h2 salami slicing & -pseudo-optimal model order … castagnotto 3 introduction: model order...

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

a.castagnotto@tum.de

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

100

101

102

103

-100

-90

-80

-70

-60

-50

-40

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)

Alessandro Castagnotto 8

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

Alessandro Castagnotto 12

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]

Alessandro Castagnotto 16

Example: H2 (pseudo) optimal MOR

-106

-104

-102

-100

-10-2

-100

-50

0

50

100

150

-106

-104

-102

-100

-10-2

193.25

193.3

193.35

193.4

193.45

193.5

193.55

193.6

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)

Alessandro Castagnotto 18

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

10-4

10-2

100

102

104

106

10-2

100

102

104

106

108

-9

-8

-7

-6

-5

-4

-3

-2

-1

x 10-6

[Panzer 14]

relativ. cheap

gradient/Hessian

Alessandro Castagnotto 19

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