h2 salami slicing & -pseudo-optimal model order … castagnotto 3 introduction: model order...
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SIAM chapter at Virginia Tech | Blacksburg VA | 05.11.2015
Alessandro CastagnottoBoris Lohmann
H2 Salami slicing & -pseudo-optimal
model order reduction
MOR
Alessandro Castagnotto 2
Who is this Italian guy talking about salamis?
Research assistant
Chair of Automatic Control
Department of Mechanical Engineering
Technische Universität München
www.rt.mw.tum.de
Research interests:
Automated, adaptive model order reduction oflarge-scale systems by Krylov-subspace methods
At VT:
Visiting Serkan Gugercin until December 24th
(office 407)
Alessandro Castagnotto
M.Sc. Mechanical Engineering
Alessandro Castagnotto 3
Introduction: model order reduction (MOR)
Source(s): nasa.gov, wikimedia.org, dailymail.co.uk
MOR
LTI system in state space
good approximation
preservation of properties
numerically efficient
Alessandro Castagnotto 4
Projective MOR
Approximation in the subspace .
Procedure:
1. Plug in in state equation2. Reduce the number of equations (cf. projection by )3. Petrov-Galerkin condition
Alessandro Castagnotto 5
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Magnit
ude
/dB
Frequency / rad/ sec
original
Interpolatory MOR (moment matching/ Krylov-subspace methods)
Moments of a transfer function.
: Interpolation frequency (shift): i-th moment about
moments about of the full order model are matched
Moment Matching by Krylov-subspaces
Choose V und W such that:
Alessandro Castagnotto 6
MOR by Krylov-subspace methods
Sylvester equations – a handy tool
A new error formulation
Cumulative reduction (CURE)
pseudo optimality - more than „pseudo“
Stability-preserving, adaptive rational Krylov
Agenda
Alessandro Castagnotto 7
Heiko Peuscher, Thomas Wolf, Boris Lohmann
Heiko K.F. Peuscher Thomas Wolf Boris Lohmann
[Panzer Model Order Reduction by Krylov Subspace Methods with Global Error
Bounds and Automatic Choice of Parameters, 2014]
[Wolf H2 Pseudo-Optimal Model Order Reduction, 2014]
… and references therein.
(Panzer)
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Sylvester equation for Krylov subspaces
Recall:
-> compact representation of the projection matrices (Krylov subspaces)-> general: covers all cases (SISO/MIMO, tangential/block Krylov)-> will be used as a tool in the following
Consider the input Krylov subspace (SISO): [Gallivan et al. 04]
Alessandro Castagnotto 9
MOR by Krylov-subspace methods
Sylvester equations – a handy tool
A new error formulation
Cumulative reduction (CURE)
pseudo optimality - more than „pseudo“
Stability-preserving, adaptive rational Krylov
Agenda
Alessandro Castagnotto 10
A new error formulation
Consider the approximation error:
Transform the error system
Ingredients:
Alessandro Castagnotto 11
MOR by Krylov-subspace methods
Sylvester equations – a handy tool
A new error formulation
Cumulative reduction (CURE)
pseudo optimality - more than „pseudo“
Stability-preserving, adaptive rational Krylov
Agenda
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Cumulative Reduction (CURE)
Adaptive choice of reduced order
„Salamitaktik“
Alessandro Castagnotto 13
New paradigm: minimize the error bound instead of the true error!
Rigorous, global error bounds
So when should we stop?
[Panzer 14]
Rigorous, global error bounds that are cheap to evaluate are still an open problem
For systems in strictly dissipative form, i.e. , you have
Alessandro Castagnotto 14
MOR by Krylov-subspace methods
Sylvester equations – a handy tool
A new error formulation
Cumulative reduction (CURE)
pseudo optimality - more than „pseudo“
Stability-preserving, adaptive rational Krylov
Agenda
Alessandro Castagnotto 15
(pseudo) optimal MOR
How should we choose the shifts (interpolation points)?
shifts
red. eigenvalues
Sources(n): Thomas Wolf
Idea by Wilson:
Optimize the subspace and then pick the pseudo optimum
Necessary conditions for local H2 optimality (SISO):(Meier-Luenberger)
Necessary and sufficient conditions for global H2 pseudo optimality (SISO)
(pseudooptimal: optimal in a subspace)
[Gugercin et al. 08]
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Example: H2 (pseudo) optimal MOR
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SISO,
Example: pseudo optimality:
Example: pole ↔ residue Example: error
pseudo-opt.local opt.global opt.
Source(s): Thomas Wolf
Alessandro Castagnotto 17
Source(s): Thomas Wolf
Pseudo-optimal ROMs can be given explicitly
Stability is preserved by construction
Advantages of H2-pseudo-optimal reduction
almost FOM-indepenent
V (Krylov subspace) is given
Choice of shifts is twice as
important
H2 pseudo-optimality makes the reduction easier in various respects
Let satisfy
The pseudo-optimal reduced order
model is given by
Algo.: Pseudo-Optimal Rational Krylov (PORK) [Wolf 14]
(cf. Hilbert projection thm/orthogonality principle)
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Stability Preserving Adaptive Rational Krylov (SPARK)
Combine advantages of CURE and PORK for optimal choice of shifts
Goal: minimize the norm of the approximation error
pseudo-optimality
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x 10-6
[Panzer 14]
relativ. cheap
gradient/Hessian
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Conclusions
Sylvester equations for Krylov subspaces
Factorization of the error
CUREd SPARK
Adaptive selection of reduced order
Stability preservation
Maximization of the reduced order norm
Current/future work
DAEs (arXiv)
MIMO2nd order systems
Alessandro Castagnotto 20
Announcements
Expected release: November 15th
sys = sss(A,B,C,D,E);
bode(sys)
step(sys)
norm(sys)
isstable(sys)...
sysr = tbr(sys,n)
sysr = rk(sys,s0)
sysr = irka(sys,s0)
2) Matrix computation seminar Dec. 1st: Fast H2-optimal model order reduction
1) MATLAB toolbox (free, open source) for sparse state-space
Alessandro Castagnotto 21
Q & A
Source(s): mytanach.com
Alessandro Castagnotto 22
References
[Castagnotto et al. 15] Stability-preserving, adaptive model order reduction of DAEs byKrylov-subspace methods (arXiv: 1508:07227)
[Gallivan et al. 04] Sylvester equations and projection-based model reduction
[Gugercin et al. 08] H2 model reduction for large-scale linear dynamical systems
[Panzer et al. 13] A greedy rational Krylov method for H2-pseudooptimal modelorder reduction with preservation of stability
[Panzer 14] Model order reduction by Krylov subspace methods with global error bounds and automatic choice of parameters
[Wolf et al. 13] H2 pseudo-optimality in model order reduction by Krylov subspace Methods
[Wolf 14] H2 pseudo-optimal model order reduction