x = Ax + Bu I(x, u) = ∞ 1

2xTQx + xTSu + 1 2uTRu

  • dt

L(x, u, µ) = I(x, u) + ∞

0 µT( ˙

x − Ax − Bu) dt   0 I 0 −I 0 0 0 0 0     ˙ x ˙ µ ˙ u   −   Q −AT S −A −B ST −BT R     x µ u   = 0 skew-symmetric/symmetric

Cortona, September 2008 – p. 8

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SLIDE 41

Associated eigenvalue problem

Cortona, September 2008 – p. 9

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SLIDE 42

Associated eigenvalue problem

λ   0 I 0 −I 0 0 0 0 0     v w y  −   Q −AT S −A −B ST −BT R     v w y   = 0

Cortona, September 2008 – p. 9

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SLIDE 43

Associated eigenvalue problem

λ   0 I 0 −I 0 0 0 0 0     v w y  −   Q −AT S −A −B ST −BT R     v w y   = 0 H = A − BR−1ST BR−1BT Q + SR−1ST −AT + SR−1BT

  • Cortona, September 2008 – p. 9
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SLIDE 44

Associated eigenvalue problem

λ   0 I 0 −I 0 0 0 0 0     v w y  −   Q −AT S −A −B ST −BT R     v w y   = 0 H = A − BR−1ST BR−1BT Q + SR−1ST −AT + SR−1BT

  • Stable invariant subspace is wanted.

Cortona, September 2008 – p. 9

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SLIDE 45

Associated eigenvalue problem

λ   0 I 0 −I 0 0 0 0 0     v w y  −   Q −AT S −A −B ST −BT R     v w y   = 0 H = A − BR−1ST BR−1BT Q + SR−1ST −AT + SR−1BT

  • Stable invariant subspace is wanted.

complete eigensystem

Cortona, September 2008 – p. 9

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SLIDE 46

Working Directly with the Pencil

Cortona, September 2008 – p. 10

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SLIDE 47

Working Directly with the Pencil

Hamiltonian ⇔ alternating pencil M − λN

Cortona, September 2008 – p. 10

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SLIDE 48

Working Directly with the Pencil

Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT

Cortona, September 2008 – p. 10

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SLIDE 49

Working Directly with the Pencil

Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT Schröder (Ph.D. 2008)

Cortona, September 2008 – p. 10

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SLIDE 50

Working Directly with the Pencil

Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT Schröder (Ph.D. 2008) Kressner/Schröder/Watkins (2008)

Cortona, September 2008 – p. 10

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SLIDE 51

Working with Hamiltonian Matrices

Cortona, September 2008 – p. 11

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SLIDE 52

Working with Hamiltonian Matrices

symplectic matrix: STJS = J

Cortona, September 2008 – p. 11

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SLIDE 53

Working with Hamiltonian Matrices

symplectic matrix: STJS = J symplectic similarity transformations

Cortona, September 2008 – p. 11

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SLIDE 54

Working with Hamiltonian Matrices

symplectic matrix: STJS = J symplectic similarity transformations

  • rthogonal symplectic transformations

Cortona, September 2008 – p. 11

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SLIDE 55

Working with Hamiltonian Matrices

symplectic matrix: STJS = J symplectic similarity transformations

  • rthogonal symplectic transformations

isotropic subspace: U TJU = 0

Cortona, September 2008 – p. 11

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SLIDE 56

Working with Hamiltonian Matrices

symplectic matrix: STJS = J symplectic similarity transformations

  • rthogonal symplectic transformations

isotropic subspace: U TJU = 0 isotropy and symplectic matrices

Cortona, September 2008 – p. 11

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SLIDE 57

Difficulty Obtaining Hessenberg Form

Cortona, September 2008 – p. 12

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SLIDE 58

Difficulty Obtaining Hessenberg Form

PVL form (1981)

Cortona, September 2008 – p. 12

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SLIDE 59

Difficulty Obtaining Hessenberg Form

PVL form (1981) the desired Hessenberg form

Cortona, September 2008 – p. 12

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SLIDE 60

Difficulty Obtaining Hessenberg Form

PVL form (1981) the desired Hessenberg form Byers (1983)

Cortona, September 2008 – p. 12

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SLIDE 61

Difficulty Obtaining Hessenberg Form

PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace?

Cortona, September 2008 – p. 12

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SLIDE 62

Difficulty Obtaining Hessenberg Form

PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace? Ammar/Mehrmann (1991)

Cortona, September 2008 – p. 12

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SLIDE 63

Difficulty Obtaining Hessenberg Form

PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) new ideas needed

Cortona, September 2008 – p. 12

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SLIDE 64

Skew-Hamiltonian matrices . . .

Cortona, September 2008 – p. 13

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SLIDE 65

Skew-Hamiltonian matrices . . . . . . are easier

Cortona, September 2008 – p. 13

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SLIDE 66

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK)

Cortona, September 2008 – p. 13

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SLIDE 67

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK) H2

Cortona, September 2008 – p. 13

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SLIDE 68

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK) H2 more and bigger invariant subspaces

Cortona, September 2008 – p. 13

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SLIDE 69

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK) H2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic.

Cortona, September 2008 – p. 13

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SLIDE 70

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK) H2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form

Cortona, September 2008 – p. 13

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SLIDE 71

Skew-Hamiltonian matrices . . . . . . are easier

skew-Hamiltonian matrix: (JK)T = −(JK) H2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form make use of H2

Cortona, September 2008 – p. 13

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SLIDE 72

Symplectic URV Decomposition

Cortona, September 2008 – p. 14

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SLIDE 73

Symplectic URV Decomposition

H = UR1V T = V R2U T

Cortona, September 2008 – p. 14

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SLIDE 74

Symplectic URV Decomposition

H = UR1V T = V R2U T R1 = S B 0 T T

  • and R2 =

−T BT −ST

  • Cortona, September 2008 – p. 14
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SLIDE 75

Symplectic URV Decomposition

H = UR1V T = V R2U T R1 = S B 0 T T

  • and R2 =

−T BT −ST

  • H2

Cortona, September 2008 – p. 14

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SLIDE 76

Symplectic URV Decomposition

H = UR1V T = V R2U T R1 = S B 0 T T

  • and R2 =

−T BT −ST

  • H2

eigenvalues of H

Cortona, September 2008 – p. 14

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SLIDE 77

Symplectic URV Decomposition

H = UR1V T = V R2U T R1 = S B 0 T T

  • and R2 =

−T BT −ST

  • H2

eigenvalues of H Benner/Mehrmann/Xu (199X)

Cortona, September 2008 – p. 14

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SLIDE 78

CLM Method

Cortona, September 2008 – p. 15

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SLIDE 79

CLM Method

Chu/Liu/Mehrmann (2004)

Cortona, September 2008 – p. 15

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SLIDE 80

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU

Cortona, September 2008 – p. 15

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SLIDE 81

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure.

Cortona, September 2008 – p. 15

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SLIDE 82

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure. span{e1} invariant under H2

Cortona, September 2008 – p. 15

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SLIDE 83

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure. span{e1} invariant under H2 ⇒ span{e1, He1} invariant under H

Cortona, September 2008 – p. 15

slide-84
SLIDE 84

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure. span{e1} invariant under H2 ⇒ span{e1, He1} invariant under H Extract 1-D isotropic invariant subspace.

Cortona, September 2008 – p. 15

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SLIDE 85

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure. span{e1} invariant under H2 ⇒ span{e1, He1} invariant under H Extract 1-D isotropic invariant subspace. Build an orthogonal symplectic similarity transformation.

Cortona, September 2008 – p. 15

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SLIDE 86

CLM Method

Chu/Liu/Mehrmann (2004) H ← U THU H2 has special structure. span{e1} invariant under H2 ⇒ span{e1, He1} invariant under H Extract 1-D isotropic invariant subspace. Build an orthogonal symplectic similarity transformation. Deflate. (many details skipped)

Cortona, September 2008 – p. 15

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SLIDE 87

Block CLM Method

Cortona, September 2008 – p. 16

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SLIDE 88

Block CLM Method

CLM works surprisingly well.

Cortona, September 2008 – p. 16

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SLIDE 89

Block CLM Method

CLM works surprisingly well. difficulties with clusters

Cortona, September 2008 – p. 16

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SLIDE 90

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM,

Cortona, September 2008 – p. 16

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SLIDE 91

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008)

Cortona, September 2008 – p. 16

slide-92
SLIDE 92

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2

Cortona, September 2008 – p. 16

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SLIDE 93

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H

Cortona, September 2008 – p. 16

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SLIDE 94

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace.

Cortona, September 2008 – p. 16

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SLIDE 95

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. This is

Cortona, September 2008 – p. 16

slide-96
SLIDE 96

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. This is more robust,

Cortona, September 2008 – p. 16

slide-97
SLIDE 97

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. This is more robust, more efficient,

Cortona, September 2008 – p. 16

slide-98
SLIDE 98

Block CLM Method

CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. This is more robust, more efficient, but we’re still working on it.

Cortona, September 2008 – p. 16

slide-99
SLIDE 99

Cortona, September 2008 – p. 17

slide-100
SLIDE 100

Cortona, September 2008 – p. 17

slide-101
SLIDE 101

Cortona, September 2008 – p. 17

slide-102
SLIDE 102

Cortona, September 2008 – p. 17

slide-103
SLIDE 103

Cortona, September 2008 – p. 17

slide-104
SLIDE 104

Cortona, September 2008 – p. 17

slide-105
SLIDE 105

Cortona, September 2008 – p. 18

slide-106
SLIDE 106

Cortona, September 2008 – p. 18

slide-107
SLIDE 107

Cortona, September 2008 – p. 18

slide-108
SLIDE 108

Cortona, September 2008 – p. 18

slide-109
SLIDE 109

Cortona, September 2008 – p. 18

slide-110
SLIDE 110

Cortona, September 2008 – p. 18