SLIDE 1
Recent Progress in the Solution of the Hamiltonian Eigenvalue Problem
David S. Watkins
watkins@math.wsu.edu
Department of Mathematics Washington State University
Calais, May 2009 – p. 1
Alternating Pencils
Calais, May 2009 – p. 2
Recent Progress in the Solution of the Hamiltonian Eigenvalue - - PowerPoint PPT Presentation
Recent Progress in the Solution of the Hamiltonian Eigenvalue - - PowerPoint PPT Presentation
Recent Progress in the Solution of the Hamiltonian Eigenvalue Problem David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Calais, May 2009 p. 1 Alternating Pencils Calais, May 2009 p. 2
SLIDE 2
SLIDE 3
Alternating Pencils
(λkAk + λk−1Ak−1 + · · · + A0)v = 0
Calais, May 2009 – p. 2
SLIDE 4
Alternating Pencils
(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd
Calais, May 2009 – p. 2
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Alternating Pencils
(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations
Calais, May 2009 – p. 2
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Alternating Pencils
(λkAk + λk−1Ak−1 + · · · + A0)v = 0 alternating, even, odd Example: anisotropic solids, Lamé equations (λ2M + λG + K)v = 0
Calais, May 2009 – p. 2
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
Calais, May 2009 – p. 2
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
Calais, May 2009 – p. 2
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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
Calais, May 2009 – p. 2
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Spectrum of an Alternating Pencil
Calais, May 2009 – p. 3
SLIDE 11
Reduction of Order
Calais, May 2009 – p. 4
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Reduction of Order
(linearization)
Calais, May 2009 – p. 4
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Reduction of Order
(linearization) same as for differential equations
Calais, May 2009 – p. 4
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Reduction of Order
(linearization) same as for differential equations w = λv,
Calais, May 2009 – p. 4
SLIDE 15
Reduction of Order
(linearization) same as for differential equations w = λv, −λMv + Mw = 0
Calais, May 2009 – p. 4
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
- =
- Calais, May 2009 – p. 4
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
Calais, May 2009 – p. 4
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
Calais, May 2009 – p. 4
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Factorization
Calais, May 2009 – p. 5
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Factorization
- G
M −M
- =
I
1 2G M
T 0 I −I 0 I
1 2G M
- Calais, May 2009 – p. 5
SLIDE 21
Factorization
- G
M −M
- =
I
1 2G M
T 0 I −I 0 I
1 2G M
- J =
- 0 I
−I 0
- Calais, May 2009 – p. 5
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
Calais, May 2009 – p. 5
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
Calais, May 2009 – p. 5
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
Calais, May 2009 – p. 5
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
Calais, May 2009 – p. 5
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Special Case
Calais, May 2009 – p. 6
SLIDE 27
Special Case
λ
- G
M −M v w
- +
K M v w
- =
- Calais, May 2009 – p. 6
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
- Calais, May 2009 – p. 6
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
- Calais, May 2009 – p. 6
SLIDE 30
H = J I
1 2G
I K M −1 I −1
2G I
- Calais, May 2009 – p. 7
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H = J I
1 2G
I K M −1 I −1
2G I
- Don’t form H explicitly.
Calais, May 2009 – p. 7
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.
Calais, May 2009 – p. 7
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.
Calais, May 2009 – p. 7
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
Calais, May 2009 – p. 7
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LQG Control Problem
Calais, May 2009 – p. 8
SLIDE 36
LQG Control Problem
˙ x = Ax + Bu
Calais, May 2009 – p. 8
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LQG Control Problem
˙ x = Ax + Bu I(x, u) = ∞ 1
2xTQx + xTSu + 1 2uTRu
- dt
Calais, May 2009 – p. 8
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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
Calais, May 2009 – p. 8
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
Calais, May 2009 – p. 8
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
Calais, May 2009 – p. 8
SLIDE 41
Associated eigenvalue problem
Calais, May 2009 – p. 9
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
Calais, May 2009 – p. 9
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
- Calais, May 2009 – p. 9
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.
Calais, May 2009 – p. 9
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
Calais, May 2009 – p. 9
SLIDE 46
Working Directly with the Pencil
Calais, May 2009 – p. 10
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Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil M − λN
Calais, May 2009 – p. 10
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Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT
Calais, May 2009 – p. 10
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Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT Schröder (Ph.D. 2008)
Calais, May 2009 – p. 10
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Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λGT Schröder (Ph.D. 2008) Kressner/Schröder/Watkins (2008)
Calais, May 2009 – p. 10
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Working with Hamiltonian Matrices
Calais, May 2009 – p. 11
SLIDE 52
Working with Hamiltonian Matrices
symplectic matrix: STJS = J
Calais, May 2009 – p. 11
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Working with Hamiltonian Matrices
symplectic matrix: STJS = J symplectic similarity transformations
Calais, May 2009 – p. 11
SLIDE 54
Working with Hamiltonian Matrices
symplectic matrix: STJS = J symplectic similarity transformations
- rthogonal symplectic transformations
Calais, May 2009 – p. 11
SLIDE 55
Working with Hamiltonian Matrices
symplectic matrix: STJS = J symplectic similarity transformations
- rthogonal symplectic transformations
isotropic subspace: U TJU = 0
Calais, May 2009 – p. 11
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Working with Hamiltonian Matrices
symplectic matrix: STJS = J symplectic similarity transformations
- rthogonal symplectic transformations
isotropic subspace: U TJU = 0 isotropy and symplectic matrices
Calais, May 2009 – p. 11
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Difficulty Obtaining Hessenberg Form
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981)
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983)
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace?
Calais, May 2009 – p. 12
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Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace? Ammar/Mehrmann (1991)
Calais, May 2009 – p. 12
SLIDE 64
Difficulty Obtaining Hessenberg Form
PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) new ideas needed
Calais, May 2009 – p. 12
SLIDE 65
Skew-Hamiltonian matrices . . .
Calais, May 2009 – p. 13
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Skew-Hamiltonian matrices . . . . . . are easier
Calais, May 2009 – p. 13
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Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK)
Calais, May 2009 – p. 13
SLIDE 68
Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK) H Hamiltonian ⇒ H2 skew Hamiltonian
Calais, May 2009 – p. 13
SLIDE 69
Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK) H Hamiltonian ⇒ H2 skew Hamiltonian more and bigger invariant subspaces
Calais, May 2009 – p. 13
SLIDE 70
Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK) H Hamiltonian ⇒ H2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic.
Calais, May 2009 – p. 13
SLIDE 71
Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK) H Hamiltonian ⇒ H2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form
Calais, May 2009 – p. 13
SLIDE 72
Skew-Hamiltonian matrices . . . . . . are easier
skew-Hamiltonian matrix: (JK)T = −(JK) H Hamiltonian ⇒ H2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form make use of H2
Calais, May 2009 – p. 13
SLIDE 73
Symplectic URV Decomposition
Calais, May 2009 – p. 14
SLIDE 74
Symplectic URV Decomposition
H = UR1V T = V R2U T
Calais, May 2009 – p. 14
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Symplectic URV Decomposition
H = UR1V T = V R2U T R1 = S B 0 T T
- and R2 =
−T BT −ST
- Calais, May 2009 – p. 14
SLIDE 76
Symplectic URV Decomposition
H = UR1V T = V R2U T R1 = S B 0 T T
- and R2 =
−T BT −ST
- H2 = U(R1R2)U T
Calais, May 2009 – p. 14
SLIDE 77
Symplectic URV Decomposition
H = UR1V T = V R2U T R1 = S B 0 T T
- and R2 =
−T BT −ST
- H2 = U(R1R2)U T
R1R2 = −ST Z −(ST)T
- Calais, May 2009 – p. 14
SLIDE 78
Symplectic URV Decomposition
H = UR1V T = V R2U T R1 = S B 0 T T
- and R2 =
−T BT −ST
- H2 = U(R1R2)U T
R1R2 = −ST Z −(ST)T
- eigenvalues of H
Calais, May 2009 – p. 14
SLIDE 79
Symplectic URV Decomposition
H = UR1V T = V R2U T R1 = S B 0 T T
- and R2 =
−T BT −ST
- H2 = U(R1R2)U T
R1R2 = −ST Z −(ST)T
- eigenvalues of H
Benner/Mehrmann/Xu (1998)
Calais, May 2009 – p. 14
SLIDE 80
CLM Method
Calais, May 2009 – p. 15
SLIDE 81
CLM Method
Chu/Liu/Mehrmann (2004)
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CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU
Calais, May 2009 – p. 15
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CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- Calais, May 2009 – p. 15
SLIDE 84
CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- span{e1} invariant under H2
Calais, May 2009 – p. 15
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CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- span{e1} invariant under H2
⇒ span{e1, He1} invariant under H
Calais, May 2009 – p. 15
SLIDE 86
CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- span{e1} invariant under H2
⇒ span{e1, He1} invariant under H Extract 1-d isotropic invariant subspace.
Calais, May 2009 – p. 15
SLIDE 87
CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- span{e1} invariant under H2
⇒ span{e1, He1} invariant under H Extract 1-d isotropic invariant subspace. Build an orthogonal symplectic similarity transformation.
Calais, May 2009 – p. 15
SLIDE 88
CLM Method
Chu/Liu/Mehrmann (2004) H ← U THU H2 = −ST Z −(ST)T
- span{e1} invariant under H2
⇒ span{e1, He1} invariant under H Extract 1-d isotropic invariant subspace. Build an orthogonal symplectic similarity transformation. Deflate. (with form of H2 preserved!)
Calais, May 2009 – p. 15
SLIDE 89
Block CLM Method
Calais, May 2009 – p. 16
SLIDE 90
Block CLM Method
CLM works surprisingly well.
Calais, May 2009 – p. 16
SLIDE 91
Block CLM Method
CLM works surprisingly well. difficulties with clusters
Calais, May 2009 – p. 16
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Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM,
Calais, May 2009 – p. 16
SLIDE 93
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008)
Calais, May 2009 – p. 16
SLIDE 94
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2
Calais, May 2009 – p. 16
SLIDE 95
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H
Calais, May 2009 – p. 16
SLIDE 96
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace.
Calais, May 2009 – p. 16
SLIDE 97
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2k eigenvalues.
Calais, May 2009 – p. 16
SLIDE 98
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2k eigenvalues. This is
Calais, May 2009 – p. 16
SLIDE 99
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2k eigenvalues. This is more robust,
Calais, May 2009 – p. 16
SLIDE 100
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2k eigenvalues. This is more robust, more efficient,
Calais, May 2009 – p. 16
SLIDE 101
Block CLM Method
CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k) invariant under H2 ⇒ span{S, HS} invariant under H Extract k-dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2k eigenvalues. This is more robust, more efficient, but we’re still working on it.
Calais, May 2009 – p. 16
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