Model Order Reduction - Universiteit Utrecht · 11/26/2009 · Model Order Reduction Wil Schilders...

Model Order Reduction

Wil SchildersNXP Semiconductors & TU Eindhoven

November 26, 2009

Utrecht UniversityMathematics Staff Colloquium

Utrecht Staff Colloquium, November 26, 2009

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Outline

Introduction and motivation

Preliminaries

Model order reduction basics

Challenges in MOR

Conclusions


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The work of mathematicians may not always be very transparant…….


4

But it is present everywhere in our business

Transient Multirate

Partition 1

Multirate

Partition 2

Transient

Collectinginfo

Collectinginfo

Partitioning Partitioning

Invisible contribution, visible success

Mathematics Mathematics

Mathematics


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Moore’s law also holds for numerical methods!


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Very advanced simulations

Subgridding mechanisms reduce the number of mesh points in a simulation. In this example (a) the full grid is 20 times larger (35e6 mesh

nodes) than (b) the subgridded version (1.75e6 mesh nodes).


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The impossible made possible…..

A complete package layout used for full wave signal integrity analysis

8 metallization layers and 40,000 devices

The FD solver used 27 million mesh nodes and 5.3 million tetrahedrons

The transient solver model of the full package had 640 million mesh cells and 3.7 billion of unknowns


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Avoiding brute force

The behaviour of this MOS transistor can be simulated by solving a system of 3 partial differential equations discrete system for at least 30000 unknowns

Insight?

Even worse: an electronic circuit consists of 104-106 MOS devices

Solution: compact device model (made by physicists/engineers based on many device simulations, ~50 unk)

Can we construct such compact models in an automated way?


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Another example

Integrated circuits need 3-D structure for wiring

10 years ago– 1-2 layers of metal, no influence on

circuit performance

Present situation:– 8-10 layers of metal– Delay of signals, parasitic effects due

to high frequencies

3-D solution of Maxwell equations leads to millions of extra unknowns

circuitCan we construct a compact model for the interconnect in

an automated way?

Gradual


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Model Order Reduction is about capturing dominant features

Strong link to numerical linear algebra

http://images.google.nl/imgres?imgurl=http://www.bmt.tue.nl/holst2006/img/vdvorst.jpg&imgrefurl=http://www.bmt.tue.nl/holst2006/&usg=__ALyM3E0gvQr45qYVwjvbSuN1-BM=&h=166&w=121&sz=5&hl=nl&start=12&itbs=1&tbnid=6xIu7dDDe41vsM:&tbnh=99&tbnw=72&prev=/images%3Fq%3Dhenk%2Bvan%2Bder%2Bvorst%26gbv%3D2%26hl%3Dnl


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Mathematical Challenges!

And, talking about motivation…..


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Outline


Preliminaries


Challenges in MOR

Conclusions


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Overall picture

Physical System ⇁ Data

ODEs PDEs

Modeling

Reduced system of ODEs

MOR

SimulationControl


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Dynamical systems


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Problem statement


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Approximation by projection


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Special case: linear dynamical systems


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Singular value decomposition


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Optimal approximation in 2-norm


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Operators and norms Impulse response

Transfer function


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Important observationModel Order Reduction methods approximate the transfer functionIn other words: not the entire internal characteristics of the problem, but only the relation between input and output (so-called external variables)No need to approximate the internal variables (“states”), but some of these may need to be kept to obtain good representation of input-output characteristics

Originalmatching somepart of the frequency

response

)(log ωH

ωlog


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Outline


Preliminaries


Challenges in MOR

Conclusions


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Overview of MOR methods


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Balanced truncation


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Interpretation of balanced truncation


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Properties of balanced truncation


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Simple example

Mesh: 50 x 100 x 50 cells @ 1u x 1u x 1u cell dimensions

80 micron Al strip

50 micron

1u x 1u

100 micron

Ideal conductor

SiO2

Voltage step(R = 50 ohm)over 1u gap

Termination(R = 10 K)

Voltage step:0.05 ps delay +

0.25 ps risetimeSINSQ Pstar

T = 5 ps


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Singular value plot

10-th order linear dynamic system fit to FDTD resultsMapping PDE’s

(Maxwell Eqs.) to equivalenthigh order (200)ODE system


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Moment matching (AWE, 1991)

Arnoldi and Lanczos methods


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Asymptotic Waveform Evaluation (AWE)

Direct calculation of moments suffers from severe problems:

– As #moments increases, the matrix in linear system becomes extremely ill-conditioned at most 6-8 poles accurately

– Approximate system often has instable poles (real part > 0)

– AWE does not guarantee passivity

But:basis for

many newdevelopments in MOR!


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Moment matching (AWE, 1991)

Arnoldi and Lanczos methods


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The Arnoldi method


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The Arnoldi method (2)


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The Lanczos method


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Two-sided Lanczos


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Moment matching using Arnoldi (1)


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Moment matching using Arnoldi (2)


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Example: MOR for transmission line (RC)

Original Reduced

Netlist size (Mb)

0.53 0.003

# ports 22 22

# internal nodes

3231 12

# resistors 5892 28

# capacitors

3065 97


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Outline


Preliminaries


Challenges in MOR

Conclusions


41

Start of a new era!

The Pade-via-Lanczos algorithm was published in 1994-1995 by Freund and Feldmann, and was a very important invention

Not only did the method (PVL) solve the problems associated with AWE

It also sparked a multitude of new developments, and a true explosion regarding the field of Model Order Reduction

AWE1990

PVL1995

• Lanczos methods• Arnoldi methods• Laguerre methods• Vector fitting• ……

1995-2009


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Topic 1: passivity

A linear state space system is passive if its transfer function H(s) is positive real

– H has no poles in C+

– H is real for real s– Real part of z*H(s)z non-negative for all s, z

Passivity is important in practice (no energy generated)

Lanczos-based methods such as PVL turned out not to be passive! problem!!!

Search started for methods that guarantee (provable) passivity

Passive MOR:– PRIMA (Passive Reduced-order Interconnect Macromodeling Algorithm)– SVD-Laguerre (Knockaert & Dezutter)– Laguerre with intermediate orthogonalisation (Heres & Schilders)


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Recent developments w.r.t. passivity

Several passivity-preserving methods have been developed

However: if the original discretized system is not passive, or if the data originate from experiments, passivity of the reduced order model cannot be guaranteed

This happens frequently in practical situations: EM software generating an equivalent circuit model that is not passive

In the frequency domain, this may not be too problematic, just an accuracy issue (S-parameters may exceed 1)

However, it may become really problematic in time domain simulations Passivity enforcement methods

Often: trading accuracy for passivity


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Topic 2: Large resistor networks

Obtained from extraction programs to model substrate and interconnect

Networks are typically extremely large, up to millions of resistors and thousands of inputs/outputs

Network typically contains:– Resistors– Internal nodes (“state variables”)– External nodes (connection to

outside world, often to diodes)

Model Order Reduction needed to drastically reduce number of internal nodes and resistors


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Reduction of resistor networks


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Reduction of resistor networks


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General idea: exploit structure


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“Simple” network

274 external nodes (pads, in red)

5384 internal nodes

8007 resistors/branches

Can we reduce this network by deleting internal nodes and resistors, still guaranteeing

accurate approximations to the path resistances between pads?

NOTE: there are strongly connected sets of nodes (independent subsets), so the problem is to reduce each of these strongly connected components individually


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Two-Connected Components

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can delete all internal nodes here


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How to reduce the strongly connected components

Graph theoretical techniques, such as graph dissection algorithms (but: rather time consuming)

Numerical methods such as re-ordering via AMD (approximate minimum degree)

“Commercially” available tools like METIS and SCOTCH (not satisfactory)


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Border Block Diagonal Form


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Block bordered diagonal form

Algorithm was developed to put each of the strongly connected components into BBD form (see figure)

Internal nodes can be deleted in the diagonal blocks, keeping only the external nodes and crucial internal nodes

The reduced BBD matrix allows extremely fast calculation of path resistances (work of Duff et al)


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Reduction results


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Results are exact

Remaining problem: how to drasticallyreduce the number of resistors

No loss of accuracy!←Spectre New →


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Topic 3: Nonlinear MOR

One of the most popular methods is proper orthogonal decomposition (POD)

It generates a matrix of snapshots (in time), then calculates the correlation matrix and its singular value decomposition (SVD)

The vectors corresponding to the largest singular values are used to form a basis for solutions

Drawback of POD: there is no model, only a basis for the solution space

Researchers are also developing nonlinear balancing methods, but so far these can only be used for systems of a very limited size (<10)

– E. Verriest, “Time variant balancing and nonlinear balanced realizations”– J. Scherpen, “SVD analysis and balanced realizations for nonlinear

systems”


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Classification of circuits

EngineerMathematician


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Fast and accurate simulation of oscillators

Challenges:

• Linear modeling of oscillators does not provide accurate solutions and in most cases is not able to capture subtle nonlinear dynamics of oscillators (injection locking, jitter, etc)

• Nonlinear models are necessary which are fast, accurate and generic

CLK

Digital CoreModem and

control

VCO + PLL

Tripler LNA + Matching IF Amp

PA+ MatchingMIXER

TestPins

DAC

XtalCLK

PLL FilterΣΔ− A

DC

RC

Cal

CLK


control

VCO + PLL


PA+ MatchingMIXER

TestPins

DAC

Xtal

PLL FilterΣΔ− A

DC

RC

Cal


control

VCO + PLL


PA+ MatchingMIXER

TestPins

DAC

Xtal

PLL FilterΣΔ− A

DC

RC

Cal


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1. PSS of oscillator:d/dt [q(xPSS)]+ j(xPSS)=0, T=TOSC

2. u1(t)= d/dt(xPSS)(t) (‘right Floquet’ eigenfunction)

3. v1(t) solves a linearized adjoint system (‘left Floquet’ eigenfunction)

4. Perturbed oscillator: d/dt [q(x)]+ j(x)= b(t) has solutionx(t)=xPSS(t+α(t))+xn(t), [α(t) phase noise]

5. α(t) satisfies a non-linear scalar differential equationd/dt (α)(t) = v1(t+α(t)).b(t), α(0)=0

Because of orthogonality only the component of b(t) along u1(t+α(t)) isimportant

Nonlinear modeling of perturbed oscillatorsProcess & Key Observations:

Involves: Time integration, Floquet Theory, Poincaré maps,

Nonlinear eigenvalue methods, Model Order Reduction!


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Example- three stage ring oscillator

153kHz ring oscillator

Unlocked osc: iinj= 6 * 10-5 *sin(1.04ω0 * t) Locked osc: iinj=6 * 10-5 * sin(1.03ω0 * t)


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MOR: Fast and accurate modeling of VCO pulling

VCO pulling due to PA and other blocks/oscillators needs to be analyzed before production

Full system simulation is CPU intensive or infeasible

Behavioural model order reduction gives fast and accurate insight in pulling/locking and coupling

0.4 0.6 0.8 1 1.2 1.4 1.6

x 109

−70

−60

−50

−40

−30

−20

−10

0

frequency

PSD(

dB)

simulationfin

Full mathematical theory of locking/unlocking mechanisms

and conditions lacking!


6161

Outline


Preliminaries


Challenges in MOR

Conclusions


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Conclusions regarding MOR

Model Order Reduction is a flourishing field of research, both within systems & control and in numerical mathematics

Strong relation to numerical linear algebra

Future developments need mathematical methods from a wide variety of fields (graph theory, Floquet theory, combinatorial optimization, differential-algebraic systems, stochastic system theory,…..)


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Some recent books on MOR

Model Order Reduction - Universiteit Utrecht · 11/26/2009 · Model Order Reduction Wil Schilders...

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