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Solving large Hamiltonian eigenvalue problems David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Adaptivity and Beyond, Vancouver, August 2005 p.1 Some Collaborators Adaptivity and Beyond,

slide-1
SLIDE 1

Solving large Hamiltonian eigenvalue problems

David S. Watkins

watkins@math.wsu.edu

Department of Mathematics Washington State University

Adaptivity and Beyond, Vancouver, August 2005 – p.1

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SLIDE 2

Some Collaborators

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 3

Some Collaborators

Volker Mehrmann

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 4

Some Collaborators

Volker Mehrmann Thomas Apel

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 5

Some Collaborators

Volker Mehrmann Thomas Apel Peter Benner

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 6

Some Collaborators

Volker Mehrmann Thomas Apel Peter Benner Heike Faßbender

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 7

Some Collaborators

Volker Mehrmann Thomas Apel Peter Benner Heike Faßbender . . .

Adaptivity and Beyond, Vancouver, August 2005 – p.2

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SLIDE 8

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials)

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 9

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 10

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates)

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 11

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable.

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 12

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable. Get quadratic eigenvalue problem.

(λ2M + λG + K)v = 0 M∗ = M > 0 G∗ = −G K∗ = K < 0

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 13

Problem: Linear Elasticity

Elastic Deformation (3D, anisotropic, composite materials) Singularities at cracks, interfaces Lamé Equations (PDE, spherical coordinates) Separate radial variable. Get quadratic eigenvalue problem.

(λ2M + λG + K)v = 0 M∗ = M > 0 G∗ = −G K∗ = K < 0

solve numerically

Adaptivity and Beyond, Vancouver, August 2005 – p.3

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SLIDE 14

Numerical Solution

Adaptivity and Beyond, Vancouver, August 2005 – p.4

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SLIDE 15

Numerical Solution

Discretize using finite elements

Adaptivity and Beyond, Vancouver, August 2005 – p.4

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SLIDE 16

Numerical Solution

Discretize using finite elements

(λ2M + λG + K)v = 0 MT = M > 0 GT = −G KT = K < 0

Adaptivity and Beyond, Vancouver, August 2005 – p.4

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SLIDE 17

Numerical Solution

Discretize using finite elements

(λ2M + λG + K)v = 0 MT = M > 0 GT = −G KT = K < 0

matrix quadratic eigenvalue problem (large, sparse)

Adaptivity and Beyond, Vancouver, August 2005 – p.4

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SLIDE 18

Numerical Solution

Discretize using finite elements

(λ2M + λG + K)v = 0 MT = M > 0 GT = −G KT = K < 0

matrix quadratic eigenvalue problem (large, sparse) Find few smallest eigenvalues (and corresponding eigenvectors).

Adaptivity and Beyond, Vancouver, August 2005 – p.4

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SLIDE 19

Other Applications

Adaptivity and Beyond, Vancouver, August 2005 – p.5

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SLIDE 20

Other Applications

Gyroscopic systems

λ2M + λG + K

Adaptivity and Beyond, Vancouver, August 2005 – p.5

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SLIDE 21

Other Applications

Gyroscopic systems

λ2M + λG + K

Quadratic regulator (optimal control)

  • CTC

−AT −A −BBT

  • − λ
  • ET

−E

  • symmetric/skew-symmetric

Adaptivity and Beyond, Vancouver, August 2005 – p.5

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SLIDE 22

Other Applications

Gyroscopic systems

λ2M + λG + K

Quadratic regulator (optimal control)

  • CTC

−AT −A −BBT

  • − λ
  • ET

−E

  • symmetric/skew-symmetric

Higher-order systems

λnAn + λn−1An−1 + · · · + λA1 + A0

Adaptivity and Beyond, Vancouver, August 2005 – p.5

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SLIDE 23

Hamiltonian Structure

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 24

Hamiltonian Structure

Our matrices are real.

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 25

Hamiltonian Structure

Our matrices are real.

λ, λ, −λ, −λ occur together.

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 26

Hamiltonian Structure

Our matrices are real.

λ, λ, −λ, −λ occur together.

spectra of Hamiltonian matrices

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 27

Hamiltonian Structure

Our matrices are real.

λ, λ, −λ, −λ occur together.

spectra of Hamiltonian matrices

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 28

Hamiltonian Structure

Our matrices are real.

λ, λ, −λ, −λ occur together.

spectra of Hamiltonian matrices

Adaptivity and Beyond, Vancouver, August 2005 – p.6

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SLIDE 29

Hamiltonian Matrices

Adaptivity and Beyond, Vancouver, August 2005 – p.7

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SLIDE 30

Hamiltonian Matrices

H ∈ R2n×2n

Adaptivity and Beyond, Vancouver, August 2005 – p.7

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SLIDE 31

Hamiltonian Matrices

H ∈ R2n×2n J =

  • I

−I

  • ∈ R2n×2n

Adaptivity and Beyond, Vancouver, August 2005 – p.7

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SLIDE 32

Hamiltonian Matrices

H ∈ R2n×2n J =

  • I

−I

  • ∈ R2n×2n

H is Hamiltonian iff JH is symmetric.

Adaptivity and Beyond, Vancouver, August 2005 – p.7

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SLIDE 33

Hamiltonian Matrices

H ∈ R2n×2n J =

  • I

−I

  • ∈ R2n×2n

H is Hamiltonian iff JH is symmetric. H =

  • A

K N −AT

  • ,

where K = KT and N = NT

Adaptivity and Beyond, Vancouver, August 2005 – p.7

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SLIDE 34

Reduction of Order

λ2Mv + λGv + Kv = 0

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 35

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv,

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 36

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv, −Mw = −λMv

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 37

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv, −Mw = −λMv

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 38

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv, −Mw = −λMv

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

Ax − λNx = 0

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 39

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv, −Mw = −λMv

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

Ax − λNx = 0

symmetric/skew-symmetric

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 40

Reduction of Order

λ2Mv + λGv + Kv = 0 w = λv, −Mw = −λMv

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

Ax − λNx = 0

symmetric/skew-symmetric Structure has been preserved.

Adaptivity and Beyond, Vancouver, August 2005 – p.8

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SLIDE 41

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 42

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.9
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SLIDE 43

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 44

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Transform:

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 45

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Transform:

A − λRT JR

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 46

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Transform:

A − λRT JR R−TAR−1 − λJ

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 47

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Transform:

A − λRT JR R−TAR−1 − λJ J−1R−T AR−1 − λI

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 48

Reduction to Hamiltonian Matrix

A − λN

(symmetric/skew-symmetric)

N = RT JR

  • J =
  • I

−I

  • always possible, sometimes easy

Transform:

A − λRT JR R−TAR−1 − λJ J−1R−T AR−1 − λI H = J−1R−T AR−1 is Hamiltonian.

Adaptivity and Beyond, Vancouver, August 2005 – p.9

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SLIDE 49

Example (our application)

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

Adaptivity and Beyond, Vancouver, August 2005 – p.10

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SLIDE 50

Example (our application)

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

N =

  • G

M −M

  • Adaptivity and Beyond, Vancouver, August 2005 – p.10
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SLIDE 51

Example (our application)

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

N =

  • G

M −M

  • N = RT JR =

Adaptivity and Beyond, Vancouver, August 2005 – p.10

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SLIDE 52

Example (our application)

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

N =

  • G

M −M

  • N = RT JR =
  • I

−1

2G

M I −I I

1 2G

M

  • Adaptivity and Beyond, Vancouver, August 2005 – p.10
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SLIDE 53

Example (our application)

  • −K

−M v w

  • − λ
  • G

M −M v w

  • = 0

N =

  • G

M −M

  • N = RT JR =
  • I

−1

2G

M I −I I

1 2G

M

  • cost: zero flops

Adaptivity and Beyond, Vancouver, August 2005 – p.10

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SLIDE 54

H = J−1R−T AR−1

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 55

H = J−1R−T AR−1

After some algebra . . .

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 56

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.11
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SLIDE 57

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Do not form the product explicitly

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 58

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Do not form the product explicitly

Krylov subspace methods

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 59

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Do not form the product explicitly

Krylov subspace methods We just need to apply the operator.

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 60

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Do not form the product explicitly

Krylov subspace methods We just need to apply the operator.

M = LLT

(done once)

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 61

H = J−1R−T AR−1

After some algebra . . .

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Do not form the product explicitly

Krylov subspace methods We just need to apply the operator.

M = LLT

(done once) backsolve

Adaptivity and Beyond, Vancouver, August 2005 – p.11

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SLIDE 62

However, . . .

Adaptivity and Beyond, Vancouver, August 2005 – p.12

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SLIDE 63

However, . . .

Adaptivity and Beyond, Vancouver, August 2005 – p.12

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SLIDE 64

However, . . . . . . we really want H−1.

Adaptivity and Beyond, Vancouver, August 2005 – p.12

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SLIDE 65

However, . . . . . . we really want H−1.

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.12
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SLIDE 66

However, . . . . . . we really want H−1.

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • H−1 =
  • I

1 2G

I (−K)−1 M I

1 2G

I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.12
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SLIDE 67

However, . . . . . . we really want H−1.

H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • H−1 =
  • I

1 2G

I (−K)−1 M I

1 2G

I

  • −K = LLT

Adaptivity and Beyond, Vancouver, August 2005 – p.12

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SLIDE 68

Shift and Invert?

Adaptivity and Beyond, Vancouver, August 2005 – p.13

slide-69
SLIDE 69

Shift and Invert?

(H − τI)−1

Adaptivity and Beyond, Vancouver, August 2005 – p.13

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SLIDE 70

Shift and Invert?

(H − τI)−1 is not Hamiltonian

Adaptivity and Beyond, Vancouver, August 2005 – p.13

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SLIDE 71

Shift and Invert?

(H − τI)−1 is not Hamiltonian

Structure is lost.

Adaptivity and Beyond, Vancouver, August 2005 – p.13

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SLIDE 72

Shift and Invert?

(H − τI)−1 is not Hamiltonian

Structure is lost. How can we recover it?

Adaptivity and Beyond, Vancouver, August 2005 – p.13

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SLIDE 73

Exploitable Structures

Adaptivity and Beyond, Vancouver, August 2005 – p.14

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SLIDE 74

Exploitable Structures

symplectic (first idea)

(H − τI)−1(H + τI)

Adaptivity and Beyond, Vancouver, August 2005 – p.14

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SLIDE 75

Exploitable Structures

symplectic (first idea)

(H − τI)−1(H + τI)

skew-Hamiltonian (easiest?)

H−2

Adaptivity and Beyond, Vancouver, August 2005 – p.14

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SLIDE 76

Exploitable Structures

symplectic (first idea)

(H − τI)−1(H + τI)

skew-Hamiltonian (easiest?)

H−2 (H − τI)−1(H + τI)−1

Adaptivity and Beyond, Vancouver, August 2005 – p.14

slide-77
SLIDE 77

Exploitable Structures

symplectic (first idea)

(H − τI)−1(H + τI)

skew-Hamiltonian (easiest?)

H−2 (H − τI)−1(H + τI)−1

Hamiltonian

H−1

Adaptivity and Beyond, Vancouver, August 2005 – p.14

slide-78
SLIDE 78

Exploitable Structures

symplectic (first idea)

(H − τI)−1(H + τI)

skew-Hamiltonian (easiest?)

H−2 (H − τI)−1(H + τI)−1

Hamiltonian

H−1 H−1(H − τI)−1(H + τI)−1

Adaptivity and Beyond, Vancouver, August 2005 – p.14

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SLIDE 79

Structured Lanczos Processes

Adaptivity and Beyond, Vancouver, August 2005 – p.15

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SLIDE 80

Structured Lanczos Processes Unsymmetric Lanczos Process

uk+1bkdk = Buk − ukakdk − uk−1bk−1dk wk+1dk+1bk = BT wk − wkdkak − wk−1dk−1bk−1

Adaptivity and Beyond, Vancouver, August 2005 – p.15

slide-81
SLIDE 81

Hamiltonian Lanczos Process

uk+1bk+1 = Hvk − ukak − uk−1bk−1 vk+1dk+1 = Huk+1

Adaptivity and Beyond, Vancouver, August 2005 – p.16

slide-82
SLIDE 82

Symplectic Lanczos Process

vk+1dk+1bk = Svk − vkdkak − vk−1dk−1bk−1 + ukdk uk+1dk+1 = S−1vk+1

Adaptivity and Beyond, Vancouver, August 2005 – p.17

slide-83
SLIDE 83

Symplectic Lanczos Process

vk+1dk+1bk = Svk − vkdkak − vk−1dk−1bk−1 + ukdk uk+1dk+1 = S−1vk+1

many interesting relationships

Adaptivity and Beyond, Vancouver, August 2005 – p.17

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SLIDE 84

Implicit Restarts

Adaptivity and Beyond, Vancouver, August 2005 – p.18

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SLIDE 85

Implicit Restarts

short Lanczos runs (breakdowns!!, no look-ahead)

Adaptivity and Beyond, Vancouver, August 2005 – p.18

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SLIDE 86

Implicit Restarts

short Lanczos runs (breakdowns!!, no look-ahead) Restart implicitly as in IRA (Sorensen 1991), ARPACK

Adaptivity and Beyond, Vancouver, August 2005 – p.18

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SLIDE 87

Implicit Restarts

short Lanczos runs (breakdowns!!, no look-ahead) Restart implicitly as in IRA (Sorensen 1991), ARPACK Restart with HR, not QR

Adaptivity and Beyond, Vancouver, August 2005 – p.18

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SLIDE 88

Remarks on Stability

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-89
SLIDE 89

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable.

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-90
SLIDE 90

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur.

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-91
SLIDE 91

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything?

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-92
SLIDE 92

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-93
SLIDE 93

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-94
SLIDE 94

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals condition numbers for eigenvalues

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-95
SLIDE 95

Remarks on Stability

Both Hamiltonian and symplectic Lanczos processes are potentially unstable. Breakdowns can occur. Are the answers worth anything? right and left eigenvectors residuals condition numbers for eigenvalues Don’t skip these tests.

Adaptivity and Beyond, Vancouver, August 2005 – p.19

slide-96
SLIDE 96

Example

Adaptivity and Beyond, Vancouver, August 2005 – p.20

slide-97
SLIDE 97

Example

λ2Mv + λGv + Kv = 0

Adaptivity and Beyond, Vancouver, August 2005 – p.20

slide-98
SLIDE 98

Example

λ2Mv + λGv + Kv = 0 n = 3423

Adaptivity and Beyond, Vancouver, August 2005 – p.20

slide-99
SLIDE 99

Example

λ2Mv + λGv + Kv = 0 n = 3423 H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.20
slide-100
SLIDE 100

Example

λ2Mv + λGv + Kv = 0 n = 3423 H =

  • I

−1

2G

I M−1 −K I −1

2G

I

  • H−1

=

  • I

1 2G

I (−K)−1 M I

1 2G

I

  • Adaptivity and Beyond, Vancouver, August 2005 – p.20
slide-101
SLIDE 101

Compare various approaches:

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-102
SLIDE 102

Compare various approaches: Hamiltonian(1)

H−1

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-103
SLIDE 103

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-104
SLIDE 104

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

symplectic

(H − τI)−1(H + τI)

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-105
SLIDE 105

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

symplectic

(H − τI)−1(H + τI)

unstructured

(H − τI)−1

+ ordinary Lanczos with implicit restarts

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-106
SLIDE 106

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

symplectic

(H − τI)−1(H + τI)

unstructured

(H − τI)−1

+ ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane.

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-107
SLIDE 107

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

symplectic

(H − τI)−1(H + τI)

unstructured

(H − τI)−1

+ ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane. Tolerance = 10−8

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-108
SLIDE 108

Compare various approaches: Hamiltonian(1)

H−1

Hamiltonian(3)

H−1(H − τI)−1(H + τI)−1

symplectic

(H − τI)−1(H + τI)

unstructured

(H − τI)−1

+ ordinary Lanczos with implicit restarts Get 6 smallest eigenvalues in right half-plane. Tolerance = 10−8 Take 20 steps and restart with 10.

Adaptivity and Beyond, Vancouver, August 2005 – p.21

slide-109
SLIDE 109

No-Clue Case (τ = 0)

Adaptivity and Beyond, Vancouver, August 2005 – p.22

slide-110
SLIDE 110

No-Clue Case (τ = 0)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Adaptivity and Beyond, Vancouver, August 2005 – p.22

slide-111
SLIDE 111

No-Clue Case (τ = 0)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 158 7 + 7

5 × 10−7

Adaptivity and Beyond, Vancouver, August 2005 – p.22

slide-112
SLIDE 112

No-Clue Case (τ = 0)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 158 7 + 7

5 × 10−7

Unstructured code must find everything twice.

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

Adaptivity and Beyond, Vancouver, August 2005 – p.22

slide-113
SLIDE 113

Conservative Shift (τ = 0.5)

Adaptivity and Beyond, Vancouver, August 2005 – p.23

slide-114
SLIDE 114

Conservative Shift (τ = 0.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

Adaptivity and Beyond, Vancouver, August 2005 – p.23

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SLIDE 115

Conservative Shift (τ = 0.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 138 7 + 2

3 × 10−5

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

Adaptivity and Beyond, Vancouver, August 2005 – p.23

slide-116
SLIDE 116

Conservative Shift (τ = 0.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 138 7 + 2

3 × 10−5

Hamiltonian(3) 174 11

3 × 10−13

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

Adaptivity and Beyond, Vancouver, August 2005 – p.23

slide-117
SLIDE 117

Conservative Shift (τ = 0.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 138 7 + 2

3 × 10−5

Hamiltonian(3) 174 11

3 × 10−13

Symplectic 156 11

2 × 10−8

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

Adaptivity and Beyond, Vancouver, August 2005 – p.23

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SLIDE 118

Aggressive Shift (τ = 1.5)

Adaptivity and Beyond, Vancouver, August 2005 – p.24

slide-119
SLIDE 119

Aggressive Shift (τ = 1.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Adaptivity and Beyond, Vancouver, August 2005 – p.24

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SLIDE 120

Aggressive Shift (τ = 1.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 96 9

1 × 10−7

Adaptivity and Beyond, Vancouver, August 2005 – p.24

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SLIDE 121

Aggressive Shift (τ = 1.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 96 9

1 × 10−7

Hamiltonian(3) 120 9

2 × 10−12

Adaptivity and Beyond, Vancouver, August 2005 – p.24

slide-122
SLIDE 122

Aggressive Shift (τ = 1.5)

Method Solves Eigvals Max. Found Resid. Hamiltonian(1) 78 9

2 × 10−10

Unstructured 96 9

1 × 10−7

Hamiltonian(3) 120 9

2 × 10−12

Symplectic 156 11

2 × 10−11

Adaptivity and Beyond, Vancouver, August 2005 – p.24

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SLIDE 123

The Last Slide

Adaptivity and Beyond, Vancouver, August 2005 – p.25

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SLIDE 124

The Last Slide

We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems.

Adaptivity and Beyond, Vancouver, August 2005 – p.25

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SLIDE 125

The Last Slide

We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method.

Adaptivity and Beyond, Vancouver, August 2005 – p.25

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SLIDE 126

The Last Slide

We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method. By exploiting structure we can solve our problems more economically.

Adaptivity and Beyond, Vancouver, August 2005 – p.25

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SLIDE 127

The Last Slide

We have developed structure-preserving implicitly-restarted Lanczos methods for Hamiltonian and symplectic eigenvalue problems. The structure-preserving methods are more accurate than a comparable non-structured method. By exploiting structure we can solve our problems more economically.

Thank you for your attention.

Adaptivity and Beyond, Vancouver, August 2005 – p.25