BRAM – Backward Reduced Adjoint MethodAn application of Proper Orthogonal Decomposition to Sensitivity Analysis of electronic circuits.

Zoran IlievskiZ.Ilievski@tue.nl

CASA Day – November 13th 2008

Talk Overview

• Motivation• Proper Orthogonal Decomposition• Error Analysis for POD• Sensitivity Analysis• BRAM: New approach• Competition• Conclusions & Further Work

Motivation:

1948

Bell Labson-off switch

Texas Instruments

First IC, a few more transistors

1958

Intel 4004

2300 transistors

1971

Pentium 4 Tens of millions of

transistors

2000

Gordon Moore, 1999:‘1 transistor made for every ant on earth’

Circuits: A reduced history

- Node Voltages, Currents

- Sources

- Dynamic, static elementsCapacitors & Inductors, Resistors

- Input selector

Circuit Network ProblemsModified Nodal Analysis: Network Equations

Network Equations:

Problem: C,G can be very large.

Circuit Transient Simulation

For certain applications we would like to reduce this system

Time-consuming calculation

Tool: Model Order Reduction

Proper Orthogonal Decomposition

Proper Orthogonal DecompositionAlso known as: 1) Principal component analysis 2) Karhunen-Loève theorem

In a sentence:

POD finds a subspace approximating a given set of data in an optimal least-squares sense.

Data set: Any collection of data, representing anything from digital images, to sound and video recordings; In fact any data you would care to compress/decompose.

POD – Proper Orthogonal Decomposition

• Set of DATA,

• Approximating subspace -- POD basis problem

where

Find d-dimensional subspace

n-dimensional data parameterization: time

or

• or continuous and discrete time • orthogonal projection• orthonormal basis POD Basis

POD basis -- Sketch of derivation (I)

•Let be an orthonormal basis of S

• Projection error (continuous time)

•Strategy: find orthonormal basis of

complete:

• Such that

POD basis -- Sketch of derivation (II)

• Maximisation with constraint

• First basis vector maximise averaged projection

• Further basis vectors

adding constraint:

orthogonal eigenvectors with

• Eigenvalue problem:

POD basis -- Circuit Simulation

• Discrete time data – snapshot matrix• Eigenvalue problem:

• POD basis:

• Optimal • Orthogonal Projection

POD and SVD

• Singular Value Decomposition (SVD)

With orthogonal, orthogonal,

• Singular Values

• Left and Right singular vectors and

• Correlation matrix -- Eigenvalue problem

System Reduction With POD

• Network equation

with

• Reduced system

with

• Galerkin projection Reduced System

• Project state:

More Details?

P. Holmes, J.L.Lumley and G. BerkoozTurbulence, Coherent Structures, Dynamical Systemsand Symmetry,Cambridge University Press (1996)

Example: Image data set

• Total Singular Values: 500 (pixel hight of image)• Rapid Decay: First 100, most of the information.• Large data reduction possible. • At least 80% (100 basis vectors) reduction is possible.

Singular Value Distribution

What is the most important principal component?

Data Set Example (1): An Electronic Circuit

• Total singular values: 6• Rapid decay.• Only 4 basis vectors needed.

Singular Value Distribution

!DATA COMPRESSION!

Error Analysis

POD Projection Errors

• Data Projection Error• Reduced System Simulation Error

• Error between original & reduced system solution:

POD Projection Errors

• Available for POD methods (An great advantage over other MOR tools)

- can be found, as seen, from the POD subspace approximation.

POD Projection Errors

Ref: “A New Look at POD” M.Rathinam, L. R. Petzold (2003)

Sensitivity Analysis

Component Parameters

IC Resistor: IC Diode:

Revisiting the Network Equation:

- Parameter vector

State Sensitivity

Matrix: Each column vector represents the sensitivity of the state vector to a parameter in the vector p

Function Sensitivity

Observation Function:

Sensitivity to parameters p:

Direct Forward Approach - DFA

A direct numerical simulation:

Integrand complexity ~

Problem Suitability: For small circuits, low number of observation functions, low number of parameters. IDEAL.

Backward Adjoint Method - BAM

Adjoint Method: Elimination of

Lagrangian Equivalent:

Lagrangian Constraint:

BAM Integrand Complexity:

DFA Integrand Complexity:

Backward Adjoint Method - BAM

• Adjoint Method is now an established approach.

• Performed well on our selected test circuits.• Qimonda – test on large circuit with 1000s of parameters, works like a charm!

• We were able to further improve the method, bring a new idea to the table.

Backward Reduced Adjoint Method - BRAM

Remaining “Burden” :

Idea: Model Order Reduction using knowledge obtained from forward problem

• Construct POD basis from the available snapshots .

• Restrict to for the backward adjoint problem.

• By how much do the right hand sides differ?

Justifications?

Forward: Backward:

• Sensitivity of POD basis w.r.t parameter p.

Note: If there was no space change, V(p) is constant and the first term would disappear

Example: Transmission Line

• Characteristics: States follow input signal.

• Parameterisation: Length (i.e. value) of one specific resistor r.

• Observation function: Energy consumption of resistor r over [0,T]

• Closer look at F:

• similar RHS as state follows input

Justified!

We have noted, by justifying the POD projector V, an equivalent analysis can be done by reducing the initial forward system.

“Total” BRAM in sight: Development still in progress

After which, as before, we 'simply' apply the adjoint method.

T-Line Example: This provided justified answers of our initial questions.

Industrial Collaboration

http://www.st.com

• STM – VERY keen and interested in BRAM.• Invitations to collaborate at STM HQ.• Recent meetings (yesterday) – agreements reached on example set.• Part of work plan for next few months.

Industrial Collaboration

http://www.st.com

• Scalable Circuit• parameterized models• parameters: dimensions, bias ...

LNA- Low noise amplifiers

• Use COMSON demonstrator platform.

http://www.comson.org

Advantage over other methods

US Patent 7216309 - Method and apparatus for model-order reduction and sensitivity analysis

Competitors:

Non-POD method.

Tue / NXP BRAM method:

POD based methods give error estimates

At a glance, you will know if reduced sensitivityanalysis for your circuit problem is appropriate.

Advantage over the USA:

Conclusions

• Adjoint method is reliable and established.• POD modification and BRAM method is justified. (for at least one network and function class)

Right hand sides function on the same subspace Sensitivity for this class is low.

• POD method enables development of an error bound estimate.• We have an advantage over our competitors.

Future work

• Further analysis of the sensitivity of the POD basis to p

Starting point: Paper by M. Rathinam & L.R. Petzold 2003

Development of adjoint method for a reduced system

expect additional terms in Langragian equivalent due to, e.g.,

• Study Errors: Promising bound estimates available for POD related techniques , see ref.

• LNA- Low noise amplifier collaboration with STMicroelectronics.

• Development of T-BRAM

END

Bernard D. H. Tellegen

Tellegen's theorem:

Born: June 24, 1900(1900-06-24)Died: August 30, 1990 (aged 90)Nationality: DutchInstitutions: Delft University Notable awards: IEEE Edison Medal

1923 Masters degree in electrical engineering1923 Joined the Philips Research Laboratories in Eindhoven.

The set of potential differences Wk is from one network and the set of currents Fk is from an entirely different network,

so long as the two networks have the same topology

Also used in biological and metabolic networks, pipeline flow networks, and chemical process networks

Consider a circuit network topology problem with b branches.

Model reduction: POD and Galerkin projection

Full dynamical system

• reduced dynamical system

with

• Project state (DATA)

w.r.t basis of Swith

• Insertion • Galerkin approach (FLOW):

SVD Example

Backward Reduced Adjoint Method – BRAM 2