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Theoretical and practical aspects of linear and nonlinear model order reduction techniques

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### Theoretical and practical aspects of linear and nonlinear model order reduction techniques

OutlineOutlineOutline

Dmitry Vasilyev

Thesis supervisor: Jacob K White

December 19, 2007

Outline

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Nonlinear

Linear

Main motivation for MOR: system-level simulation

Q: How to reduce the cost of simulating the big system?

System to simulate

Device 2

Device 1

A: Reduce the complexity of each sub-system, i.e.

approximate input-output behavior of the system by a system of lower complexity.

Device 3

…

…

Device 10

The goal of MOR in a nutshell

Main motivation for MOR: system-level simulation

Modern processor or

system-on-a chip

> Millions of transistors

> Kilometers of interconnects> Linear and nonlinear devices

Figures thanks to Mike Chou, Michał Rewienski

Model reduction problem

inputs

outputs

Many (> 104)

internal states

inputs

outputs

few (<100)

internal states

- Reduction should be automatic
- Must preserve input-output properties

Differential equation model

- Original complex model:

- Model can represent:
- Finite-difference spatial discretization of PDEs
- Circuits with linear capacitors and inductors

Need accurate input-output behavior

Nonlinear model reduction problem

- Original complex model:

- Requirements for reduced model
- Want q << n (cost of simulation is q3)
- f r should be fast to compute
- Want yr(t) to be close to y(t)

- Reduced model:

Linear Model

- state

A – stable, n xn (large)

- vector of inputs

- vector of outputs

- Model can represent:
- Spatial discretization of linear PDEs
- Circuits with linear elements

Transfer function of LTI system

Transfer function:

Laplace transform of the output

Laplace transform of the input

Matrix-valued rational function of s

Outline

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Linear MOR problem

n – large

(thousands)!

q – small (tens)

Need the reduction to be automatic and preserve

input-output properties (transfer function)

Approximation error

- Wide-band applications: model should have small worst-case error

maximal difference

over all frequencies

ω

Approximation error

- Narrow-band approximation: need good approximation only near particular frequency:

Elmore delay:

preserved if the first derivative at zero frequency is matched.

ω

frequency response

Linear MOR methods roadmap

Linear MOR

Projection-based

Non projection-based

Most widely used.

Will be the central topic of this work.

Projection-based linear MOR

- Important: reduced system depends only on column spans of V and U
- D = Dr, preserves response at infinite frequency

Linear MOR methods roadmap

Linear MOR

Projection-based

Non projection-based

V, U =

eig{PQ, QP}

Balancing-based (TBR)

V, U =

K((si I-A)-1,B), K((si I-A)-T,CT)

Krylov-subspace

methods

Proper Orthogonal

Decomposition methods

V=U = {x(t1)… x(tq )}

Linear MOR methods roadmap

Linear MOR

Projection-based

Non projection-based

Will describe

next.

Balancing-based (TBR)

V, U =

K((si I-A)-1,B), K((si I-A)-T,CT)

Krylov-subspace

methods

Proper Orthogonal

Decomposition methods

V=U = {x(t1)… x(tq )}

TBR idea

u

y

LTI SYSTEM

t

t

input

output

P (controllability)

Which states are easier to reach?

Q (observability)

Which states produce more output?

X (state)

- Reduced model retains most controllable and most observable states
- Such states must be both very controllable and very observable

Balanced truncation reduction (TBR)

Compute controllability and observability gramians P and Q : (~n3)AP + PAT + BBT =0 ATQ + QA + CTC = 0

Reduced model keeps the dominant eigenspaces of PQ : (~n3)

PQui= λiuivTiPQ = λivTi

Reduced system: (VTAU, VTB, CU, D)

Very expensive. P and Q are dense even for sparse models

TBR benefits

- Guaranteed stability
- In practice provides more reduction than Krylov
- H-infinity error bound => ideal for wide-band approximations

Hankel

singular values

Linear MOR methods roadmap

Linear MOR

Projection-based

Non projection-based

Twice better error

bound than TBR

[Glover ’84]

Hankel-optimal MOR

Singular perturbation

approximation

Match at zero frequency

instead of infinity [Liu ‘89]

Transfer function

fitting methods

Promising topic

of ongoing research [Sou ‘05]

Outline

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Nonlinear MOR framework

- Consider original (large) system:

- Projection of the nonlinear operator f(x):

substitute x ≈ Uz and project residual onto VT

Problem: evaluation of V Tf(Uz)is still expensive

Nonlinear MOR framework

Problem: evaluation of V Tf(Uz)is still expensive

Two solutions:

Use Taylor series off

Use TPWL approximation

Taylor series for nonlinear MOR

Problem: evaluation of V Tf(Uz)is still expensive

Two solutions:

Use Taylor series off

Use TPWL approximation

- Accurate only near expansion point or weakly nonlinear systems
- Storing of dense tensors is expensive; limits the series to orders no more than 3.

Nonlinear MOR framework

Problem: evaluation of V Tf(Uz)is still expensive

Two solutions:

Use Taylor series off

Use TPWL approximation

Will be discussed next

Trajectory piecewise linear (TPWL) approximation of f( ) [Rewieński, 2001]

Training trajectory

x0

x2

x1

…

wi(x)is zero outside circle

xs

Simulating trajectory

Projection and TPWL approximation yields efficient f r( )

q x1

Air

Ai

q

U

=

Air

q

n

n

Evaluating fTPWLr( ) requires only O(sq2) operations

TPWL approximation of f( ). Extraction algorithm

- Compute A1
- Obtain V1 and U1using linear reduction for A1
- Simulate training input, collect and reduce linearizations Air = W1TAiV1f r (xi)=W1Tf(xi)

Initial system position

x0

x2

x1

…

xs

Training trajectory

Non-reduced state space

Outline

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Krylov-subspace methods

Balanced-truncation method

The matter of this contributionWhat are projection options for TPWL?

Can we use it?

Used in the original work

[Rewienski ‘02]

Numerical results – nonlinear RLC transmission line

System response for input current i(t) = (sin(2π/10)+1)/2

- Input:

training

input

testing

input

Voltage at node 1 [V]

Time [s]

Numerical results –RLC transmission line

TBR-based TPWL beat

Krylov-based

4-th order TBR TPWL reaches the limit of TPWL representation

Error in transient

||yr – y||2

Order of the reduced model

Micromachined switch example

Finite-difference

model of order 880

non-symmetric

indefinite Jacobian

Model description [Hung ‘97]

TPWL-TBR results– MEMS switch example

Errors in transient

Unstable!

Odd order models unstable!

Even order models beat Krylov

||yr – y||2

Why???

Order of reduced system

Explanation of even-odd effect – Problem statement

Consider two LTI systems:

Perturbed:

Initial:

TBR reduction

TBR reduction

~

Projection basis V

Projection basis V

Define our problem:

How perturbation in the initial system

affects TBR projection matrices?

Perturbation behavior of TBR basis is similar to symmetric eigenvalue problem

Eigenvectors ofM0 :

Eigenvectors ofM0 + Δ:

Mixing of eigenvectors (assuming small perturbations):

ciklarge when λi0 ≈ λk0

Hankel singular values, MEMS beam example

This is the key to the problem.

Singular values are arranged in pairs!

# of the Hankel singular value

Explaining even-odd behavior

The closer Hankel singular

values lie to each other, the

more corresponding eigenvectors

of V tend to intermix!

- Analysis implies simple recipe for using TBR
- Pick reduced order to ensure that
- Remaining Hankel singular values are small enough
- The last kept and the first removed Hankel singular values are well separated

Helps to ensure that linearizations are stable

Summary

- We used TBR-based linear reduction procedure to generate TPWL reduced models
- Order reduced 5 times while maintaining comparable accuracy with Krylov TPWL method (efficiency improved 125 times!)
- Simple recipe found which helps to ensure stability.

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Balanced truncation reduction (TBR)

Compute controllability and observability gramians P and Q : (~n3)AP + PAT + BBT =0 ATQ + QA + CTC = 0

Reduced model keeps the dominant eigenspaces of PQ : (~n3)

PQui= λiuivTiPQ = λivTi

Reduced system: (VTAU, VTB, CU, D)

Very expensive. P and Q are dense even for sparse models

Most reduction algorithms effectively separately approximate dominant eigenspaces of Pand Q:

- Arnoldi [Grimme ‘97]:U = colsp{A-1B, A-2B, …}, V=U, approx. Pdomonly
- Padé via Lanczos [Feldman and Freund ‘95]colsp(U) = {A-1B, A-2B, …}, - approx. Pdomcolsp(V) = {A-TCT, (A-T )2CT, …},- approx. Qdom
- Frequency domain POD [Willcox ‘02], Poor Man’s TBR [Phillips ‘04]

colsp(U) = {(jω1I-A)-1B, (jω2I-A)-1B, …}, - approx.Pdom

colsp(V) = {(jω1I-A)-TCT, (jω2I-A)-TCT, …},- approx.Qdom

However, what matters is the product PQ

RC line (symmetric circuit)

V(t) – input

i(t) - output

- Symmetric Jacobian, B=CT,

P=Qall controllable states are observable and vice versa

RLC line (nonsymmetric circuit)

Vector of states:

- P and Q are no longer equal!
- By keeping only mostly controllable and/or only mostly observable states, we may not find dominant eigenvectors of PQ

Lightly damped RLC circuit

y(t) = i1

R = 0.008,

L = 10-5

C = 10-6

N=100

- Exact low-rank approximations of P and Q of order < 50 leads to PQ≈ 0

“iterate”

AISIAD model reduction algorithmIdea of AISIAD approximation:

Approximate eigenvectors using power iterations:

Uiconverges to dominant eigenvectors ofPQ

Need to find the product (PQ)Ui

How?

Approximation of the product Ui+1=qr(PQUi), AISIAD algorithm

Vi≈ qr(QUi)

Ui+1≈ qr(PVi)

Approximate using solution of Sylvester equation

Approximate using solution of Sylvester equation

Modified AISIAD approximation

Right-multiply by Vi

^

X

H, qxq

X

Approximate!

M, nxq

We can take advantage of various

methods, which approximate P and Q

Solving Sylvester equation

- Applicable to any stable A
- Requires solving q times

Schur decomposition of H :

Solution can be accelerated via fast MVP

Original method suggests IRA, needs A>0 [Zhou ‘02]

Modified AISIAD algorithm

LR-sqrt

^

^

- Obtain low-rank approximations of Pand Q
- Solve AXi+XiH+ M = 0, => Xi≈ PVi where H=ViTATVi, M = P(I - ViViT)ATVi + BBTVi
- Perform QR decomposition of Xi =UiR
- Solve ATYi+YiF+ N = 0, => Yi≈ QUi where F=UiTAUi, N = Q(I - UiUiT)AUi + CTCUi
- Perform QR decomposition of Yi =Vi+1 Rto get new iterate.
- Go to step 2 and iterate.
- Bi-orthogonalize VandUand construct reduced model:

^

^

(VTAU, VTB, CU, D)

RLC line example results

H-infinity norm of reduction error

(worst-case discrepancy over all frequencies)

N = 1000,

1 input

2 outputs

Summary of the modified AISIAD

- Fast approximation to TBR
- Especially useful if gramians do not share common dominant eigenspace
- Improved accuracy and extended applicability over AISIAD
- Generalized to the systems in descriptor form

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Features of the method

Reduction quality

Hankel-optimal

TBR

Krylov-subspace methods

Graph-based reduction:

manipulates RC network by removing nodes and inserting new elements

Cost of reduction

Linear RC network description

State-space model in the frequency domain:

vs

vm

Vector of node voltages (state):

Jm

vk

symmetric

Jk

External ports

Conductance matrix is analogous, ground node is excluded.

Low-frequency approximation for reduced circuit

- Consider removing a single internal node (Nth), partition matrices and vectors:

- Substitute vN in the system equations (one step of Gaussian elimination):

- Where

Node elimination

- Problem: last capacitance term is negative! Potentially inserting a negative capacitor???
- The term was ignored in the TICER algorithm [Sheehan ‘99]. Leads to inconsistent diagonal update.

Added

conductance

Capacitance-like

Node elimination – Theorem 1

- Claim: keeping the exact Taylor series is OK:

Gnew

Cnew

- Proof: Define projection:

Congruence transform

Model is always

stable and passive

Node elimination criteria

- When is it safe to eliminate a node?

- Denominator expansion:

(used in TICER)

- Numerator term ~s2 (element-by-element)

(overlooked in TICER)

Using these criteria the reduced order will be chosen on-the-fly

Resulting algorithm:

- Given the initial circuit and maximal frequency of interest
- Using lowest-degree ordering (minimize fill-ins)
- Perform the elimination of the “qualified” nodes by inserting new capacitors and resistors:

(for every nodes i and j which were connected via the node N)

- Until no nodes satisfy elimination conditions.

Results: testing substitution rules

Testing only substitution rules, 1-CDF of the reduction error

tested more than

30,000 circuits

Results: testing elimination conditions

Same elimination rules, same average reduction

different elimination criteria:

Narrower distribution

Better worst-case accuracy

Summary of the new method

- Improved accuracy and error control over TICER by using correct Taylor series and elimination criteria
- Preserves stability and passivity
- Generalized to parameter-dependent case
- Fastest, though conservative

- Motivation
- Overview of existing methods:
- Linear MOR
- Nonlinear MOR
- TBR-based trajectory piecewise-linear method
- Modified AISIAD linear reduction method
- Graph-based model reduction for RC circuits
- Case study: microfluidic channel
- Conclusions

Electro-osmotic flow in the 3D U-shaped microchannel

- Inside the carrying fluid, marker fluid spreads governed by 3D convection-diffusion equation:

u(t) = Cin(t)

(input)

r1

(outputs)

y2(t)

w

- Using mapped-domain finite-difference volume discretization, obtained model has 2842 unknowns (large)

y1(t) =<Cout>(t)

y3(t)

V

Linear case

- In case of constant mobility and diffusivity the model is linear:

Linear reduction techniques are extremely efficient for such models

[Vasilyev, Rewienski, White ‘06]

Nonlinear microchannel problem

For arbitrary nonlinearity in convection and diffusion coefficients and TPWL, this problem is verychallenging!

[Vasilyev, Rewienski, White ‘06]

However, the problem becomes more tractable, if one considers a quadratic problem

This is the case for affine μand D:

μ(C) = μ0C+ μ1

D(C) = D0C+ D1

Quadratic model of microchannel system

- Affine mobility and diffusivity leads to quadratic model:

- Use orthogonal projection V = U, V TV = I

Reduced quadratic system

Quadratic microchannel problem - result

Projected reduced quadratic model of size 60 approximates original system of size 2842 quite well:

Krylov-subspace basis,

Quadratic reduction

Conclusions

- Performed applicability analysis of TBR-based TPWL models based on matrix perturbation theory
- Developed modified AISIAD method which is aimed at approximating TBR for the cases where gramians do not necessarily share common dominant eigenspaces
- Developed graph-based parameterized RC reduction method and improved nominal reduction

I extend my sincere thanks to:

Prof. Jacob White – my supervisor,

Profs. Luca Daniel and Alexandre Megretski,

Profs. Karen Willcox, John Kassakian, John Wyatt, Dr. Yehuda Avniel, Dr. Joel Phillips, Dr. Mark Reichelt

My groupmates: Anne, Bo, Brad, Carlos, Dave, Jay, Jung Hoon, Homer, Kin, Laura, Lei, Michał, Shihhsien, Steve, Tarek, Tom, Xin, Zhenhai

My wife, Patrycja

Thank you! Спасибо!Grazie! Dziękuje!

Komapsumnida!Xie Xie! Dua Netjer en ek!

For systems in the descriptor form

Generalized Lyapunov equations:

Lead to similar

approximate power iterations

mAISIAD and low-rank square root

Low-rank gramians

(cost varies)

mAISIAD

LR-square root

(inexpensive step)

(more expensive)

For the majority of non-symmetric cases,

mAISIAD works better than low-rank square root

RLC line example results

H-infinity norm of reduction error

(worst-case discrepancy over all frequencies)

N = 1000,

1 input

2 outputs

Steel rail cooling profile benchmark

Taken from Oberwolfach benchmark collection, N=1357 7 inputs, 6 outputs

mAISIAD is useless for symmetric models

For symmetric systems (A = AT, B = CT) P=Q, therefore

mAISIAD is equivalent to LRSQRT for P,Q of order q

^

^

RC line example

Cost of the algorithm

- Cost of the algorithm is directly proportional to the cost of solving a linear system:(where sjj is a complex number)
- Cost does not depend on the number of inputs and outputs

(non-descriptor case)

(descriptor case)

Lightly damped RLC circuit

Top 5 eigenvectorsof Q

Top 5 eigenvectors of P

Union of eigenspaces of P and Q

does not necessarily approximate

dominant eigenspace of PQ .

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