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Model Order Reduction
Wil SchildersNXP Semiconductors & TU Eindhoven
November 26, 2009
Utrecht UniversityMathematics Staff Colloquium
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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…….
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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
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Mathematical Challenges!
And, talking about motivation…..
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Outline
Introduction and motivation
Preliminaries
Model order reduction basics
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
Introduction and motivation
Preliminaries
Model order reduction basics
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
Introduction and motivation
Preliminaries
Model order reduction basics
Challenges in MOR
Conclusions
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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
Digital CoreModem and
control
VCO + PLL
Tripler LNA + Matching IF Amp
PA+ MatchingMIXER
TestPins
DAC
Xtal
PLL FilterΣΔ− A
DC
RC
Cal
Digital CoreModem and
control
VCO + PLL
Tripler LNA + Matching IF Amp
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!
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Outline
Introduction and motivation
Preliminaries
Model order reduction basics
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