Dealing with Hamiltonian Structure:
Challenges and Successes
David S. Watkins
[email protected]
Department of Mathematics
Washington State University
Cortona, September 2008 – p.
Alternating Pencils
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Example: anisotropic solids, Lamé equations
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Example: anisotropic solids, Lamé equations
(λ2 M + λG + K)v = 0
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Example: anisotropic solids, Lamé equations
(λ2 M + λG + K)v = 0
large, sparse matrices
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Example: anisotropic solids, Lamé equations
(λ2 M + λG + K)v = 0
large, sparse matrices
compute a few smallest eigenvalues
Cortona, September 2008 – p.
Alternating Pencils
(λk Ak + λk−1 Ak−1 + · · · + A0 )v = 0
alternating, even, odd
Example: anisotropic solids, Lamé equations
(λ2 M + λG + K)v = 0
large, sparse matrices
compute a few smallest eigenvalues
symmetry of spectrum
Cortona, September 2008 – p.
Spectrum of an Alternating Pencil
Cortona, September 2008 – p.
Reduction of Order
Cortona, September 2008 – p.
Reduction of Order
(linearization)
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
w = λv,
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
w = λv,
−λM v + M w = 0
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
w = λv,
−λM v + M w = 0
λ
G M
−M 0
v
w
+
K 0
0 M
v
w
=
0
0
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
w = λv,
−λM v + M w = 0
λ
G M
−M 0
v
w
+
K 0
0 M
v
w
=
0
0
structure is preserved
Cortona, September 2008 – p.
Reduction of Order
(linearization) same as for differential equations
w = λv,
−λM v + M w = 0
λ
G M
−M 0
v
w
+
K 0
0 M
v
w
=
0
0
structure is preserved
Mackey/Mackey/Mehl/Mehrmann
Cortona, September 2008 – p.
Factorization
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Factorization
G M
−M 0
=
I 0
1
2G M
T 0 I
−I 0
I 0
1
2G M
Cortona, September 2008 – p.
Factorization
G M
−M 0
=
I 0
1
2G M
J=
T 0 I
−I 0
0 I
−I 0
I 0
1
2G M
Cortona, September 2008 – p.
Factorization
G M
−M 0
=
I 0
1
2G M
J=
T 0 I
−I 0
0 I
−I 0
I 0
1
2G M
N = LT JL
Cortona, September 2008 – p.
Factorization
G M
−M 0
=
I 0
1
2G M
J=
N = LT JL
A − λN ⇒
T 0 I
−I 0
0 I
−I 0
I 0
1
2G M
J T L−T AL−1 − λI
Cortona, September 2008 – p.
Factorization
G M
−M 0
=
I 0
1
2G M
J=
T 0 I
−I 0
0 I
−I 0
I 0
1
2G M
N = LT JL
A − λN ⇒ J T L−T AL−1 − λI
Hamiltonian matrix: (JH)T = JH
Cortona, September 2008 – p.
Factorization
G M
−M 0
=
I 0
1
2G M
J=
T 0 I
−I 0
0 I
−I 0
I 0
1
2G M
N = LT JL
A − λN ⇒ J T L−T AL−1 − λI
Hamiltonian matrix: (JH)T = JH
Hamiltonian matrix ⇔ alternating pencil
Cortona, September 2008 – p.
Special Case
Cortona, September 2008 – p.
Special Case
λ
G M
−M 0
v
w
+
K 0
0 M
v
w
=
0
0
Cortona, September 2008 – p.
Special Case
λ
G M
−M 0
G M
−M 0
=
v
w
+
I 0
1
2G M
K 0
0 M
T 0 I
−I 0
v
w
=
0
0
I 0
1
2G M
Cortona, September 2008 – p.
Special Case
λ
G M
−M 0
v
w
+
K 0
0 M
T v
w
=
0
0
I 0
0 I
I 0
= 1
1
G
M
−I
0
2
2G M
1
I 2G
K
0
I
0
H =J
0 I
0 M −1
− 21 G I
G M
−M 0
Cortona, September 2008 – p.
H =J
I
0
1
2G
I
K
0
0 M −1
I
0
− 21 G I
Cortona, September 2008 – p.
H =J
I
0
1
2G
I
K
0
0 M −1
I
0
− 21 G I
Don’t form H explicitly.
Cortona, September 2008 – p.
H =J
I
0
1
2G
I
K
0
0 M −1
I
0
− 21 G I
Don’t form H explicitly.
Use a Krylov subspace method.
Cortona, September 2008 – p.
H =J
I
0
1
2G
I
K
0
0 M −1
I
0
− 21 G I
Don’t form H explicitly.
Use a Krylov subspace method.
But we want the smallest eigenvalues.
Cortona, September 2008 – p.
H =J
I
0
1
2G
I
K
0
0 M −1
I
0
− 21 G I
Don’t form H explicitly.
Use a Krylov subspace method.
But we want the smallest eigenvalues.
−1
1
I 0
K
0
I −2G
−1
H = 1
JT
0 M
0 I
2G I
Cortona, September 2008 – p.
LQG Control Problem
Cortona, September 2008 – p.
LQG Control Problem
ẋ = Ax + Bu
Cortona, September 2008 – p.
LQG Control Problem
ẋ = Ax + Bu
R ∞ 1 T
1 T
T
I(x, u) = 0 2 x Qx + x Su + 2 u Ru dt
Cortona, September 2008 – p.
LQG Control Problem
ẋ = Ax + Bu
R ∞ 1 T
1 T
T
I(x, u) = 0 2 x Qx + x Su + 2 u Ru dt
R∞ T
L(x, u, µ) = I(x, u) + 0 µ (ẋ − Ax − Bu) dt
Cortona, September 2008 – p.
LQG Control Problem
ẋ = Ax + Bu
R ∞ 1 T
1 T
T
I(x, u) = 0 2 x Qx + x Su + 2 u Ru dt
R∞ T
L(x, u, µ) = I(x, u) + 0 µ (ẋ − Ax − Bu) dt
 
  

T
x
Q −A
S
ẋ
0 I 0
 −I 0 0   µ̇  −  −A
0
−B   µ  = 0
u
S T −B T R
u̇
0 0 0
Cortona, September 2008 – p.
LQG Control Problem
ẋ = Ax + Bu
R ∞ 1 T
1 T
T
I(x, u) = 0 2 x Qx + x Su + 2 u Ru dt
R∞ T
L(x, u, µ) = I(x, u) + 0 µ (ẋ − Ax − Bu) dt
 
  

T
x
Q −A
S
ẋ
0 I 0
 −I 0 0   µ̇  −  −A
0
−B   µ  = 0
u
S T −B T R
u̇
0 0 0
skew-symmetric/symmetric
Cortona, September 2008 – p.
Associated eigenvalue problem
Cortona, September 2008 – p.
Associated eigenvalue problem


 
T


v
Q −A
S
v
0 I 0
λ  −I 0 0   w − −A
0
−B   w  = 0
y
S T −B T R
y
0 0 0
Cortona, September 2008 – p.
Associated eigenvalue problem


 
T


v
Q −A
S
v
0 I 0
λ  −I 0 0   w − −A
0
−B   w  = 0
y
S T −B T R
y
0 0 0
H=
−1 T
−1
T
A − BR S
BR B
Q + SR−1 S T −AT + SR−1 B T
Cortona, September 2008 – p.
Associated eigenvalue problem


 
T


v
Q −A
S
v
0 I 0
λ  −I 0 0   w − −A
0
−B   w  = 0
y
S T −B T R
y
0 0 0
H=
−1 T
−1
T
A − BR S
BR B
Q + SR−1 S T −AT + SR−1 B T
Stable invariant subspace is wanted.
Cortona, September 2008 – p.
Associated eigenvalue problem


 
T


v
Q −A
S
v
0 I 0
λ  −I 0 0   w − −A
0
−B   w  = 0
y
S T −B T R
y
0 0 0
H=
−1 T
−1
T
A − BR S
BR B
Q + SR−1 S T −AT + SR−1 B T
Stable invariant subspace is wanted.
complete eigensystem
Cortona, September 2008 – p.
Working Directly with the Pencil
Cortona, September 2008 – p. 1
Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil
M − λN
Cortona, September 2008 – p. 1
Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil
M − λN
symplectic ⇔ palindromic pencil
G − λGT
Cortona, September 2008 – p. 1
Working Directly with the Pencil
Hamiltonian ⇔ alternating pencil
M − λN
symplectic ⇔ palindromic pencil
G − λGT
Schröder (Ph.D. 2008)
Cortona, September 2008 – p. 1
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. 1
Working with Hamiltonian Matrices
Cortona, September 2008 – p. 1
Working with Hamiltonian Matrices
symplectic matrix:
S T JS = J
Cortona, September 2008 – p. 1
Working with Hamiltonian Matrices
symplectic matrix: S T JS = J
symplectic similarity transformations
Cortona, September 2008 – p. 1
Working with Hamiltonian Matrices
symplectic matrix: S T JS = J
symplectic similarity transformations
orthogonal symplectic transformations
Cortona, September 2008 – p. 1
Working with Hamiltonian Matrices
symplectic matrix: S T JS = J
symplectic similarity transformations
orthogonal symplectic transformations
isotropic subspace:
U T JU = 0
Cortona, September 2008 – p. 1
Working with Hamiltonian Matrices
symplectic matrix: S T JS = J
symplectic similarity transformations
orthogonal symplectic transformations
isotropic subspace: U T JU = 0
isotropy and symplectic matrices
Cortona, September 2008 – p. 1
Difficulty Obtaining
Hessenberg Form
Cortona, September 2008 – p. 1
Difficulty Obtaining
Hessenberg Form
PVL form (1981)
Cortona, September 2008 – p. 1
Difficulty Obtaining
Hessenberg Form
PVL form (1981)
the desired Hessenberg form
Cortona, September 2008 – p. 1
Difficulty Obtaining
Hessenberg Form
PVL form (1981)
the desired Hessenberg form
Byers (1983)
Cortona, September 2008 – p. 1
Difficulty Obtaining
Hessenberg Form
PVL form (1981)
the desired Hessenberg form
Byers (1983)
getting an isotropic Krylov subspace?
Cortona, September 2008 – p. 1
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. 1
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. 1
Skew-Hamiltonian matrices . . .
Cortona, September 2008 – p. 1
Skew-Hamiltonian matrices . . .
. . . are easier
Cortona, September 2008 – p. 1
Skew-Hamiltonian matrices . . .
. . . are easier
skew-Hamiltonian matrix:
(JK)T = −(JK)
Cortona, September 2008 – p. 1
Skew-Hamiltonian matrices . . .
. . . are easier
skew-Hamiltonian matrix:
(JK)T = −(JK)
H2
Cortona, September 2008 – p. 1
Skew-Hamiltonian matrices . . .
. . . are easier
skew-Hamiltonian matrix:
(JK)T = −(JK)
H2
more and bigger invariant subspaces
Cortona, September 2008 – p. 1
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. 1
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. 1
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 H 2
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
H = U R1 V T = V R2 U T
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
H = U R1 V T = V R2 U T
T
S B
−T
B
R1 =
and R2 =
T
0 T
0 −S T
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
H = U R1 V T = V R2 U T
T
S B
−T
B
R1 =
and R2 =
T
0 T
0 −S T
H2
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
H = U R1 V T = V R2 U T
T
S B
−T
B
R1 =
and R2 =
T
0 T
0 −S T
H2
eigenvalues of H
Cortona, September 2008 – p. 1
Symplectic U RV Decomposition
H = U R1 V T = V R2 U T
T
S B
−T
B
R1 =
and R2 =
T
0 T
0 −S T
H2
eigenvalues of H
Benner/Mehrmann/Xu (199X)
Cortona, September 2008 – p. 1
CLM Method
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
span{e1 } invariant under H 2
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
span{e1 } invariant under H 2
⇒ span{e1 , He1 } invariant under H
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
span{e1 } invariant under H 2
⇒ span{e1 , He1 } invariant under H
Extract 1-D isotropic invariant subspace.
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
span{e1 } invariant under H 2
⇒ span{e1 , He1 } invariant under H
Extract 1-D isotropic invariant subspace.
Build an orthogonal symplectic similarity
transformation.
Cortona, September 2008 – p. 1
CLM Method
Chu/Liu/Mehrmann (2004)
H ← U T HU
H 2 has special structure.
span{e1 } invariant under H 2
⇒ 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. 1
Block CLM Method
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM,
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ span{S, HS} invariant under H
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ span{S, HS} invariant under H
Extract k-dimensional isotropic invariant subspace.
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ span{S, HS} invariant under H
Extract k-dimensional isotropic invariant subspace.
This is
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ span{S, HS} invariant under H
Extract k-dimensional isotropic invariant subspace.
This is more robust,
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ span{S, HS} invariant under H
Extract k-dimensional isotropic invariant subspace.
This is more robust, more efficient,
Cortona, September 2008 – p. 1
Block CLM Method
CLM works surprisingly well.
difficulties with clusters
Block CLM, Mehrmann/Schröder/Watkins (2008)
S invariant under H 2
⇒ 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.
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