part i: overview thanos antoulas - startseite tu ilmenau · introduction and model reduction...

Model reduction of large-scale systems

Part I: Overview

Thanos Antoulas

Rice University and Jacobs University

email: [email protected]: www.ece.rice.edu/ aca

1st Elgersburg School on Mathematical System TheoryTU Ilmenau, 30 March - 3 April 2009

Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 1 / 40

Outline

1 Introduction and model reduction problem

The big picture

Problem formulation

Projections

2 Motivating examples

3 Overview of approximation methods

SVD-based methods

Krylov methods

4 Summary – Challenges – References


Introduction and model reduction problem

Outline


The big picture

Problem formulation

Projections



SVD-based methods

Krylov methods



Introduction and model reduction problem The big picture

Recall: the big picture

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Modeling

Model reduction

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Simulation

Controlreduced # of ODEs

ODEs PDEs

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Physical System and/or Data


Introduction and model reduction problem Problem formulation

Dynamical systems

x1(·)x2(·)

...xn(·)

u1(·) −→u2(·) −→

...um(·) −→

−→ y1(·)−→ y2(·)

...−→ yp(·)

We consider (implicit) state equations

Σ : f(x(t) ,x(t), u(t)) = 0, y(t) = h(x(t), u(t))

with state (internal vartiable) x(·) of dimension n� m,p.


Introduction and model reduction problem Problem formulation

Problem statement

Given: dynamical system

Σ = (f,h) with: u(t) ∈ Rm, x(t) ∈ Rn, y(t) ∈ Rp.

Problem: Approximate Σ with:

Σ = (f, h) with : u(t) ∈ Rm, x(t) ∈ Rk , y(t) ∈ Rp, k � n :

(1) Approximation error small - global error bound

(2) Preservation of stability/passivity

(3) Procedure must be computationally efficient


Introduction and model reduction problem Projections

Approximation by projection

Unifying feature of approximation methods: projections.

Let V, W ∈ Rn×k , such that W∗V = Ik ⇒ Π = VW∗ is a projection.

Define x = W∗x ⇒ x ≈ Vx. Then

Σ :

{ddt x(t) = W∗f(Vx(t), u(t))

y(t) = h(Vx(t), u(t))

Thus Σ is ”good” approximation of Σ, if x− Πx is small.


Introduction and model reduction problem Projections

Special case: linear dynamical systems

Σ: Ex(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t)

Σ =

(E,A BC D

)Problem: Approximate Σ by projection: Π = VW∗

Σ =

(E, A BC D

)=

(W∗EV,W∗AV W∗B

CV D

), k � n

Norms:• H∞-norm:worst output error‖y(t)− y(t)‖ for ‖u(t)‖ = 1.

• H2-norm: ‖h(t)− h(t)‖

E,An

n

C

B

D

⇒ E, Ak

k

C

B

D

Σ: : Σ

= Vn

k

W∗


Motivating examples

Outline


The big picture

Problem formulation

Projections



SVD-based methods

Krylov methods



Motivating examples

Motivating Examples: Simulation/Control

1. Passive devices • VLSI circuits• Thermal issues• Power delivery networks

2. Data assimilation • North sea forecast• Air quality forecast

3. Neural models • Simulations4. CVD reactor • Bifurcations5. Mechanical systems: •Windscreen vibrations

• Buildings6. Optimal cooling • Steel profile7. MEMS: Micro Electro-

-Mechanical Systems • Elf sensor8. Nano-Electronics • Plasmonics


Motivating examples

Passive devices: VLSI circuits

1960’s: IC 1971: Intel 4004 2001: Intel Pentium IV10µ details 0.18µ details2300 components 42M components64KHz speed 2GHz speed

2km interconnect7 layers


Motivating examples

Passive devices: VLSI circuits

Typical GateDelay

0.1

1.0D

elay

(ns)

1.01.3 0.8 0.30.5Technology (μm)

0.1 0.08

Average WiringDelay

≈ 0.25 μm

Today’s Technology: 65 nm

65nm technology: gate delay < interconnect delay!

Conclusion: Simulations are required to verify that internal electromagneticfields do not significantly delay or distort circuit signals. Thereforeinterconnections must be modeled.

⇒ Electromagnetic modeling of packages and interconnects⇒ resultingmodels very complex: using PEEC methods (discretization of Maxwell’sequations): n ≈ 105 · · · 106 ⇒ SPICE: inadequate

• Source: van der Meijs (Delft)Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 12 / 40

Motivating examples

Power delivery network for VLSI chips

VDD

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AA AA AA AA

AA AA AA AA

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R L

C C

#states ≈ 8 · 106

#inputs/outputs ≈ 1 · 106

1


Motivating examples

Mechanical systems: carsCar windscreen simulation subject to acceleration load.

Problem: compute noise at points away from the window.

PDE: describes deformation of a structure of a specific material; FEdiscretization: 7564 nodes (3 layers of 60 by 30 elements). Material:glass with Young modulus 7·1010 N/m2; density 2490 kg/m3; Poissonratio 0.23⇒ coefficients of FE model determined experimentally.The discretized problem has dimension: 22,692.

Notice: this problem yields 2nd order equations:

Mx(t) + Cx(t) + Kx(t) = f(t).

• Source: Meerbergen (Free Field Technologies)


Motivating examples

Mechanical Systems: Buildings, Earthquake prevention

Taipei 101: 508m Damper between 87-91 floors 730 ton damper

Building Height Control mechanism Damping frequencyDamping mass

CN Tower, Toronto 533 m Passive tuned mass damperHancock building, Boston 244 m Two passive tuned dampers 0.14Hz, 2x300tSydney tower 305 m Passive tuned pendulum 0.1,0.5z, 220tRokko Island P&G, Kobe 117 m Passive tuned pendulum 0.33-0.62Hz, 270tYokohama Landmark tower 296 m Active tuned mass dampers (2) 0.185Hz, 340tShinjuku Park Tower 296 m Active tuned mass dampers (3) 330tTYG Building, Atsugi 159 m Tuned liquid dampers (720) 0.53Hz, 18.2t


Motivating examples

Neural models: How a cell distinguishes between inputs

Image from Neuromart (Rice-Baylor archive of neural morphology)

Goal: • simulation of systems containing a few million neurons• simulations currently limited to systems with ≈10K neurons


Motivating examples

Example: neuron simulationPropagation of action potentials

The Hodgkin-Huxley model: C · dVdt = Iext − ICa − INa − IL

V is the potential across the membrane. The currents ICa , INa und IL (with given potentials ECa , ENa , EL) are:

ICa = gCa · (V − ECa), INa = gNa · (V − ENa) and IL = gL · (V − EL),

The conductivities gCa und gNa depend on time, while gL is constant. H-H postulated the existence of gate variables, whichrepresent the dynamics of the ion channels (i.e. the active channels). Experimental results led to three gate variables: n, m, h.Together with the ion conductivity g, gate variables describe the current which flows through the membrane:

ICa = gCa · n4 · (V − ECa) and INa = gNa · m3 · h · (V − ENa).

The three gate variables are described as follows:dndt = αn(V ) · (1− n)− βn(V ) · n, dm

dt = αm(V ) · (1− m)− βm(V ) · m, dhdt = αh(V ) · (1− h)− βh(V ) · h,

where αn(V ), αm(V ), αh(V ), βn(V ), βm(V ), βh(V ), are exponential functions.

Example. The model size is dictated by synapse density and total dendritic length. For instance, with approximately one synapseper micron on a cell with 5mm of dendrite we obtain 5000 compartments. Under usual conditions there are 10 equations percompartment (1 voltage and 9 gating) ⇒ 50,000 coupled equations.


Overview of approximation methods

Outline


The big picture

Problem formulation

Projections



SVD-based methods

Krylov methods




Large-scale systems

What is the problem with very large systems?

⇓

• Storage

• Computational speed

• Accuracy

• System theoretic properties



Approximation methods: OverviewPPPPPPPPq

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Krylov

• Realization• Interpolation• Lanczos• Arnoldi

SVD

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Nonlinear systems Linear systems• POD methods • Balanced truncation• Empirical Gramians • Hankel approximation

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Krylov/SVD Methods


Overview of approximation methods SVD-based methods

Approximation methods: a prototype problem

The Singular value decomposition: SVD

A = UΣV ∗ ∈ Rn×m

Singular values: Σ = diag (σ1, · · · , σn), σ1 ≥ · · · ≥ σn ≥ 0

⇒ σi =√λi(A∗A)

left singular vectors: U = (u1 u2 · · · un), UU∗ = Inright singular vectors: V = (v1 v2 · · · vm), VV ∗ = ImDyadic decomposition:

A = σ1u1v∗1 + σ2u2v∗2 + · · ·+ σnunv∗n

σ1: 2-induced norm of A



Reminder: 2-norm

Vectors:

x =

x1...

xn

⇒ ‖ x ‖2 =√

x21 + · · ·+ x2

n

Matrices/Operators: induced 2-norm

A

⇒ ‖ A ‖2 = max‖x‖=1 ‖ Ax ‖2 = σ1



Optimal approximation in the 2-norm

Given: A ∈ Rn×m

find: X ∈ Rn×m, rank X = k < rank ACriterion: norm(error) is minimized, whereerror: E = A− X , norm: 2-norm

Theorem (Schmidt-Mirsky, Eckart-Young)

minrankX≤k

‖ A− X ‖2 = σk+1(A)

Minimizer (non-unique): truncation of dyadic decomposition of A:

X# = σ1u1v∗1 + σ2u2v∗2 + · · ·σkukv∗k



Remarks.

(a) Importance of Schmidt-Mirsky: establishes a relationship betweenthe rank k of the approximant, and the (k + 1)st singular value of A.

(b) Other minimizers:

X (η1, · · · , ηk ) :=∑k

i=1 (σi − ηi)uiv∗i ,

where 0 ≤ ηi ≤ σk+1.

(c) The problem of minimizing the 2-induced norm of A− X over allmatrices X of rank at most k , is non-convex.

(d) The SVD cannot be used to approximate unitary matrices, whereσk = 1, for all k .

(e) Problem can also be solved in the Frobenius norm.



SVD Approximation methods: two static examples

Supernova Clown

Singular values provide trade-off between accuracy and complexity


Overview of approximation methods Krylov methods

Projectors for Krylov and rational Krylov methods

Given:

Σ =

(E,A BC D

)by projection: Π = VW∗, Π2 = Π obtain

Σ =

(E, A BC D

)=

(W∗EV,W∗AV W∗B

CV D

), where k < n.

Krylov (Lanczos, Arnoldi): let E = I and

V =[B, AB, · · · , Ak−1B

]∈ Rn×k

W∗ =

C

CA...

CAk−1

∈ Rk×n

⇒ W∗ = (W∗V)−1W∗

then the Markov parameters match:

CAiB = CAiB

Rational Krylov: let

V =[

(λ1E− A)−1B · · · (λk E− A)−1B]∈ Rn×k

W∗ =

C(λk+1E− A)−1

C(λk+2E− A)−1

.

.

.C(λ2k E− A)−1

∈ Rk×n

⇒ W∗ = (W∗V)−1W∗

then the moments of G match those of G at λi :

G(λi) = D + C(λiE−A)−1B = D + C(λiE− A)−1B = G(λi)


Overview of approximation methods Krylov methods

Properties of Krylov methods

(a) Number of operations: O(kn2) or O(k2n) vs. O(n3)⇒ efficiency

(b) Only matrix-vector multiplications are required. No matrix factorizationsand/or inversions. No need to compute transformed model and then truncate.

(c) Drawbacks

• global error bound?• Σ may not be stable.

Q: How to choose the projection points?


Summary – Challenges – References

Outline


The big picture

Problem formulation

Projections



SVD-based methods

Krylov methods




Approximation methods: SummaryPPPPPPPPq

��

Krylov

• Realization• Interpolation• Lanczos• Arnoldi

SVD

@@@R

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Nonlinear systems Linear systems• POD methods • Balanced truncation• Empirical Gramians • Hankel approximation

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Krylov/SVD Methods

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rProperties

• numerical efficiency

• n� 103

• choice of matching moments

Properties

• Stability

• Error bound

• n ≈ 103



(Some) Challenges in complexity reduction

Model reduction of uncertain systems

Model reduction of differential-algebraic (DAE) systems

Domain decomposition methods

Parallel algorithms for sparse computations in model reduction

Development/validation of control algorithms based on reducedmodels

Model reduction and data assimilation (weather prediction)

Active control of high-rise buildings

MEMS and multi-physics problems

VLSI design

Molecular Dynamics (MD) simulations

CAD tools for nanoelectronics



ReferencesPassivity preserving model reduction

Antoulas SCL (2005)Sorensen SCL (2005)Ionutiu, Rommes, Antoulas IEEE TCAD (2008)

Optimal H2 model reductionGugercin, Antoulas, Beattie SIMAX (2008)

Low-rank solutions of Lyapunov equationsGugercin, Sorensen, Antoulas, Numerical Algorithms (2003)Sorensen (2006)

Model reduction from dataMayo, Antoulas LAA (2007)Lefteriu, Antoulas, Tech. Report (2008)

General reference: Antoulas, SIAM (2005)


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