Model Order Reduction of Parameterized Interconnect Networks via a Two-Directional

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Model Order Reduction of

Model Order Reduction of

Parameterized Interconnect Networks

Parameterized Interconnect Networks

via a Two

via a Two

-

-

Directional

Directional

Arnoldi

Arnoldi

Process

Process

Zhaojun Bai, Yangfeng Su, Xuan Zeng Yung-Ta Li

joint work with

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Problem statement

Problem statement

Parameterized linear interconnect networks

where

Problem: find a reduced-order system

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Application:

Application:

variational

variational

RLC circuit

RLC circuit

Image source: OIIC

Complex Structure of Interconnects

spacing parameters

L. Daniel et al., IEEE TCAD 2004 J. Philips, ICCAD 2005

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Application:

Application:

micromachined

micromachined

disk resonator

disk resonator

Disk resonator

PML region Silicon wafer Electrode

Cutaway schematic (left) and SEM picture (right) of a micromachined disk resonator. Through modulated electrostatic attraction between a disk and a surrounding ring of electrodes, the disk is driven into

mechanical resonance. Because the disk is anchored to a silicon wafer, energy leaks from the disk to the substrate, where radiates away as elastic waves. To study this energy loss, David Bindel has constructed finite element models in which the substrate is modeled by a perfectly matched absorbing layer. Resonance poles are approximated by eigenvalues of a large, sparse complex-symmetric matrix pencil. For more

details, see D. S. Bindel and S. Govindjee, "Anchor Loss Simulation in Resonators," International Journal for

Numerical Methods in Engineering, vol 64, issue 6. Resonator micrograph courtesy of Emmanuel Quévy.

thickness

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Outline

Outline

1. Transfer function and multiparameter moments

2. MOR via subspace projection

- projection subspace and moment-matching

3. Projection matrix computation

- “2D’’ Krylov subspace and Arnoldi process

4. Numerical examples

- affine model of one geometric parameter

- affine model of multiple geometric parameters - polynomial type model

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Transfer function

State space model:

One geometric parameter for clarity of presentation

Transfer function::

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Multiparameter

Multiparameter

moments and 2D recursion

moments and 2D recursion

Power series expansion:

Multiparameter moments:

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Outline

Outline

1. Transfer function and multiparameter moments

2. MOR via subspace projection

- projection subspace and moment matching

3. Projection matrix computation

- “2D’’ Krylov subspace and Arnoldi process

4. Numerical examples

- affine model of one geometric parameter

- affine model of multiple geometric parameters - polynomial type model

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MOR via subspace projection

MOR via subspace projection

• A proper projection subspace: • Orthogonal projection:

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MOR via subspace projection

MOR via subspace projection

Goals:

1.

1. Keep the affined form in the state space equationsKeep the affined form in the state space equations 2.

2. Preserve the stability and the passivityPreserve the stability and the passivity 3.

3. Maximize the number of matched momentsMaximize the number of matched moments Goal 1 is guaranteed via orthogonal projection

Goal 1 is guaranteed via orthogonal projection

The number of matched moments is

The number of matched moments is

decided by the projection subspace

decided by the projection subspace

Goal 2 can be achieved if the original system complies

Goal 2 can be achieved if the original system complies

with a certain passive form

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Projection subspace and moment

Projection subspace and moment

-

-

matching

matching

Projection subspace:

for (1)

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Outline

Outline

1. Transfer function and multiparameter moments

2. MOR via subspace projection

- projection subspace and moment matching

3. Projection matrix computation

- “2D’’ Krylov subspace and Arnoldi process

4. Numerical examples

- affine model of one geometric parameter

- affine model of multiple geometric parameters - polynomial type model

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Krylov

Krylov

subspace

subspace

(1,i)th Krylov subspace:

] 1 [ ] 1 [ r [1] ] 2 [ r [1] ] 3 [ r [1] ] 4 [ r [1] ] 5 [ r [1] ] 6 [ r

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Krylov

Krylov

subspace

subspace

(2,i)th Krylov subspace::

] 2 [ ] 1 [ r [2] ] 2 [ r [2] ] 3 [ r [2] ] 4 [ r [2] ] 5 [ r [2] ] 6 [ r

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Efficiently compute

Efficiently compute

] 2 [ ] 1 [ r [2] ] 2 [ r [2] ] 3 [ r [2] ] 4 [ r [2] ] 5 [ r [2] ] 6 [ r Computed by 2D Arnoldi process

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Projection subspace

Projection subspace 2D2D KrylovKrylov subspacesubspace

Define

We have the linear recurrence

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2D

2D KrylovKrylov subspace and 2D subspace and 2D ArnoldiArnoldi decompositiondecomposition

• two-directional Arnoldi decomposition:

where

• two properties:

Computed by 2D Arnoldi process

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Generate V by 2D

Generate V by 2D

Arnoldi

Arnoldi

process

process

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Outline

Outline

1. Transfer function and multiparameter moments

2. MOR via subspace projection

- projection subspace and moment-matching

3. Projection matrix computation

- “2D’’ Krylov subspace and Arnoldi process

4. Numerical examples

- affine model of one geometric parameter

- affine model of multiple geometric parameters - polynomial type model

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RLC circuit with one parameter

RLC network: 8-bit bus with 2 shield lines

variations on and

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RLC circuit

: Relative error

: Relative error

Compare PIMTAP and CORE [X Li et al., ICCAD 2005]

0 1 2 3 4 5 6 7 8 9 10 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 Relative error: | h − h ^ | / | h | CORE (p,q)=(40,1) CORE (p,q)=(80,1) PIMTAP (p,q)=(40,1) CORE (p,q)=(40,1) CORE (p,q)=(80,1) PIMTAP (p,q)=(40,1)

order of the ROM=76

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RLC circuit

: Numerical stability

: Numerical stability

0 1 2 3 4 5 6 7 8 9 10 5 6 7 8 9 10 11 Frequency (GHz) |h(s)| original CORE (p,q)=(80,2) PIMTAP (p,q)=(40,2) CORE (p,q)=(80,2) PIMTAP (p,q)=(40,2)

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0 2 4 6 8 10 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 Frequency (GHz)

Relative error: abs( Horiginal − Hreduced )/ abs( Horiginal )

bus8b: λ = 0.06

PIMTAP (p,q)=(40,1) PIMTAP (p,q)=(40,2) PIMTAP (p,q)=(40,3) PIMTAP (p,q)=(40,4)

RLC circuit

: PIMTAP for q=1,2,3,4

: PIMTAP for q=1,2,3,4

4

10−

2

10− q=1,2

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Parametric thermal model

Parametric thermal model

[[RudnyiRudnyi et al. 2005]et al. 2005]

Thermal model with parameters and

Power series expansion of the transfer function on

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Parametric thermal model

Parametric thermal model

Stack the vectors via the following ordering:

#

#

(0,0,0) (1,0,0) Æ (0,1,0) Æ (0,0,1) (2,0,0) Æ (1,1,0) Æ(1,0,1)Æ(0,2,0)Æ(0,1,1)Æ(0,0,2) (0,0,0) Æ(1,0,0) Æ(0,1,0) Æ (0,0,1) Æ (2,0,0) Æ …… Æ (0,0,2) The sequence :

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Polynomial type model

Polynomial type model

State space model:

Transfer function:

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Projection subspace

Projection subspace 2D2D KrylovKrylov subspacesubspace

Define

We have the linear recurrence

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Outline

Outline

1. Transfer function and multiparameter moments

2. MOR via subspace projection

- projection subspace and moment matching

3. Projection matrix computation

- “2D’’ Krylov subspace and Arnoldi process

4. Numerical examples

- affine model of one geometric parameter

- affine model of multiple geometric parameters - polynomial type model

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Concluding remarks:

Concluding remarks:

1.

PIMTAP is a moment matching based approach

2.

PIMTAP is designed for systems with a

low-dimensional parameter space

3.

Systems with a high-dimensional parameter space are dealt by parameter reduction and PMOR

4.

A rigorous mathematical definition of projection subspaces is given for the design of Pad’e-like approximation of the transfer function

5.

The orthonormal basis of the projection subspace is computed adaptively via a novel 2D Arnoldi process

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Current projects:

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References

References

1.

Y.T. Li, Z. Bai, Y. Su, and X. Zeng, “Parameterized Model Order Reduction via a Two-Directional Arnoldi Process,’’ ICCAD 2007, pp. 868-873.

2.

Y.T. Li, Z. Bai, Y. Su, and X. Zeng, “Model Order

Reduction of Parameterized Interconnect Networks via a Two-Directional Arnoldi Process,’’ IEEE TCAD, in revision.

3.

Y.T Li, Z. Bai, and Y. Su, “A Two-Directional Arnoldi Process and Applications,’’ Journal of

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Figure

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