Reduced Order Modeling of Parameterized and Distributed Systems

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Reduced Order Modeling of Parameterized and Distributed Systems. Luca Daniel, M.I.T. Parameterized Model Order Reduction. Problem Classification Reducing Linear Systems Moment Matching with Linear parameters Moment Matching with NON-linear parameters Quasi Convex Optimization approach

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### Reduced Order Modeling of Parameterized and Distributed Systems

Luca Daniel, M.I.T.

Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters

4

10

3

10

|Z(f)|

2

10

1

10

• Field solvers can produce instance impedance vs. frequency curves.

0

10

6

7

8

9

10

10

10

10

10

10

frequency

Motivation. Example: Analysis of RF micro-inductor
• How are the substrate eddy currents affecting the quality factor of the inductor?
• How are the displacement currents affecting the resonance of the inductor?
• Need to capture all 2nd order effects
Motivation Example: MOR of RF micro-inductor

I

LNA

Q

LO

• “Model Order Reduction” can help verify how the inductor performance (Q, resonance position, etc) affect the transceiver performance (distortion, interference rejection etc.)
• Modeling Requirements:
• as accurate as a field solver
• automatic and robust
• model compatible with circuit simulators
Motivation Example: Parameterized MOR of RF micro-inductor

I

LNA

Q

LO

d

• “Model Order Reduction” can help verify how the resonator or inductor performance (noise, quality factor, etc)affect the transceiver performance (distortion, interference rejection)

W

• “Parameterized Model Order Reduction” can help verify how the transceiver performance changeswhen I change wire widths and wire separation
MotivationExample. Analysis of Integrated Power Electronics

How is the magnetic fringing field from the core effecting eddy current losses?

Analysis tools can produce for instance resistance vs. frequency curves.

Rac

Spattered laminated NiFe core, electroplated windings [Daniel96]

frequency [Hz]

Micro-inductor in a DC/DC power converter

Vin(t)

Vout(t)

• “Model Order Reduction” can help verifying how the 2nd order effects of the power inductor (e.g. power loss vs. freq. curve)influences the power converter functionality (dynamics, overall power efficiency)
• “Parameterized Model Order Reduction” can help verify how the functionality and the efficiency of the overall power converter changeswhen I change wire widths and wire separations?

1M equations

10 equations

Field Solvers discretize geometry and produce large systems

d

W

PMOR

• PMOR produces a dynamical model:
• automatically
• match port impedance
• small (10-15 ODEs)
Parameterized Model Order ReductionProblem Classification

matrix size

Non-Linear Systems

Linear Time Invariant

non-linearly

parameterized

linearly

parameterized

linearity

# parameters

LO

LNA

Parameterized Model Order Reduction.Applications

interconnect

MEMS

RF inductors

matrix size

Packages

Linear Time Invariant

Non-Linear Systems

linearly

parameterized

non-linearly

parameterized

linearity

# parameters

LO

LNA

Parameterized Model Order Reduction.Previous work

interconnect

MEMS

RF inductors

matrix size

Packages

Linear Time Invariant

Non-Linear Systems

linearly

parameterized

non-linearly

parameterized

Moment Matching: Pullela97, Weile99, Gunupudi00, Prud’homme02,

Daniel02, Li05

linearity

• statistical data mining
• Liu DAC99
• CMU Rutenbar02

# parameters

Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters
• In interconnect design often would like:
• reliable functionality:
• minimize capacitive cross-talk,
• minimize inductive cross-talk,
• minimize electromagnetic interference
• high speed:
• minimize resistance
• minimize capacitance
• low cost:
• minimize area
• Need to explore tradeoff space and find optimal design!
The traditional design methodology
• The traditional design flow:
• REPEAT
• design all interconnect wires
• extract accurately parasitics all at once
• UNTIL noise and timing are within specs
• such procedure is not ideal for optimization!
• each iteration is very time consuming
Alternative design methodologies
• Pre-characterize standard interconnect structures (e.g. busses):
• using parasitic extraction and table lookup
• or building parameterized and accurate low order models
• However... if the model construction is fast enough can also:
• build the interconnect structure model "on the fly" during layout
• accounting for any topology in surrounding topologies already committed to layout
• then use optimizer to choose the best parameter for optimal tradeoff design.

accounting for surrounding topology

Example: an interconnect bus
• We construct a multi-parameter model of the bus parameterized in wire width W and separation d

W

d

…………..

Conductance matrix

Parasitic extraction produces large state space models
• E.g. subdividing wires in short sections and using for instance Nodal Analysis

…………..

Large linear dynamical system

Our goal
• Given a large parameterized linear system:

Construct a linear system model with:

• smaller complexity
• same fidelity
• small reduction cost

20 x 20

BackgroundNon-parameterized Model order reduction
• Given a large linear system model:

500,000 x 500,000

BackgroundModel order reduction (cont.)

Taylor series expansion:

• change basis and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point

U

qxn

nxn

nxq

nxq

qxq

• Discretizing wires and using Nodal Analysis

…………..

More in general...[D. TCAD04] [D. PhD04]
• It is a p-variables Taylor series expansion

Uq

Once again change basis:

• use first few vectors of the Taylor expansion,
• matching first few derivatives with respect to each parameter
Parameterized moment matching (cont.)

Congruence transformations on each of the matrices

qxn

qxq

nxn

nxq

W0=1um

d0 =1um

W0=1um

d0=1um

W=0.25um

W=0.2um

W=4um

W=8um

W=0.25um

W=0.2um

W=4um

W=8um

d=0.25um

d=2um

Example: model step responses for different W and d.

…………..

N=16 wires

h=1.2um

L=1mm

W

d

W0=1um

d0 =1um

W=0.25um

W=0.2um

W=4um

W=8um

W=0.25um

W=0.2um

W=4um

W=8um

W0=1um

d0 =1um

d=2um

d=0.25um

Example: model crosstalk responsesfor different W and d.

…………..

N=16 wires

h=1.2um

L=1mm

W

d

Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters

LO

LNA

interconnect

MEMS

RF inductors

matrix size

Linear Time Invariant

Non-Linear Systems

linearly

parameterized

non-linearly

parameterized

Moment Matching

Pullela97, Weile99, Prud’homme 02

linearity

• statistical data mining
• Liu DAC99
• CMU Rutenbar02

# parameters

Example: RF inductor
• Design parameters:
• wire dimensions and separations,
• number of turns
• type of substrate and distance from wires
• Performance parameters:
• inductance
• quality factor Q
• resonance frequency
• power
• area
• Effects captured:
• displacement currents affect resonance
• skin effect in wires affect Q
• proximity effect in wires affect Q
• dielectrics affect Q and resonance
• substrate eddy currents affect Q and resonance
• interference with other devices

d

W

n= 3

current and charge

conservation

resistive effect

magnetic coupling

charge-voltage

relation

• PEEC subdivides the volumes into small thin filaments

conductor k

conductor h

Enforcing this equation in each filament: produces branch equations

• PEEC subdivides the surface into small panels

panel j

panel i

Enforcing this equation in each panel: produces branch equations

thin volume filaments

with constant current

small surface panels

with constant charge

PEEC Discretization Basis Functions [Ruehli MTT74, MIT course 6.336J and 16.920J]
• PEEC discretizes volumes in short thin filaments, small surface panels
• PEEC discretization gives branch equations:
PEEC Discretization Example: On-Chip RF Inductor [D. BMAS03]

overall dimensions = 600um x 600um

wire thickness 1um

picture not to scale

W

wire separation d = 1um-5um

wire width W = 1um-5um

d

x100um

x100um

Mesh (Loop) Analysis [FastHenry94, Kamon Trans Packaging98]

Imposing current conservation with mesh (loop) analysis (KVL)

Example of a Field Solver output: package powergrid admittance amplitude

* 3 proximity templates per cross-section

- 20 non-uniform thin filaments per cross-section

From Field Solvers to a Dynamical Linear System Model

Imposing current conservation with mesh (loop) analysis (KVL)

Multiply out and introduce state

Case 1

geometrical

parameters

Case 2

Laplace parameter

(frequency)

Discretization produces a HUGE “nonlinearly parameterized” dynamical linear system [D. BMAS03]

small surface panels

with constant charge

thin volume filaments

with constant current

E

(W,d,s)

A

(W,d,s)

Fit a low order polynomial (e.g. quadratic) to the evaluated matrices R and L

A(W,d) ≈ A0,0+ WA1,0 + d A0,1 + W2A2,0 + Wd A1,1 + d2 A0,2

E(W,d) ≈ E0,0+ WE1,0 + d E0,1 + W2E2,0 + Wd E1,1 + d2 E0,2

[A0,0+ WA1,0 + d A0,1 + W2A2,0 + Wd A1,1 + d2 A0,2 +...

sE0,0+ sWE1,0 + sd E0,1 + sW2E2,0 + sWd E1,1 + sd2 E0,2] x= b u

A polynomial interpolation approach [D. et al BMAS03]

[ s E (W,d)- A(W,d) ] x= b u

A polynomial interpolation approach [D. et al BMAS03]
• Selected a grid of 9 evaluation points for different combination of parameters

(W,d) = (1um,1um), (1um,3um), (1um,5um),

(3um,1um), (3um,3um), (3um,5um),

(5um,1um), (5um,3um), (5um,5um)

• Used the Volume Integral Equation code to generate system matrices Ek= E(Wk ,dk) and A k= A (Wk ,dk) for each combination of parameters

Ak

Ek

A polynomial interpolation approach [D. et al BMAS03] 4. Calculating Interpolation coefficients
• Need to calculate 6 polynomial coefficients
• Hence need at least 6 equations imposing fit in 6 evaluation points
• However in general it is better to use more evaluation points than the minimum.
• For instance here we used the 9 evaluation points above and solved with a least square method

E0

s1

E1

s2

E2

s3

E3

s4

E4

s5

E5

s6

E6

s7

E7

s8

E8

s9

E9

s10

E10

s11

E11

• Transformed the polynomial parameterized system into a linearly parameterized system introducing new parameters

[A0,0+ WA1,0 + d A0,1 + W2A2,0 + Wd A1,1 + d2 A0,2 +...

sE0,0+ sWE1,0 + sd E0,1 + sW2E2,0 + sWd E1,1 + sd2 E0,2] x= b u

• Used the previously developed model reduction for linearly parameterized systems
More in general...[D. TCAD04] [D. PhD04]
• It is a p-variables Taylor series expansion

Uq

Once again change basis:

• use first few vectors of the Taylor expansion,
• matching first few derivatives with respect to each parameter
Computational Complexity (time and memory)

E is DENSE !

Moment matching bottleneck:

gettingEb, E2b, ...

pFFT: an O(NlogN) Matrix-Vector productfor the Mixed Potential Integral equations [Phillips97]
• Matrix-vector product Eb physical interpretation:
• given N charges on N panels
• calculate the resulting N potentials

(1) Project charges into a 3D grid. Complexity O(N)

(1)

(2)

(2) Calculate grid potentials, it is a convolution use FFT. Complexity O(n log n).

(3)

(3) Interpolate potentials from grid. Complexity O(N)

Picture by J. Phillips

grid points n ≈ N panels

PEEC Discretization Example: On-Chip RF Inductor [D. BMAS03]

overall dimensions = 600um x 600um

wire thickness 1um

picture not to scale

W

wire separation d = 1um-5um

wire width W = 1um-5um

d

x100um

x100um

Wire width = 5um

separation = 1um, 2um, 3um, 4um, 5um

frequency [ x10GHz ]

Results: Inductance vs. frequency

Wire width = 1um

separation = 1um, 2um, 3um, 4um, 5um

L

L

__ original system (order 420)

--- reduced model (order 12)

__ original system (order 420)

--- reduced model (order 12)

Worst case error

in resonance position = 3%

frequency [ x10GHz ]

Results: Quality factor (Q=wL/R) vs. frequency

Wire width = 1um

separation = 1um, 2um, 3um, 4um, 5um

Wire width = 5um

separation = 1um, 2um, 3um, 4um, 5um

Q

Q

__ original system (order 420)

--- reduced model (order 12)

Worst case error

in amplitude = 4%

__ original system (order 420)

--- reduced model (order 12)

frequency [ x10GHz ]

frequency [ x10GHz ]

Open issues in the PMOR Matrix Reduction step
• Model order grows as O(pm) where p = # parameters and m = # derivatives matched for each parameter
• however model order is linear in # of parameters when matching only one derivative per parameter (m = 1) and still produces good accuracy in our experiments.
• furthermore, for higher accuracy instead of increasing # of matched derivatives, can instead match multiple points (or combine the two approaches) [Li, Liu, Nassif, Pileggi DATE05]

W or d

Limitations and future workuse better interpolation!

Wire width = 1um

separation = 1um, 2um, 3um, 4um, 5um

Q

__ original system (order 420)

--- reduced model (order 12)

very little error in the

points used for fitting

1um, 3um, 5um

so the critical step

is the fitting!!

Worst case errors far

from fitting points 2um, 4um

(3% in resonance position)

frequency [ x10GHz ]

Case #2. The parameter is frequency.Already discussed: Distributed Linear Systems

Examples:

• full-wave MPIE
• MPIE using layered-media Green functions(e.g. for handling substrate or dielectrics)
• frequency-dependent basis functions
• frequency dependent discretizations
Polinomial interpolation for frequency [Phillips96]
• Polynomial approximation e.g. Taylor expansion, or a polynomial interpolation for E(s)
• Performance: Fast and accurate in the frequency band of interest
• Problem: Can not be used in a time domain circuit simulator because does not guarantee stability and passivity
Positive real transfer functionin the complex plane for different frequencies

original system

Active

region

Passive region

Why does polynomial interpolation failwhen applied to the Laplace parameter ‘s’?
• Although accurate in the frequency band of interest
• Polynomial interpolation is unlikely to preserve GLOBAL properties such as positive realness because it is GLOBALLY not well-behaved

original system

Active

region

Passive region

reduced system

• Proof: just choose accuracy of interpolation smaller than minimum distance from imaginary axis
Using global uniformly convergent interpolants[Daniel02]
• If E(s) is strictly positive real, a GLOBALLY and UNIFORMLY convergent interpolant will eventually get close enough (for a large enough order M of the interpolant) and be positive-real as well.

original system

Active

region

Passive region

Conclusions
• Parameterized reduction can enable automatic design of circuit components that can only be described by field solvers
• All methods presented are based on Krylov subspace moment matching projection
• Key observation: all these methods are compatible with O(NlogN) field solver based matrix-vector
• But of course all these methods also have all the arguable issues associated with moment matching
• Linearly parameterized (use p-variable Taylor series expansion
• e.g. interconnect bus
• Non-linearly parameterized (use polynomial interpolation)
• e.g. RF inductor (no substrate)
• Distributed systems i.e. non-linear in s (use globally convergent interpolant implemented with FFT)
• e.g. substrate layered green functions or high order basis functions
Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Example utilization of PMOR
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters

TH3D-3

### Accelerated Optical Topography Using Parameterized Model Order Reduction

Junghoon Lee, Dimitry Vasilyev, Anne Vithayathil, Luca Daniel, and Jacob White

Research Laboratory of Electronics

Massachusetts Institute of Technology

http://www.rle.mit.edu/cpg

Non-destructive Inspection of Fabricated Structures
• Spectroscopic ellipsometry
• Shine light of =200~800nm
• Measure the scattered light
• Estimate the geometric parameters, e.g. w and h

etched structure

scattered field measurements

plasma-processing.com/insitu.htm

www.sopra-sa.com

Model Based Approach
• Determine parameters p by solving
• Solve nonlinear least squares problem
• Algorithms are iterative in nature
• Scattering model evaluated for many p’s
• Need an efficient model of parameters p
• Up to a dozen parameters
• width (w), height (h), left/right sidewall angle, top curvature, etc.
Model Possibilities
• Tabulated library of scattered fields
• Table grows exponentially as the number of parameters grow
• Full EM simulation for scattered field
• Too computationally expensive (too slow)
• New approach: automatic extraction of parameterized low-order model
• Automatically extract parameterized low-order model from pre-determined test structures
• Use model in optimization during in-line process diagnosis

integral of

incident wave

discretized

integral operators

scattered

field

weights for

RWG basis

Surface Integral Formulation
• Discretized surface integral formulation
• PMCHW formulation on dielectric interfaces
• Discretization and RWG basis functions
• Method of moments
• Resulting system of equations

coefficient matrices

Polynomial Fitting
• Polynomial fitting to find explicit dependency
• y(p;): rough function of p and 
• A(p;), C(p;): smooth function of p and 
• Polynomial fit of A and C
• Accurate since they are smooth
Parameterized Moment Matching
• Projection representation: approximate xf as
• Many different ways to choose V
• Moment matching, balanced realization, POD, etc.
• We used moment matching: match Taylor coefficients of yf(p;) and yi(p;)
Parameterized Moment Matching
• Fitting with moment matching: for the constant term
• Overall reduced system

True: 97 nm

Predicted: 97.3 nm

error

error

True: 105 nm

Predicted: 104.9 nm

width

Accelerated Optimization with Reduced Model

original: 576

reduced: 30

original: 540

reduced: 50

Conclusions
• Optical Inspection Problem:
• Parameterized reduced model using polynomial fitting plus projection
• Accelerates optimization
• The method achieves good accuracy in practice
• Acknowledgements

MARCO designated Interconnect Focus Centers,

SRC, Singapore-MIT Alliance, and NSF

Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters
A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction

Parameterized Reduced Linear Models

Field Solver Linear Models

Linear analog components

e.g. interconnect, packages, RF inductors

convex set

quasi-convex set

Review: Quasi-Convex setupfor NON-parameterized MOR

Stability:

Passivity:

Standard problem.

Use for example by the ellipsoid algorithm

Parameterized MOR setup
• Construct guaranteed stable parameterized reduced model
• Let design parameter p explicitly enter (a,b,c)
• More difficult to check stability
Parameterized stability
• Need to check a(z,p) > 0for all z, for all p
• Cannot use the trick in the non-parameterized case
• Make a(z,p) a multivariate trigonometric polynomial

trig poly

ordinary poly

trig poly

trig polys

with

• Check positivity of multivariate trigonometric polynomial by using sum-of-squares argument
Positivity of multivariate trig. poly

Given a(z,p), solving the semi-definite program

• If feasible and y* < 0, then
• Otherwise, claim a(z,p)  0 at some point
Example 7: Parameterized RF inductor
• Construct reduced order models in the design space

Circle: training points

Triangle: test points

• Identify dominant poles z*(p)of each ROM
• Construct “non-dominant” systems such that

where K(p), A(p) are scalars functions

G1(z,p) is the “non-dominant” system

• Construct parameterized ROM of each “non-dominant” system
• Interpolate the scalars functions
• Combine both parameterized models to obtain the final model
Example 6: Parameterized RF inductor

Quality factor for W=16.5 µm,

D = 1,5,18,20 µm

Circle: training points

----- full model

___ our QCO PMOR

Triangle: test points

Parameterized RF inductor (cont.)

Quality factor for W=16.5 µm, D = 1,5,18,20 µm

Frequency of the peak Q factor for W = 16.5 µm

Summary of Quasi-Convex Optimization basedParameterized Model Order Reduction
• QCO competes reasonably well with existing alg (e.g. PRIMA) for reducing large systems
• But in addition: QCO can reduce models with frequency dependent matrices
• QCO is very flexible in imposing constraints such as stability and passivity
• QCO can be extended to parameterized MOR problems
Parameterized Model Order Reduction
• Problem Classification
• Reducing Linear Systems
• Moment Matching with Linear parameters
• Moment Matching with NON-linear parameters
• Quasi Convex Optimization approach
• Reducing NON-linear systems
• Moment Matching with NON-linear parameters

### “Field Solver Accurate” and AutomaticParameterized Reduced Order Modeling of NonLinear Analog and MEMs Components

Luca Daniel, M.I.T.

luca@mit.edu

with contributions from Brad Bond

www.rle.mit.edu

LO

LNA

Parameterized Model Order Reduction.Applications

interconnect

MEMS

RF inductors

matrix size

Packages

Linear Time Invariant

Non-Linear Systems

linearly

parameterized

non-linearly

parameterized

linearity

# parameters

Case 3. Reduction of NonLinear Systems

Parameterized NonLinear Reduction

PDE Field Solvers or Circuit Simulators

Non-Linear analog components

e.g. MEMs, VCO, LNA

Parameterized Model Order Reduction.Linear Systems
• Moment Matching approaches- (Pullela97, Gunupudi00, Prud’homme02, Daniel02, Li05)
• Can handle nonlinear dependence on parameter
• Can handle extremely large systems

Truncated Balance Realizations- (Heydari01, Phillips04)

- Does not handle extremely large systems

• Optimization based approaches - (Sou05)
• Good for fitting data from measurements
• Cannot construct large order reduced models
• Statistical Data Mining - (Liu02)
• Can handle nonlinear parameter dependence and nonlinear systems
• Does not work well for extremely large systems

Previous work on Non-Parameterized

MOR 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]
Background – TPWL [Reiwenski01]

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

Use training trajectories to pick linearization points

y(t)

x2

State Space

Time Domain Simulation

t

x1

Linearization at current state xi

Background – TPWL: Weighting / Simulation[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

=

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 – TPWLConstructing the projection matrix V

Use moments from EACH linear model to construct V

Introduction

Background

NonLinear Parameterized Model Order Reduction (NLPMOR)

Constructing the system

How to pick linearization points

How to construct V

Examples

Results

Conclusions

Future Work

Outline
NLPMOR – Constructing the System

Start with system possessing nonlinear dependence on state x and parameters sj

By linear approximation, polynomial fitting to data points…

Obtain Linear dependence on new parameters

Single linear model

Linearize to get weighted sum of linear models

Weighting basis functions

Non-Linear Parameterized Reduction[Bond, Daniel ICCAD05]

Use training trajectories to pick linearization points

y(t)

x2

State Space

time

x1

s2

Train again at different points in parameter space

Repeat at different points in parameter space to populate state space

s1

Parameter Space

Non-Linear Parameterized Reduction[Bond, Daniel ICCAD05]

Populate relevant regions of state-space with linear models from training at different inputs or different points in parameter-space

x2

State Space

x1

s2

s1

Parameter Space

NLPMOR – Constructing V: 2 Options

Option 1

Single variable Taylor series expansion of x for each model, as in TPWL

“MOR V”

OR

Option 2

p-variable Taylor series expansion of x as in PMOR, but for each model

“PMOR V”

Non-Linear Parameterized ReductionConstructing V

For qth order model with p parameters and k linear models, total number of vectors for V is O(kpm)

Keeping all vectors would results in a huge “reduced” model

Hence we perform an SVD on V and keep only the q most important vectors

O(kpm)

q

SVD

N

N

NLPMOR – 4 Algorithms

Single p-space point training

Multiple p-space point training

MOR

V

TPWL

TPWL

ALG2

ALG2

PMOR

V

ALG1

ALG1

ALG3

ALG3

Original Model

Linearize in parameters sj

MOR

Train to pick linearization points xi

PMOR

Train at single p-space point

Train at multiple p-space points

MOR

V

PMOR

V

MOR

V

PMOR

V

Parameter Space

Outline
• Introduction
• Background
• NLPMOR
• Examples
• Circuit
• Beam
• Results
• Conclusions
• Future Work
Analog Circuit Example

Nonlinear terms

Picture by Michał Rewienski

Analog circuit with nonlinear components distributed throughout

Using constitutive relations for each element, apply KCL at each node to obtain state-space model

Analog Circuit Example – State Space System

Capacitor values

Saturation current

Turn on voltage

Resistor values

Single equation at node j

State-Space System

Possible parameters

Analog Circuit Example – One possible parameterized model

Training point

Voltage (V)

Evaluation point

Full Model

ALG1 ROM

ALG1

Time (s)

Selected parameter: α

Obtained linear dependence on parameters

Single point training

Multiple point training

MOR

V

PMOR

V

Micromachined Switch Example

Governing Equations

Discretize

Picture by Michał Rewienski

Micromachined Switch ExampleState Space System

Possible Parameters

Material Properties

State-Space System

Beam height

Beam width

Micromachined Switch ExampleOne possible parameterized model

u(t) = 5.52 , t > 0

Full Model

ALG2 ROM

Training point

Evaluation point

ALG2

System parameterized in

Single training

Multiple training

MOR

V

PMOR

V

Outline
• Introduction
• Background
• NLPMOR
• Examples
• Results
• Algorithm Comparison
• Algorithm Cost
• Conclusions
• Future Work
Analog Circuit Example – Algorithm 1

ALG1

Single training

Multiple training

Full Model

ALG1 ROM

MOR

V

PMOR

V

1.4

1

0.9

- Training points

- Evaluation points

0.5e-10

1e-10

1.5e-10

Results – Benefit of PMOR V

TPWL

ALG1

Error

Id

TPWL

Id0

Accuracy

Trained at Id0

ALG1

TPWL – MOR V

ALG1 – PMOR V

Id

Analog Circuit

Analog Circuit

TPWL ROM

ALG1 ROM

Single p-space point training

Multiple p-space point training

MOR

V

PMOR

V

Id0

Analog Circuit Example

ALG2

Single training

Multiple training

This model is parameterized in Id and 1/R, and was created using ALG2

MOR

V

Full Model

ALG2 ROM

PMOR

V

2

1.5

1

0.5

- Training points

- Evaluation points

0.5e-10

1e-10

1.5e-10

2e-10

Results – Benefit of multiple training points in p-space

TPWL

ALG2

% Error

TPWL

Id1

Id2

ALG2

Id0

Accuracy

TPWL – trained at Id0

ALG2 – trained at Id1, Id2

Id0

Id1

Id

Id2

MEMs Switch

Analog Circuit

TPWL ROM

ALG2 ROM

% Error

time (s)

Single p-space point training

Multiple p-space point training

MOR

V

PMOR

V

Results – Algorithm Comparisons

TPWL

ALG2

ALG1

ALG3

Each algorithm covers a different region of p-space

Single p-space point training

Multiple p-space point training

S2

s2B

MOR

V

PMOR

V

s2A

All algorithms produce model with same reduced order and same number of linear pieces

S1

s1A

s1B

Parameter Space Validity of Models

Results – Algorithm Comparisons

Cost of constructing V

(in system solves)

Training cost

(trajectories made per input)

Algorithm

TPWL

TPWL

O(km)

1

ALG 1

O(kpm)

Single point training

Multiple point training

1

ALG 2

rp

O(krpm)

MOR

V

ALG2

ALG 3

rp

O(krppm)

ALG1

PMOR

V

ALG3

Each algorithm has a different cost

Cheap

Majority of cost lies in making vectors for V

Majority of cost lies in training

p - # of parameters

k - # of linear models per trajectory

r - # of parameter values used

m - # of moments matched per parameter

Expensive in both training and creating V

NLPMOR

Captures nonlinear effects

Preserves parameter dependence

Algorithm choice is situation/system dependent

TPWL, ALG1 can provide high accuracy locally in parameter space

ALG2, ALG3 can provide global accuracy in parameter space

ALG1, ALG3 : expensive to create V

ALG2, ALG3 : expensive to train

Conclusions

TPWL

Single point training

Multiple point training

MOR

V

ALG2

ALG1

PMOR

V

ALG3

Try other approaches for Projection Matrix

e.g. TBR, Quasi-Convex Optimization

Try other approaches for capturing nonlinearity

e.g. Volterra series, or piece-wise polynomial

Future Work in Parameterized Model Order Reduction of Non-Linear Systems
Summary of the course
• Model Order Reduction of Linear Systems
• Krylov + TBR two step procedure
• 1st step: Krylov Moment Matching Projection
• PVL or PRIMA (for passive systems)
• 2nd step Truncated Balance Realization (TBR) or Passive-TRB
• Distributed Passive Systems:
• Quasi Convex Optimization
• Laguerre interpolation
• Model Order Reduction of NonLinear Systems
• Weakly Nonlinear: use Volterra series + moment matching
• Strongly Nonlinear: use TPWL + moment matching (and/or TBR)
• Parameterized Model Order Reduction
• Linear: moment matching OR quasi-convex optimization
• NonLinear: TWPL + moment matching