ADI-based Galerkin-Methods for Algebraic Lyapunov and ......2010/06/08 · ADI-based Galerkin-Methods for Algebraic Lyapunov and Riccati Equations Peter Benner Max-Planck-Institute

ADI-based Galerkin-Methods for AlgebraicLyapunov and Riccati Equations

Peter Benner

Max-Planck-Institute for Dynamics ofComplex Technical Systems

Computational Methods in Systems andControl Theory Group

Magdeburg, Germany

Technische Universitat ChemnitzFakultat fur Mathematik

Mathematik in Industrie und TechnikChemnitz, Germany

joint work with Jens Saak (TU Chemnitz)

Universite du Littoral Cote d’Opale, CalaisJune 8, 2010

ADI forLyapunov and

Riccati Equations

Peter Benner

Large-ScaleMatrix Equations

ADI for Lyapunov

Newton-ADI forAREs

Software

Conclusions andOpen Problems

References

Overview

1 Large-Scale Matrix EquationsMotivation

2 ADI Method for Lyapunov EquationsLow-Rank ADI for Lyapunov equationsFactored Galerkin-ADI Iteration

3 Newton-ADI for AREsLow-Rank Newton-ADI for AREsApplication to LQR ProblemGalerkin-Newton-ADINumerical ResultsQuadratic ADI for AREsAREs with High-Rank Constant Term

4 Software

5 Conclusions and Open Problems

6 References2/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

Large-Scale Matrix EquationsLarge-Scale Algebraic Lyapunov and Riccati Equations

Algebraic Riccati equation (ARE) for A,G = GT ,W = W T ∈ Rn×n

given and X ∈ Rn×n unknown:

0 = R(X ) := ATX + XA− XGX + W .

G = 0 =⇒ Lyapunov equation:

0 = L(X ) := ATX + XA + W .

Typical situation in model reduction and optimal control problems forsemi-discretized PDEs:

n = 103 – 106 (=⇒ 106 – 1012 unknowns!),

A has sparse representation (A = −M−1S for FEM),

G ,W low-rank with G ,W ∈ BBT ,CTC, whereB ∈ Rn×m, m n, C ∈ Rp×n, p n.

Standard (eigenproblem-based) O(n3) methods are notapplicable!3/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References




0 = R(X ) := ATX + XA− XGX + W .


0 = L(X ) := ATX + XA + W .


n = 103 – 106 (=⇒ 106 – 1012 unknowns!),




ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References




0 = R(X ) := ATX + XA− XGX + W .


0 = L(X ) := ATX + XA + W .


n = 103 – 106 (=⇒ 106 – 1012 unknowns!),




ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References




0 = R(X ) := ATX + XA− XGX + W .


0 = L(X ) := ATX + XA + W .


n = 103 – 106 (=⇒ 106 – 1012 unknowns!),




ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References




0 = R(X ) := ATX + XA− XGX + W .


0 = L(X ) := ATX + XA + W .


n = 103 – 106 (=⇒ 106 – 1012 unknowns!),




ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References




0 = R(X ) := ATX + XA− XGX + W .


0 = L(X ) := ATX + XA + W .


n = 103 – 106 (=⇒ 106 – 1012 unknowns!),




ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

Large-Scale Matrix EquationsLow-Rank Approximation

Consider spectrum of ARE solution (analogous for Lyapunovequations).

Example:

Linear 1D heat equation withpoint control,

Ω = [ 0, 1 ],

FEM discretization using linearB-splines,

h = 1/100 =⇒ n = 101.

Idea: X = XT ≥ 0 =⇒

X = ZZT =n∑

k=1

λkzkzTk ≈ Z (r)(Z (r))T =

r∑k=1

λkzkzTk .

=⇒ Goal: compute Z (r) ∈ Rn×r directly w/o ever forming X !4/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

Large-Scale Matrix EquationsLow-Rank Approximation

Consider spectrum of ARE solution (analogous for Lyapunovequations).

Example:

Linear 1D heat equation withpoint control,

Ω = [ 0, 1 ],

FEM discretization using linearB-splines,

h = 1/100 =⇒ n = 101.

Idea: X = XT ≥ 0 =⇒

X = ZZT =n∑

k=1

λkzkzTk ≈ Z (r)(Z (r))T =

r∑k=1

λkzkzTk .

=⇒ Goal: compute Z (r) ∈ Rn×r directly w/o ever forming X !4/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

MotivationLinear-quadratic Optimal Control

Numerical solution of linear-quadratic optimal control problem forparabolic PDEs via Galerkin approach, spatial FEM discretization

LQR Problem (finite-dimensional)

Min J (u) =1

2

∞∫0

(yTQy + uTRu

)dt for u ∈ L2(0,∞; Rm),

subject to Mx = −Sx + Bu, x(0) = x0, y = Cx ,with stiffness S ∈ Rn×n, mass M ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n.

Solution of finite-dimensional LQR problem: feedback control

u∗(t) = −BTX∗x(t) =: −K∗x(t),

where X∗ = XT∗ ≥ 0 is the unique stabilizing1 solution of the ARE

0 = R(X ) := CTC + ATX + XA− XBBTX ,

with A := −M−1S , B := M−1BR−12 , C := CQ−

12 .

1X is stabilizing ⇔ Λ (A− BBT X ) ⊂ C−.5/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

MotivationLinear-quadratic Optimal Control

Numerical solution of linear-quadratic optimal control problem forparabolic PDEs via Galerkin approach, spatial FEM discretization

LQR Problem (finite-dimensional)

Min J (u) =1

2

∞∫0

(yTQy + uTRu

)dt for u ∈ L2(0,∞; Rm),

subject to Mx = −Sx + Bu, x(0) = x0, y = Cx ,with stiffness S ∈ Rn×n, mass M ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n.

Solution of finite-dimensional LQR problem: feedback control

u∗(t) = −BTX∗x(t) =: −K∗x(t),

where X∗ = XT∗ ≥ 0 is the unique stabilizing1 solution of the ARE

0 = R(X ) := CTC + ATX + XA− XBBTX ,

with A := −M−1S , B := M−1BR−12 , C := CQ−

12 .

1X is stabilizing ⇔ Λ (A− BBT X ) ⊂ C−.5/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References

MotivationModel Reduction by Balanced Truncation

Linear, Time-Invariant (LTI) Systems

Σ :

x(t) = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,y(t) = Cx + Du, C ∈ Rp×n, D ∈ Rp×m.

(A,B,C ,D) is a realization of Σ (nonunique).

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References


Linear, Time-Invariant (LTI) Systems

Σ :

x(t) = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,y(t) = Cx + Du, C ∈ Rp×n, D ∈ Rp×m.

(A,B,C ,D) is a realization of Σ (nonunique).

Model Reduction Based on Balancing

Given P,Q ∈ Rn×n symmetric positive definite (spd), and acontragredient transformation T : Rn → Rn,

TPTT = T−TQT−1 = diag(σ1, . . . , σn), σ1 ≥ . . . ≥ σn ≥ 0.

Balancing Σ w.r.t. P,Q:

Σ ≡ (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D) ≡ Σ.

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References


Model Reduction Based on Balancing

Given P,Q ∈ Rn×n symmetric positive definite (spd), and acontragredient transformation T : Rn → Rn,

TPTT = T−TQT−1 = diag(σ1, . . . , σn), σ1 ≥ . . . ≥ σn ≥ 0.

Balancing Σ w.r.t. P,Q:

Σ ≡ (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D) ≡ Σ.

For Balanced Truncation: P/Q = controllability/observabilityGramian of Σ, i.e., for asymptotically stable systems, P,Q solve dualLyapunov equations

AP + PAT + BBT = 0, ATQ + QA + CTC = 0.

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References


Basic Model Reduction Procedure

1 Given Σ ≡ (A,B,C ,D) and balancing (w.r.t. given P,Q spd)transformation T ∈ Rn×n nonsingular, compute

(A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

([A11 A12

A21 A22

],

[B1

B2

],[

C1 C2

],D

)2 Truncation reduced-order model:

(A, B, C , D) = (A11,B1,C1,D).

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References


Basic Model Reduction Procedure

1 Given Σ ≡ (A,B,C ,D) and balancing (w.r.t. given P,Q spd)transformation T ∈ Rn×n nonsingular, compute

(A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

([A11 A12

A21 A22

],

[B1

B2

],[

C1 C2

],D

)2 Truncation reduced-order model:

(A, B, C , D) = (A11,B1,C1,D).

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


Motivation

ADI for Lyapunov

Newton-ADI forAREs

Software


References


Implementation: SR Method

1 Given Cholesky (square) or (low-rank approximation to) full-rank(maybe rectangular, “thin”) factors of P,Q

P = STS , Q = RTR.

2 Compute SVD

SRT = [ U1, U2 ]

[Σ1

Σ2

] [V T

1

V T2

].

3 SetW = RTV1Σ

−1/21 , V = STU1Σ

−1/21 .

4 Reduced-order model is

(A, B, C , D) := (W TAV ,W TB,CV ,D) (≡ (A11,B1,C1,D).)

6/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI

FactoredGalerkin-ADIIteration

Newton-ADI forAREs

Software


References

ADI Method for Lyapunov EquationsBackground

Recall Peaceman Rachford ADI:Consider Au = s where A ∈ Rn×n spd, s ∈ Rn. ADI Iteration Idea:Decompose A = H + V with H,V ∈ Rn×n such that

(H + pI )v = r(V + pI )w = t

can be solved easily/efficiently.

ADI Iteration

If H,V spd ⇒ ∃pk , k = 1, 2, . . . such that

u0 = 0(H + pk I )uk− 1

2= (pk I − V )uk−1 + s

(V + pk I )uk = (pk I − H)uk− 12

+ s

converges to u ∈ Rn solving Au = s.

7/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

ADI Method for Lyapunov EquationsBackground

Recall Peaceman Rachford ADI:Consider Au = s where A ∈ Rn×n spd, s ∈ Rn. ADI Iteration Idea:Decompose A = H + V with H,V ∈ Rn×n such that

(H + pI )v = r(V + pI )w = t

can be solved easily/efficiently.

ADI Iteration

If H,V spd ⇒ ∃pk , k = 1, 2, . . . such that

u0 = 0(H + pk I )uk− 1

2= (pk I − V )uk−1 + s

(V + pk I )uk = (pk I − H)uk− 12

+ s

converges to u ∈ Rn solving Au = s.

7/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

ADI Method for Lyapunov EquationsMotivation

The Lyapunov operator

L : P 7→ AX + XAT

can be decomposed into the linear operators

LH : X 7→ AX LV : X 7→ XAT .

In analogy to the standard ADI method we find the

ADI iteration for the Lyapunov equation [Wachspress 1988

P0 = 0(A + pk I )Xk− 1

2= −W − Pk−1(AT − pk I )

(A + pk I )XTk = −W − XT

k− 12

(AT − pk I )

8/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Low-Rank ADI for Lyapunov equations

For A ∈ Rn×n stable, B ∈ Rn×m (w n), consider Lyapunovequation

AX + XAT = −BBT .

ADI Iteration: [Wachspress 1988]

(A + pk I )Xk− 12

= −BBT − Xk−1(AT − pk I )

(A + pk I )XkT = −BBT − Xk− 1

2(AT − pk I )

with parameters pk ∈ C− and pk+1 = pk if pk 6∈ R.

For X0 = 0 and proper choice of pk : limk→∞

Xk = X superlinear.

Re-formulation using Xk = YkYTk yields iteration for Yk ...

9/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Low-Rank ADI for Lyapunov equations

For A ∈ Rn×n stable, B ∈ Rn×m (w n), consider Lyapunovequation

AX + XAT = −BBT .

ADI Iteration: [Wachspress 1988]

(A + pk I )Xk− 12

= −BBT − Xk−1(AT − pk I )

(A + pk I )XkT = −BBT − Xk− 1

2(AT − pk I )

with parameters pk ∈ C− and pk+1 = pk if pk 6∈ R.

For X0 = 0 and proper choice of pk : limk→∞

Xk = X superlinear.

Re-formulation using Xk = YkYTk yields iteration for Yk ...

9/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Low-Rank ADI for Lyapunov equationsLyapunov equation 0 = AX + XAT + BBT .

Setting Xk = YkYTk , some algebraic manipulations =⇒

Algorithm [Penzl ’97/’00, Li/White ’99/’02, B. 04, B./Li/Penzl ’99/’08]

V1 ←p−2Re (p1)(A + p1I )−1B, Y1 ← V1

FOR k = 2, 3, . . .

Vk ←q

Re (pk )Re (pk−1)

`Vk−1 − (pk + pk−1)(A + pk I )−1Vk−1

Ýk ←

ˆYk−1 Vk

˜Yk ← rrlq(Yk , τ) % column compression

At convergence, YkmaxYTkmax≈ X , where (without column compression)

Ykmax =[

V1 . . . Vkmax

], Vk = ∈ Cn×m.

Note: Implementation in real arithmetic possible by combining two steps.

10/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Low-Rank ADI for Lyapunov equationsLyapunov equation 0 = AX + XAT + BBT .

Setting Xk = YkYTk , some algebraic manipulations =⇒

Algorithm [Penzl ’97/’00, Li/White ’99/’02, B. 04, B./Li/Penzl ’99/’08]

V1 ←p−2Re (p1)(A + p1I )−1B, Y1 ← V1

FOR k = 2, 3, . . .

Vk ←q

Re (pk )Re (pk−1)

`Vk−1 − (pk + pk−1)(A + pk I )−1Vk−1

Ýk ←

ˆYk−1 Vk

˜Yk ← rrlq(Yk , τ) % column compression

At convergence, YkmaxYTkmax≈ X , where (without column compression)

Ykmax =[

V1 . . . Vkmax

], Vk = ∈ Cn×m.

Note: Implementation in real arithmetic possible by combining two steps.

10/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Numerical ResultsOptimal Cooling of Steel Profiles

Mathematical model: boundary controlfor linearized 2D heat equation.

c · ρ ∂∂t

x = λ∆x , ξ ∈ Ω

λ∂

∂nx = κ(uk − x), ξ ∈ Γk , 1 ≤ k ≤ 7,

∂

∂nx = 0, ξ ∈ Γ7.

=⇒ m = 7, p = 6.

FEM Discretization, different modelsfor initial mesh (n = 371),1, 2, 3, 4 steps of mesh refinement ⇒n = 1357, 5177, 20209, 79841.

2

34

9 10

1516

22

34

43

47

51

55

60 63

8392

Source: Physical model: courtesy of Mannesmann/Demag.

Math. model: Troltzsch/Unger 1999/2001, Penzl 1999, Saak 2003.

11/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Numerical ResultsOptimal Cooling of Steel Profiles

Solve dual Lyapunov equations needed for balanced truncation, i.e.,

APMT + MPAT + BBT = 0, ATQM + MTQA + CTC = 0,

for 79, 841. Note: m = 7, p = 6.25 shifts chosen by Penzl’s heuristic from 50/25 Ritz values of A oflargest/smallest magnitude, no column compression performed.New version in MESS (Matrix Equations Sparse Solvers) requires nofactorization of mass matrix!Computations done on Core2Duo at 2.8GHz with 3GB RAM and32Bit-MATLAB R©.

CPU times: 626 / 356 sec.

12/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Recent Numerical ResultsScaling / Mesh IndependenceComputations by Martin Kohler

A ∈ Rn×n ≡ FDM matrix for 2D heat equation on [0, 1]2 (Lyapackbenchmark demo l1, m = 1).

16 shifts chosen by Penzl’s heuristic from 50/25 Ritz values of A oflargest/smallest magnitude.

Computations using 2 dual core Intel Xeon 5160 with 16 GB RAM.

13/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References





CPU Timesn CMESS Lyapack MESS

100 0.023 0.124 0.158625 0.042 0.104 0.227

2,500 0.159 0.702 0.98910,000 0.965 6.22 5.64440,000 11.09 71.48 34.5590,000 34.67 418.5 90.49

160,000 109.3 out of memory 219.9250,000 193.7 out of memory 403.8562,500 930.1 out of memory 1216.7

1,000,000 2220.0 out of memory 2428.613/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References





Note: for n=1,000,000, first sparse LU needs ∼ 1, 100 sec., usingUMFPACK this reduces to 30 sec.

13/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Factored Galerkin-ADI IterationLyapunov equation 0 = AX + XAT + BBT

Projection-based methods for Lyapunov equations with A + AT < 0:1 Compute orthonormal basis range (Z), Z ∈ Rn×r , for subspaceZ ⊂ Rn, dimZ = r .

2 Set A := ZTAZ , B := ZTB.3 Solve small-size Lyapunov equation AX + X AT + BBT = 0.4 Use X ≈ ZXZT .

Examples:

Krylov subspace methods, i.e., for m = 1:

Z = K(A,B, r) = spanB,AB,A2B, . . . ,Ar−1B

[Saad ’90, Jaimoukha/Kasenally ’94, Jbilou ’02–’08].

K-PIK [Simoncini ’07],

Z = K(A,B, r) ∪ K(A−1,B, r).

14/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References




Examples:

Krylov subspace methods, i.e., for m = 1:

Z = K(A,B, r) = spanB,AB,A2B, . . . ,Ar−1B

[Saad ’90, Jaimoukha/Kasenally ’94, Jbilou ’02–’08].

K-PIK [Simoncini ’07],

Z = K(A,B, r) ∪ K(A−1,B, r).

14/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References




Examples:

ADI subspace [B./R.-C. Li/Truhar ’08]:

Z = colspan[

V1, . . . , Vr

].

Note:1 ADI subspace is rational Krylov subspace [J.-R. Li/White ’02].2 Similar approach: ADI-preconditioned global Arnoldi method

[Jbilou ’08].

14/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Factored Galerkin-ADI IterationNumerical examples

FEM semi-discretized control problem for parabolic PDE:

optimal cooling of rail profiles,

n = 20, 209, m = 7, p = 6.

Good ADI shifts

CPU times: 80s (projection every 5th ADI step) vs. 94s (no projection).

Computations by Jens Saak.15/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Factored Galerkin-ADI IterationNumerical examples

FEM semi-discretized control problem for parabolic PDE:

optimal cooling of rail profiles,

n = 20, 209, m = 7, p = 6.

Bad ADI shifts

CPU times: 368s (projection every 5th ADI step) vs. 1207s (no projection).

Computations by Jens Saak.15/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

LR-ADI


Newton-ADI forAREs

Software


References

Factored Galerkin-ADI IterationNumerical examples: optimal cooling of rail profiles, n = 79, 841, m = 7, p = 6.

MESS w/o Galerkin projection and column compression

Rank of solution factors: 532 / 426

MESS with Galerkin projection and column compression

Rank of solution factors: 269 / 20516/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI

Application toLQR Problem

Galerkin-Newton-ADI

NumericalResults

Quadratic ADIfor AREs

High-Rank W

Software


References

Newton-ADI for AREsNewton’s Method for AREs [Kleinman ’68, Mehrmann ’91, Lancaster/Rodman ’95,

B./Byers ’94/’98, B. ’97, Guo/Laub ’99]

Consider 0 = R(X ) = CTC + ATX + XA− XBBTX .

Frechet derivative of R(X ) at X :

R′X : Z → (A− BBTX )TZ + Z (A− BBTX ).

Newton-Kantorovich method:

Xj+1 = Xj −(R′Xj

)−1

R(Xj), j = 0, 1, 2, . . .

Newton’s method (with line search) for AREs

FOR j = 0, 1, . . .

1 Aj ← A− BBTXj =: A− BKj .

2 Solve the Lyapunov equation ATj Nj + NjAj = −R(Xj).

3 Xj+1 ← Xj + tjNj .

END FOR j17/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


B./Byers ’94/’98, B. ’97, Guo/Laub ’99]






)−1

R(Xj), j = 0, 1, 2, . . .


FOR j = 0, 1, . . .




END FOR j17/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


B./Byers ’94/’98, B. ’97, Guo/Laub ’99]






)−1

R(Xj), j = 0, 1, 2, . . .


FOR j = 0, 1, . . .




END FOR j17/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


B./Byers ’94/’98, B. ’97, Guo/Laub ’99]






)−1

R(Xj), j = 0, 1, 2, . . .


FOR j = 0, 1, . . .




END FOR j17/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Factored Galerkin-ADI IterationProperties and Implementation

Convergence for K0 stabilizing:

Aj = A− BKj = A− BBTXj is stable ∀ j ≥ 0.limj→∞ ‖R(Xj)‖F = 0 (monotonically).limj→∞ Xj = X∗ ≥ 0 (locally quadratic).

Need large-scale Lyapunov solver; here, ADI iteration:linear systems with dense, but “sparse+low rank” coefficientmatrix Aj :Aj = A − B · Kj

= sparse − m ·

m n =⇒ efficient “inversion” usingSherman-Morrison-Woodbury formula:

(A−BKj +p(j)k I )−1 = (In +(A + p

(j)k I )−1B(Im−Kj (A + p

(j)k I )−1B)−1Kj )(A + p

(j)k I )−1

.

BUT: X = XT ∈ Rn×n =⇒ n(n + 1)/2 unknowns!

18/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





= sparse − m ·


(A−BKj +p(j)k I )−1 = (In +(A + p


(j)k I )−1B)−1Kj )(A + p

(j)k I )−1

.


18/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





= sparse − m ·


(A−BKj +p(j)k I )−1 = (In +(A + p


(j)k I )−1B)−1Kj )(A + p

(j)k I )−1

.


18/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





= sparse − m ·


(A−BKj +p(j)k I )−1 = (In +(A + p


(j)k I )−1B)−1Kj )(A + p

(j)k I )−1

.


18/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Low-Rank Newton-ADI for AREs

Re-write Newton’s method for AREs

ATj Nj + NjAj = −R(Xj)

⇐⇒

ATj (Xj + Nj)︸︷︷︸

=Xj+1

+ (Xj + Nj)︸︷︷︸=Xj+1

Aj = −CTC − XjBBTXj︸︷︷︸=:−WjW T

j

Set Xj = ZjZTj for rank (Zj) n =⇒

ATj

(Zj+1Z

Tj+1

)+(Zj+1Z

Tj+1

)Aj = −WjW

Tj

Factored Newton Iteration [B./Li/Penzl 1999/2008]

Solve Lyapunov equations for Zj+1 directly by factored ADI iterationand use ‘sparse + low-rank’ structure of Aj .

19/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Low-Rank Newton-ADI for AREs

Re-write Newton’s method for AREs

ATj Nj + NjAj = −R(Xj)

⇐⇒


=Xj+1

+ (Xj + Nj)︸︷︷︸=Xj+1

Aj = −CTC − XjBBTXj︸︷︷︸=:−WjW T

j

Set Xj = ZjZTj for rank (Zj) n =⇒

ATj

(Zj+1Z

Tj+1

)+(Zj+1Z

Tj+1

)Aj = −WjW

Tj

Factored Newton Iteration [B./Li/Penzl 1999/2008]

Solve Lyapunov equations for Zj+1 directly by factored ADI iterationand use ‘sparse + low-rank’ structure of Aj .

19/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Application to LQR ProblemFeedback Iteration

Optimal feedbackK∗ = BTX∗ = BTZ∗Z

T∗

can be computed by direct feedback iteration:

jth Newton iteration:

Kj = BTZjZTj =

kmax∑k=1

(BTVj,k)V Tj,k

j→∞−−−−→ K∗ = BTZ∗Z

T∗

Kj can be updated in ADI iteration, no need to even form Zj ,need only fixed workspace for Kj ∈ Rm×n!

Related to earlier work by [Banks/Ito 1991].

20/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Newton-ADI for AREsGalerkin-Newton-ADI

Basic ideas

Hybrid method of Galerkin projection methods for AREs[Jaimoukha/Kasenally ’94, Jbilou ’06, Heyouni/Jbilou ’09]

and Newton-ADI, i.e., use column space of current Newtoniterate for projection, solve projected ARE, and prolongate.

Independence of good parameters observed for Galerkin-ADIapplied to Lyapunov equations fix ADI parameters for allNewton iterations.

21/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Newton-ADI for AREsGalerkin-Newton-ADI

Basic ideas

Hybrid method of Galerkin projection methods for AREs[Jaimoukha/Kasenally ’94, Jbilou ’06, Heyouni/Jbilou ’09]

and Newton-ADI, i.e., use column space of current Newtoniterate for projection, solve projected ARE, and prolongate.

Independence of good parameters observed for Galerkin-ADIapplied to Lyapunov equations fix ADI parameters for allNewton iterations.

21/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Numerical ResultsLQR Problem for 2D Geometry

Linear 2D heat equation with homogeneous Dirichlet boundaryand point control/observation.FD discretization on uniform 150× 150 grid.n = 22.500, m = p = 1, 10 shifts for ADI iterations.Convergence of large-scale matrix equation solvers:

22/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


FDM for 2D heat/convection-diffusion equations on [0, 1]2 (Lyapackbenchmarks, m = p = 1) symmetric/nonsymmetric A ∈ Rn×n,n = 10, 000.

15 shifts chosen by Penzl’s heuristic from 50/25 Ritz/harmonic Ritzvalues of A.

Computations using Intel Core 2 Quad CPU of type Q9400 at2.66GHz with 4 GB RAM and 64Bit-MATLAB.

23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





Newton-ADI

step rel. change rel. residual ADI1 1 9.99e–01 2002 9.99e–01 3.41e+01 233 5.25e–01 6.37e+00 204 5.37e–01 1.52e+00 205 7.03e–01 2.64e–01 236 5.57e–01 1.56e–02 237 6.59e–02 6.30e–05 238 4.02e–04 9.68e–10 239 8.45e–09 1.09e–11 23

10 1.52e–14 1.09e–11 23

CPU time: 76.9 sec.23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





Newton-ADI

step rel. change rel. residual ADI1 1 9.99e–01 2002 9.99e–01 3.41e+01 233 5.25e–01 6.37e+00 204 5.37e–01 1.52e+00 205 7.03e–01 2.64e–01 236 5.57e–01 1.56e–02 237 6.59e–02 6.30e–05 238 4.02e–04 9.68e–10 239 8.45e–09 1.09e–11 23

10 1.52e–14 1.09e–11 23

CPU time: 76.9 sec.

Newton-Galerkin-ADI

step rel. change rel. residual ADI1 1 3.56e–04 202 5.25e–01 6.37e+00 103 5.37e–01 1.52e+00 64 7.03e–01 2.64e–01 105 5.57e–01 1.57e–02 106 6.59e–02 6.30e–05 107 4.03e–04 9.79e–10 108 8.45e–09 1.43e–15 10

CPU time: 38.0 sec.23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





0 2 4 6 8 1010

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

rela

tive

ch

an

ge

in

LR

CF

iteration index

no projection

ever step

every 5th step

0 2 4 6 8 1010

−15

10−10

10−5

100

105

rela

tive

re

sid

ua

l n

orm

iteration index

no projection

ever step

every 5th step

23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





Newton-ADI


CPU time: 185.9 sec.

23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





Newton-ADI



Newton-Galerkin-ADI

step rel. change rel. residual ADI it.1 1 1.78e–02 352 3.11e–01 3.72e+00 153 2.88e–01 9.62e–01 204 3.41e–01 1.68e–01 155 1.22e–01 5.25e–03 206 3.89e–03 2.96e–06 157 2.30e–06 6.14e–13 20

CPU time: 75.7 sec.

23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References





1 2 3 4 5 6 7 810

−6

10−5

10−4

10−3

10−2

10−1

100

rela

tive

ch

an

ge

in

LR

CF

iteration index

no projection

ever step

every 5th step

1 2 3 4 5 6 7 810

−15

10−10

10−5

100

105

rela

tive

re

sid

ua

l n

orm

iteration index

no projection

ever step

every 5th step

23/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Control problem for 3d Convection-Diffusion Equation

FDM for 3D convection-diffusion equation on [0, 1]3

proposed in [Simoncini ’07], q = p = 1

non-symmetric A ∈ Rn×n , n = 10 648

Test system:

INTEL Xeon 5160 3.00GHz ; 16 GB RAM; 64Bit-MATLAB (R2010a) usingthreaded BLAS; stopping tolerance: 10−10

24/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Newton-ADI

NWT rel. change rel. residual ADI

1 1.0 · 100 9.3 · 10−01 100

2 3.7 · 10−02 9.6 · 10−02 94

3 1.4 · 10−02 1.1 · 10−03 98

4 3.5 · 10−04 1.0 · 10−07 97

5 6.4 · 10−08 1.3 · 10−10 97

6 7.5 · 10−16 1.3 · 10−10 97

CPU time: 4 805.8 sec.

NG-ADI inner= 5, outer= 1


1 1.0 · 100 5.0 · 10−11 80




1 1.0 · 100 7.4 · 10−11 71




1 1.0 · 100 6.5 · 10−13 100


Test system:

INTEL Xeon 5160 3.00GHz ; 16 GB RAM; 64Bit-MATLAB (R2010a) usingthreaded BLAS; stopping tolerance: 10−10

24/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Numerical ResultsScaling of CPU times / Mesh Independence

Ω

(0, 1)

(0, 0)

(1, 1)

(1, 0)

Γc

∂tx(ξ, t) = ∆x(ξ, t) in Ω

∂νx = b(ξ) · u(t)− x on Γc

∂νx = −x on ∂Ω \ Γc

x(ξ, 0) = 1

Note:Here b(ξ) = 4 (1− ξ2) ξ2 for ξ ∈ Γc and 0 otherwise, thus ∀t ∈ R>0,we have u(t) ∈ R.

⇒ Bh = MΓ,h · b.

25/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Ω

(0, 1)

(0, 0)

(1, 1)

(1, 0)

Γc




x(ξ, 0) = 1

Note:Here b(ξ) = 4 (1− ξ2) ξ2 for ξ ∈ Γc and 0 otherwise, thus ∀t ∈ R>0,we have u(t) ∈ R.

⇒ Bh = MΓ,h · b.

25/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Ω

(0, 1)

(0, 0)

(1, 1)

(1, 0)

Γc




x(ξ, 0) = 1

Consider: output equation y = Cx , where

C : L2(Ω) → Rx(ξ, t) 7→ y(t) =

∫Ω

x(ξ, t) dξ.

⇒ Ch = 1 ·Mh.

25/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Ω

(0, 1)

(0, 0)

(1, 1)

(1, 0)

Γc




x(ξ, 0) = 1

Consider: output equation y = Cx , where

C : L2(Ω) → Rx(ξ, t) 7→ y(t) =

∫Ω

x(ξ, t) dξ,⇒ Ch = 1 ·Mh.

25/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Ω

(0, 1)

(0, 0)

(1, 1)

(1, 0)

Γc




x(ξ, 0) = 1

Cost Function:

J (u) =

∫ ∞0

y2(t) + u2(t) dt.

25/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Simplified Low Rank Newton-Galerkin ADI

generalized state space form implementation

Penzl shifts (16/50/25) with respect to initial matrices

projection acceleration in every outer iteration step

projection acceleration in every 5-th inner iteration step

Test system:

INTEL Xeon 5160 @ 3.00 GHz; 16 GB RAM; 64Bit-MATLAB(R2010a) using threaded BLAS (romulus)stopping criterion tolerances: 10−10

26/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Computation Times

discretization level problem size time in seconds3 81 4.87 · 10−2

4 289 2.81 · 10−1

5 1 089 5.87 · 10−1

6 4 225 2.637 16 641 2.03 · 10+1

8 66 049 1.22 · 10+2

9 263 169 1.05 · 10+3

10 1 050 625 1.65 · 10+4

11 4 198 401 1.35 · 10+5

Test system:


26/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


3 4 5 6 7 8 9 10 1110

−4

10−2

100

102

104

106

108

1010

Scaling of CPU time

refinement level

tim

e in s

econds

Test system:


26/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Quadratic ADI for AREs0 = R(X ) = AT X + XA− XBBT X + W

Basic QADI iteration [Wong/Balakrishnan et al. ’05–’08]`(A− BBTXk)T + pk I

´Xk+ 1

2= −W − Xk((A− pk I )“

(A− BBTXTk+ 1

2)T + pk I

”Xk+1 = −W − XT

k+ 12(A− pk I )

Derivation of complicated Cholesky factor version, but requires square andinvertible Cholesky factors.

Idea of low-rank Galerkin-QADI [B./Saak ’09]

V1 ←p−2Re (p1)(A− B(BTY0)Y T

0 + p1I )−TB, Y1 ← V1

FOR k = 2, 3, . . .

Vk ← Vk−1 − (pk + pk−1)(A− B(BTYk−1)Y Tk−1 + pk I )−TVk−1

Yk ←h

Yk−1

qRe (pk )

Re (pk−1)Vk

iYk ← rrlq(Yk , τ) % column compression

If desired, project ARE onto range(Yk), solve and prolongate.

27/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

Quadratic ADI for AREs0 = R(X ) = AT X + XA− XBBT X + W

Basic QADI iteration [Wong/Balakrishnan et al. ’05–’08]`(A− BBTXk)T + pk I

´Xk+ 1

2= −W − Xk((A− pk I )“

(A− BBTXTk+ 1

2)T + pk I

”Xk+1 = −W − XT

k+ 12(A− pk I )

Derivation of complicated Cholesky factor version, but requires square andinvertible Cholesky factors.

Idea of low-rank Galerkin-QADI [B./Saak ’09]

V1 ←p−2Re (p1)(A− B(BTY0)Y T

0 + p1I )−TB, Y1 ← V1

FOR k = 2, 3, . . .

Vk ← Vk−1 − (pk + pk−1)(A− B(BTYk−1)Y Tk−1 + pk I )−TVk−1

Yk ←h

Yk−1

qRe (pk )

Re (pk−1)Vk

iYk ← rrlq(Yk , τ) % column compression

If desired, project ARE onto range(Yk), solve and prolongate.

27/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

AREs with High-Rank Constant Term

Consider ARE

0 = R(X ) = W + ATX + XA− XBBTX

with rank (W ) 6 n, e.g., stabilization of flow problems described byNavier-Stokes eqns. requires solution of

0 = R(X ) = Mh − STh XMh −MhXSh −MhXBhB

Th XMh,

where Mh = mass matrix of FE velocity test functions.

Example: von Karman vortex street, Re= 500

uncontrolled:

controlled using ARE:

t = 1 t = 5

t = 8 t = 1028/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References

AREs with High-Rank Constant TermSolution: remove W from r.h.s. of Lyapunov eqns. in Newton-ADI

One step of Newton-Kleinman iteration for ARE:


=Xj+1

+Xj+1Aj = −W − (XjB)︸︷︷︸=KT

j

BTXj︸︷︷︸=Kj

for j = 1, 2, . . .

Subtract two consecutive equations =⇒

ATj Nj + NjAj = −NT

j−1BBTNj−1 for j = 1, 2, . . .

See [Banks/Ito ’91, B./Hernandez/Pastor ’03, Morris/Navasca ’05] for

details and applications of this variant.

But: need BTN0 = K1 − K0!

Assuming K0 is known, need to compute K1.

29/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Solution idea:

K1 = BTX1

= BT

Z ∞0

e(A−BK0)T t(W + KT0 K0)e(A−BK0)t dt

=

Z ∞0

g(t) dt ≈NX`=0

γ`g(t`),

where g(t) = (“e(A−BK0)tB

”T

(W + KT0 K0)

é(A−BK0)t .

[Borgggaard/Stoyanov ’08]:

evaluate g(t`) using ODE solver applied to x = (A−BK0)x + adjoint eqn.

29/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Low-RankNewton-ADI


Galerkin-Newton-ADI

NumericalResults


High-Rank W

Software


References


Better solution idea:(related to frequency domain POD [Willcox/Peraire ’02])

K1 = BTX1 (Notation: A0 := A− BK0)

= BT · 1

2π

Z ∞−∞

(ωIn − A0)−H(W + KT0 K0)(ωIn − A0)−1 dω

=

Z ∞−∞

f (ω) dω ≈NX`=0

γ`f (ω`),

where f (ω) = ( −`(ωIn + A0)−1B

´T(W + KT

0 K0)´

(ωIn − A0)−1.

Evaluation of f (ω`) requires

1 sparse LU decmposition (complex!),

2m forward/backward solves,

m sparse and 2m low-rank matrix-vector products.

Use adaptive quadrature with high accuracy, e.g. Gauß-Kronrod (MAT-LAB’s quadgk).

29/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Software

Lyapack [Penzl 2000]

MATLAB toolbox for solving

– Lyapunov equations and algebraic Riccati equations,

– model reduction and LQR problems.

Main work horse: Low-rank ADI and Newton-ADI iterations.

30/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Software






MESS – Matrix Equations Sparse Solvers [Saak/Mena/B. 2008]

Extended and revised version of Lyapack.

Includes solvers for large-scale differential Riccati equations (based onRosenbrock and BDF methods).

Many algorithmic improvements:

– new ADI parameter selection,– column compression based on RRQR,– more efficient use of direct solvers,– treatment of generalized systems without factorization of the mass matrix.

C version CMESS under development (Martin Kohler).

30/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Software












30/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Software












30/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Conclusions and Open Problems

Galerkin projection can significantly accelerate ADI iteration forLyapunov equations.

Low-rank Galerkin-QADI may become a viable alternative toNewton-ADI.

High-rank constant terms in ARE can be handled using quadraturerules.

Software is available in MATLAB toolbox Lyapack and itssuccessor MESS.

31/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

Conclusions and Open Problems

Galerkin projection can significantly accelerate ADI iteration forLyapunov equations.

Low-rank Galerkin-QADI may become a viable alternative toNewton-ADI.

High-rank constant terms in ARE can be handled using quadraturerules.

Software is available in MATLAB toolbox Lyapack and itssuccessor MESS.

To-Do list:

– computation of stabilizing initial guess.(If hierarchical grid structure is available, a multigrid approach ispossible, other approaches based on “cheaper” matrix equationsunder development.)

– Implementation of coupled Riccati solvers for LQG controllerdesign and balancing-related model reduction.

31/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

References

1 H.T. Banks and K. Ito.A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems.SIAM J. Cont. Optim., 29(3):499–515, 1991.

2 U. Baur and P. Benner.Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic.Computing, 78(3):211–234, 2006.

3 P. Benner.Solving large-scale control problems.IEEE Control Systems Magazine, 14(1):44–59, 2004.

4 P. Benner.Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems.Logos–Verlag, Berlin, Germany, 1997.Also: Dissertation, Fakultat fur Mathematik, TU Chemnitz–Zwickau, 1997.

5 P. Benner.Editorial of special issue on ”Large-Scale Matrix Equations of Special Type”.Numer. Lin. Alg. Appl., 15(9):747–754, 2008.

6 P. Benner and R. Byers.Step size control for Newton’s method applied to algebraic Riccati equations.In J.G. Lewis, editor, Proc. Fifth SIAM Conf. Appl. Lin. Alg., Snowbird, UT, pages 177–181. SIAM, Philadelphia, PA,1994.

7 P. Benner and R. Byers.An exact line search method for solving generalized continuous-time algebraic Riccati equations.IEEE Trans. Automat. Control, 43(1):101–107, 1998.

8 P. Benner, J.-R. Li, and T. Penzl.Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems.Numer. Lin. Alg. Appl., 15(9):755–777, 2008.Reprint of unpublished manuscript, December 1999.32/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

References

9 P. Benner and J. Saak.A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations.Preprint SPP1253-090, DFG Priority Programme 1253 ”Optimization with Partial Differential Equations”, January 2010.Available at http://www.am.uni-erlangen.de/home/spp1253/wiki/images/2/28/Preprint-SPP1253-090.pdf

10 P. Benner, R.-C. Li, and N. Truhar.On the ADI method for Sylvester equations.J. Comput. Appl. Math., 233:1035–1045, 2009.

11 J. Borggaard and M. Stoyanov.A reduced order solver for Lyapunov equations with high rank matricesProc. 18th Intl. Symp. Mathematical Theory of Network and Systems, MTNS 2008, 11 pages, 2008.

12 J. Burns, E. Sachs, and L. Zietsman.Mesh independence of Kleinman-Newton iterations for Riccati equations on Hilbert space.SIAM J. Control Optim., 47(5):2663–2692, 2008.

13 L. GrasedyckNonlinear multigrid for the solution of large-scale Riccati equations in low-rank andH-matrix format.Numer. Lin. Alg. Appl., 15(9):779–807, 2008.

14 L. Grasedyck, W. Hackbusch, and B.N. Khoromskij.Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices.Computing, 70:121–165, 2003.

15 C.-H. Guo and A.J. Laub.On a Newton-like method for solving algebraic Riccati equations.SIAM J. Matrix Anal. Appl., 21(2):694–698, 2000.

16 M. Heyouni and K. Jbilou.An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation.ETNA, 33:53–62, 2009.

17 I.M. Jaimoukha and E.M. Kasenally.Krylov subspace methods for solving large Lyapunov equations.SIAM J. Numer. Anal., 31:227–251, 1994.33/35

http://www.am.uni-erlangen.de/home/spp1253/wiki/images/2/28/Preprint-SPP1253-090.pdf

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

References

18 K. Jbilou.Block Krylov subspace methods for large continuous-time algebraic Riccati equations.Numer. Algorithms, 34:339–353, 2003.

19 K. Jbilou.An Arnoldi based algorithm for large algebraic Riccati equations.Appl. Math. Lett., 19(5):437–444, 2006.

20 K. Jbilou.ADI preconditioned Krylov methods for large Lyapunov matrix equations.Lin. Alg. Appl., 432(10):2473–2485, 2010.

21 D.L. Kleinman.On an iterative technique for Riccati equation computations.IEEE Trans. Automat. Control, AC-13:114–115, 1968.

22 P. Lancaster and L. Rodman.The Algebraic Riccati Equation.Oxford University Press, Oxford, 1995.

23 J.-R. Li and J. White.Low rank solution of Lyapunov equations.SIAM J. Matrix Anal. Appl., 24(1):260–280, 2002.

24 V. Mehrmann.The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution.Number 163 in Lecture Notes in Control and Information Sciences. Springer-Verlag, Heidelberg, July 1991.

25 T. Penzl.A cyclic low rank Smith method for large sparse Lyapunov equations.SIAM J. Sci. Comput., 21(4):1401–1418, 2000.

34/35

ADI forLyapunov and

Riccati Equations

Peter Benner


ADI for Lyapunov

Newton-ADI forAREs

Software


References

References

26 T. Penzl.Lyapack Users Guide.Technical Report SFB393/00-33, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern,TU Chemnitz, 09107 Chemnitz, FRG, 2000.Available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html.

27 Y. Saad.Numerical Solution of Large Lyapunov Equation.In M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, editors, Signal Processing, Scattering, Operator Theory andNumerical Methods, pages 503–511. Birkhauser, Basel, 1990.

28 J. Saak, H. Mena, and P. Benner.Matrix Equation Sparse Solvers (MESS): a MATLAB Toolbox for the Solution of Sparse Large-Scale Matrix Equations.In preparation.

29 V. Simoncini.A new iterative method for solving large-scale Lyapunov matrix equations.SIAM J. Sci. Comput., 29:1268–1288, 2007.

30 E.L. Wachspress.Iterative solution of the Lyapunov matrix equation.Appl. Math. Letters, 107:87–90, 1988.

31 K. Willcox and J. Peraire.Balanced model reduction via the proper orthogonal decomposition.AIAA J., 40(11):2323, 2002.

32 N. Wong, V. Balakrishnan, C.-K. Koh, and T.S. Ng.Two algorithms for fast and accurate passivity-preserving model order reduction.IEEE Trans. CAD Integr. Circuits Syst., 25(10):2062–2075, 2006.

33 N. Wong and V. Balakrishnan.Fast positive-real balanced truncation via quadratic alternating direction implicit iteration.IEEE Trans. CAD Integr. Circuits Syst., 26(9):1725–1731, 2007.35/35

http://www.tu-chemnitz.de/sfb393/sfb00pr.html

Documents

ADI-based Galerkin-Methods for Algebraic Lyapunov and ......2010/06/08 · ADI-based Galerkin-Methods for Algebraic Lyapunov and Riccati Equations Peter Benner Max-Planck-Institute