ADI-based Galerkin-Methods for Algebraic Lyapunov and Riccati Equations Peter Benner Max-Planck-Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Group Magdeburg, Germany Technische Universit¨ at Chemnitz Fakult¨ at f¨ ur Mathematik Mathematik in Industrie und Technik Chemnitz, Germany joint work with Jens Saak (TU Chemnitz) Universit´ e du Littoral Cˆ ote d’Opale, Calais June 8, 2010
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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
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
Motivation
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
References
Numerical ResultsLQR Problem for 2D Geometry
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Low-RankNewton-ADI
Application toLQR Problem
Galerkin-Newton-ADI
NumericalResults
Quadratic ADIfor AREs
High-Rank W
Software
Conclusions andOpen Problems
References
Numerical ResultsLQR Problem for 2D Geometry
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.
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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.
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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.
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
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
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
References
References
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ADI forLyapunov and
Riccati Equations
Peter Benner
Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
References
References
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Riccati Equations
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Large-ScaleMatrix Equations
ADI for Lyapunov
Newton-ADI forAREs
Software
Conclusions andOpen Problems
References
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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.
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