Introduction to model order reduction
Download
1 / 64

Introduction to Model Order Reduction - PowerPoint PPT Presentation


  • 131 Views
  • Uploaded on

Introduction to Model Order Reduction. Luca Daniel Massachusetts Institute of Technology [email protected] http://onigo.mit.edu/~dluca/2006PisaMOR. www.rle.mit.edu/cpg. Course Outline. Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE Solvers Model Order reduction

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Introduction to Model Order Reduction' - sawyer-herrera


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Introduction to model order reduction

Introduction to Model Order Reduction

Luca Daniel

Massachusetts Institute of Technology

[email protected]

http://onigo.mit.edu/~dluca/2006PisaMOR

www.rle.mit.edu/cpg


Course Outline

Numerical Simulation

Quick intro to PDE Solvers

Quick intro to ODE Solvers

Model Order reduction

Linear systems

Common engineering practice

Optimal techniques in terms of model accuracy

Efficient techniques in terms of time and memory

Non-Linear Systems

Parameterized Model Order Reduction

Linear Systems

Non-Linear Systems

Today


Model order reduction of non linear systems
Model Order Reduction of Non-Linear Systems

  • Introduction

  • Reduction of weakly nonlinear systems (Volterra Series)

  • Reduction of strongly nonlinear systems Trajectory PieceWise Linear (TPWL) and Polynomial (PWP)

    • with moment matching

    • with Truncated Balance Realization


Jet engine inlet example

  • Inlet density disturbance d:





Reduction of nonlinear systems
Reduction of NonLinear Systems

NonLinear Reduction

PDE Field Solvers or Circuit Simulators

Non-Linear analog components

e.g. MEMs, VCO, LNA


Projection framework

Change of variables

Projection Framework

Equation Testing


Projection framework

for nonlinear dynamic systems

in

  • Substitute:

  • Reduced system:

  • A problem:

q=10

n=104

n=104

q=10

  • Using VTF(Ux) is too expensive!


Nonlinear problem is much harder
Nonlinear Problem is MUCH Harder

  • In what basis should we project?

    • No simple frequency domain insight

    • No eigenmodes

    • No Krylov subspace

  • How do you represent the projection?

    • New problem for nonlinear


How to get the change of variable matrix
How to get the change of variable matrix

  • Analysis of linearized models [Ma88]

  • Sampling of time-simulation data + principal components analysis [Sirovich87]

  • Nonlinear balancing [Scherpen93],

  • Time-derivatives [Gunupudi99], …


Model Order Reduction

for NonLinear Systems

  • Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]*****

  • Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01]

  • Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]


Introduction to Model Order Reduction

III.2 – Reduction of Weakly Non Linear Systems

Luca Daniel

Thanks to Joel Phillips, Cadence Berkeley Labs


  • Approximate nonreduced model:

MOR for nonlinear dynamic systems

(J. Phillips, Y. Chen, F. Wang):

  • Linear, quadratic reduced order models:


Polynomial approximations

Expand nonlinearity in multi-dimensional polynomial series

Differential equation becomes

To match first few terms in functional series expansion, only need first few polynomial terms

Each term is a -dimensional

tensor, represented as an matrix

Polynomial Approximations

[note : not static models; frequency effects retained]

where

etc.


Projection of polynomial terms

Draw from reduced space as

Identity for Kronecker products

Project tensors

Gives reduced model

Projection of polynomial terms

etc.


Reduced polynomial models

Projection procedure produces reduced model in same polynomial form

Key tensor components are compressed to lower dimensionality

Kronecker forms provide convenient general notation

How to get projection spaces V?

Reduced polynomial models


Variational analysis

Introduce variational parameter polynomial form

Consider parametrized system

Expand response in powers of

Substitute into differential equation

Compare and collect terms for each power of

Variational Analysis


Variational analysis1

Coupled set of differential equations polynomial form

Each set computes response at one order

Each set is linear in its own state variable

First set is just linearized model

Variational Analysis

etc.


Model reduction

First order equation set driven by input of low rank

Krylov subspace for projection

Second order equation set driven by input

Key insight : recall is well-approximated by thus

Model Reduction


Model reduction1

Second order equation set is driven by

Generate second Krylov space using

Compute models by projection

each equation order set separately, or

project original polynomial system using UNION of

number of matched moments of order-i given by size of appropriate Krylov subspace

Model Reduction


Computational considerations

Size of Krylov spaces grows quickly with functional series order

potentially large models

intrinsic in polynomial descriptions (e.g. Volterra series)

Practical for weak nonlinearities needing only few terms in functional series

example: RF datapath

strong nonlinearities included in bias conditions

Many local optimizations, extensions are possible

two-sided projectors

multiple reduction steps

Computational Considerations


Rf mixer example

-20 order

-60

-80

0

50

100

150

200

MHz

RF mixer example

dB


Summary weakly nonlinear mor

Projection + polynomial series + variational analysis = rigorous nonlinear reduction procedure

Compact Kronecker notation is very useful

Is there a way to reduce number of terms in higher-order Krylov space?

Smaller models faster

Diagnose when models based on first order system are failing?

Follow-on reduction?

Summary weakly nonlinear MOR


Model Order Reduciton rigorous nonlinear reduction procedure

for NonLinear Systems

  • Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]*****

  • Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01]

  • Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]


Iii 3 reduction of strongly nonlinear systems via trajectory piecewise linear method

III.3 - Reduction of Strongly NonLinear Systems via Trajectory PieceWise Linear Method

Luca Daniel

Massachusetts Institute of Technology

Thanks to Michał Rewieński, Jacob White and Brad Bond

www.rle.mit.edu/cpg


Piecewise-linear representation Trajectory PieceWise Linear Method

[M. Rewienski, J. White ICCAD01]

  • Linearizations around xi , i=0,…, s-1

  • Weighted combination of the models:

  • Project linearized models:


Background tpwl reiwenski01
Background – TPWL [Reiwenski01] Trajectory PieceWise Linear Method

x8

x1

x7

x6

x5

x2

x4

x14

x10

xa

xb

x0

x11

x9

x13

x3

x15

Model i only valid near xi

x2

#linearizations =#samplesn

n = 104

#samples = 100

10010000 = LARGE

Use collection of linear models

x1


Background tpwl picking linearization points
Background – TPWL: Picking Linearization Points Trajectory PieceWise Linear Method

Use training trajectories to pick linearization points

y(t)

x2

State Space

Time Domain Simulation

t

x1

Linearization at current state xi


Quasi-piecewise-linear MOR Trajectory PieceWise Linear Method– computing weights

  • For i=0,…,(s-1) compute:

  • Compute

  • For i=0,…,(s-1) compute:

  • Normalize wi.


Background tpwl reduction of the linearized systems

= Trajectory PieceWise Linear Method

Model from linearization 1

Model from linearization k

Model from linearization 2

K2

A1

K1

A2

Ak

Kk

Background – TPWLReduction of the Linearized Systems


Background tpwl constructing the projection matrix v

= Trajectory PieceWise Linear Method

Background – TPWLConstructing the projection matrix V

Use moments from EACH linear model to construct V


Background tpwl weighting simulation riewinski01 tiwary05 dong05
Background – TPWL: Weighting / Simulation Trajectory PieceWise Linear Method[Riewinski01,Tiwary05,Dong05]

Linearization 3

x2

Use weighting functions to combine linear models during simulation

also well

approximated

C – poorly

approximated

Well approximated

Current state

Linearization 2

x1

Linearization 1


Test example of a nonlinear circuit: Trajectory PieceWise Linear Method


Computational results – circuit example Trajectory PieceWise Linear Method

  • Input:

training

input

testing

input


Computational results - circuit example, SSS Trajectory PieceWise Linear Method


DISCRETIZE Trajectory PieceWise Linear Method

Example of a microswitch


Computational results - microswitch example Trajectory PieceWise Linear Method

  • Input:

training

input

testing

input


Computational results - microswitch example Trajectory PieceWise Linear Method


Computational results – microswitch example Trajectory PieceWise Linear Method

training

input

testing

input


Simulation with the piecewise-linear model Trajectory PieceWise Linear Method


Generate the Trajectory PieceWise Linear Methodreduced linear model

Use the reduced linear system

for simulation

x2

x1

x0

A

x3

Fast approximate simulation

x2

x1


Computational results - model order reduction Trajectory PieceWise Linear Method

*Matlab implementation


Computational results – OpAmp example Trajectory PieceWise Linear Method

OpAmp:

  • 70 MOSFETs

  • 13 resistors

  • 9 capacitors

  • 51 nodes

training input

testing

input


Computational results – OpAmp example Trajectory PieceWise Linear Method

OpAmp:

  • 70 MOSFETs

  • 13 resistors

  • 9 capacitors

  • 51 nodes


Computational results – OpAmp example, SSS Trajectory PieceWise Linear Method



Computational results inlet– shock propagation in 1D

  • Burgers equation:


Questions yet to be answered: inlet

  • How should we pick training inputs?

  • How “often” should we compute linearized models along the training trajectory?

  • What are the errors of representing a nonlinear systems as a trajectory piecewise linear system?

  • What are “the best” reduced order bases for the trajectory PWL model?


Conclusions inlet

  • TPWL macromodels can be accurate for a number of strongly nonlinear dynamical systems, outperforming reduced models based on single-state polynomial expansions.

  • TPWL models use simple, cost-efficient representation of system’s nonlinearity, resulting in very short simulation times.

  • TPWL models may be extracted very efficiently by using a fast approximate simulator.


Model order reduction of non linear systems1
Model Order Reduction of Non-Linear Systems inlet

  • Introduction

  • Reduction of weakly nonlinear systems (Volterra Series)

  • Reduction of strongly nonlinear systems (TPWL) Trajectory Pieace Wise Linear and Polynomial

    • with moment matching

    • with Truncated Balance Realization


III.4 Reduction of Strongly NonLinear Systems inletvia Trajectory-PieceWise Linear (TPWL) + Truncated Balance Realization (TBR)

Luca Daniel

Massachusetts Institute of Technology

Thanks to Dmitry Vasilyev, Michał Rewieński, Jacob White


Obtaining projection basis

Krylov-subspace methods inlet

Fast

Don’t guarantee accuracy

Balanced-truncation methods

Expensive (~n3)

Guarantee accuracy

Obtaining projection basis

For example,

V=W=

colspan(A-1B, A-2B, … , A-q B)

Let’s now use this!


Our approach
Our Approach: inlet

  • Use TPWL to handle nonlinearity

    • Before we used Krylov-subspace linear reduction (less accurate)

  • Here we use TBR for projection matrices W and V

x0

x2

x1

xn


Tbr reduction

Hankel operator inlet

TBR reduction

u

y

LTI SYSTEM

t

t

Past

input

Future

output

X (state)

P (controllability)

Which states are easier to reach?

Q (observability)

Which states produces more output?

TBR algorithm includes into projection basis

most controllable and most observable states


Numerical results rlc transmission line
Numerical results – inletRLC 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


Reducing cost of tbr reduction combined krylov tbr algorithm
Reducing cost of TBR reduction - Combined Krylov-TBR algorithm

Krylov reduction (Wi , Vi):

Ai = WiTAVi

Bi = WiTB

Ci = CVi

Initial Model:

(A B C), n

Intermediate Model:

(Ai Bi Ci), ni

TBR reduction (Wt , Vt):

Ar = WtTAVt

Br = WtTB

Cr = CVt

Reduced Model:

(Ar Br Cr), q


Micromachined device example
Micromachined device example algorithm

FD model

non-symmetric

indefinite Jacobian


Performance of krylov tbr tpwl mor extraction procedures
Performance of Krylov- TBR TPWL MOR extraction procedures* algorithm

Cost of Krylov-TBR almost equals Krylov

*Matlab implementation


Comparing accuracies of krylov tpwl method and tbr based tpwl algorithm
Comparing accuracies of Krylov TPWL method and TBR-based TPWL algorithm

5x reduction in order – 125x improvement in efficiency

*Testing input equal to training input


Future work
Future work TPWL algorithm

  • Are TBR-based TPWL models valid for unstable linearizations?

  • What about systems having the following form (i.e. circuits with nonlinear capacitors):


Conclusions
Conclusions TPWL algorithm

  • 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!)

  • Combined Krylov-TBR reduction allows to extract TPWL models at low cost


Course Outline TPWL algorithm

Numerical Simulation

Quick intro to PDE Solvers

Quick intro to ODE Solvers

Model Order reduction

Linear systems

Common engineering practice

Optimal techniques in terms of model accuracy

Efficient techniques in terms of time and memory

Non-Linear Systems

Parameterized Model Order Reduction

Linear Systems

Non-Linear Systems

Today

Tomorrow


ad