Parameterized Model Order Reduction via a TwoDirectional Arnoldi Process

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Parameterized Model Order Reductionvia a
Two-Directional Arnoldi Process
Yung-Ta Li
joint work with
Zhaojun Bai, Yanfeng Su, and Xuan Zeng
ICCAD, San Jose, Nov 8, 2007
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Problem statement

• Parameterized linear dynamical system

where
Problem find a reduced-order system
where
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Application variational RLC circuit
MNA (Modified Nodal Analysis)
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Application micromachined disk resonator
Electrode
Disk resonator
thickness
Silicon wafer
PML region
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.
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Outline

• Transfer function and multiparameter moments
• MOR via subspace projection- projection subspace
  and moment-matching
• Projection matrix computation- 2D Krylov
  subspace and Arnoldi process
• Numerical examples- one geometric parameter-
  multiple geometric parameters
• Concluding remarks

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Transfer function

• State space model

One geometric parameter for clarity of
presentation
Transfer function
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Multiparameter moments and 2D recursion

• Power series expansion

Multiparameter moments
Moment generating vectors
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Outline

• Transfer function and multiparameter moments
• MOR via subspace projection - projection
  subspace and moment matching
• Projection matrix computation- 2D Krylov
  subspace and Arnoldi process
• Numerical examples - one geometric parameter -
  multiple geometric parameters
• Concluding remarks

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MOR via subspace projection

• A proper projection subspace

• Orthogonal projection

• System matrices of the reduced-order system

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MOR via subspace projection

• Goals
• Keep the affined form in the state space
  equations
• Preserve the stability and the passivity
• Maximize the number of matched moments

Goals 1 and 2 are guaranteed via orthogonal
projection
The number of matched moments is decided by the
projection subspace
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Projection subspace and moment-matching

• Projection subspace

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Moment-matching and Pade-like approximant
p4, q2
matched moments
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Comparison to the prior methods
Compare
under the requirements
A
B
C
D
the order of the reduced-order model ( no
deflations among projection subspace)
passivity preservation
Yes
Yes
No
Yes
rational transfer function
Yes
Yes
No
Yes
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Outline

• Transfer function and multiparameter moments
• MOR via subspace projection - projection
  subspace and moment matching
• Projection matrix computation- 2D Krylov
  subspace and Arnoldi process
• Numerical examples- one geometric parameter -
  multiple geometric parameters
• Concluding remarks

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Krylov subspace

• (1,i)th Krylov subspace

Use Arnoldi process to compute an orthonormal
basis
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Krylov subspace

• (2,i)th Krylov subspace

How to efficiently compute an orthonormal basis
?
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Efficiently compute
Computed by 2D Arnoldi process
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Projection subspace 2D Krylov
subspace

• Define

We have the recursion
where
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2D Krylov subspace and 2D Arnoldi decomposition

• (j,i)th Krylov subspace

• two-directional Arnoldi decomposition

where
Computed by 2D Arnoldi process

• two properties

be used to construct projection matrix
vectors in the projection subspace
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Generate V by 2D Arnoldi process

• Use V to construct reduced-order models by
  orthogonal projection

PIMTAP (Parameterized Interconnect Macromodeling
via a Two-directional Arnoldi Process)
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Outline

• Transfer function and multiparameter moments
• MOR via subspace projection- projection subspace
  and moment-matching
• Projection matrix computation- 2D Krylov
  subspace and Arnoldi process
• Numerical examples- one geometric parameter-
  multiple geometric parameters
• Concluding remarks

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RLC circuit with one parameter

• RLC network 8-bit bus with 2 shield lines
• 
  

• Structure-preserving MOR method SPRIM
  Freund, ICCAD 2004

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RLC circuit Relative error

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

CORE (p,q)(40,1)
CORE (p,q)(80,1)
order of ROM81
PIMTAP (p,q)(40,1)
order of the ROM76
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RLC circuit Numerical stability
CORE (p,q)(80,2)
PIMTAP (p,q)(40,2)
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RLC circuit PIMTAP for q1,2,3,4
q1,2
q3,4
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Parametric thermal model Rudnyi et al. 2005

• Thermal model with parameters
  and

Power series expansion of the transfer function
on
satisfies the recursion
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Parametric thermal model

• Stack the vectors via the following
  ordering

(0,0,0)
(1,0,0) ? (0,1,0) ? (0,0,1)
(2,0,0) ? (1,1,0) ?(1,0,1)?(0,2,0)?(0,1,1)?(0,0,2)
The sequence
(0,0,0) ?(1,0,0) ?(0,1,0) ? (0,0,1) ? (2,0,0) ?
? (0,0,2)
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Outline

• Transfer function and multiparameter moments
• MOR via subspace projection - projection
  subspace and moment matching
• Projection matrix computation- 2D Krylov
  subspace and Arnoldi process
• Numerical examples- one geometric parameter -
  multiple geometric parameters
• Concluding remarks

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Concluding remarks

• PIMTAP is a moment matching based approach
• PIMTAP is designed for systems with a
  low-dimensional parameter space
• Systems with a high-dimensional parameter space
  are deal by parameter reduction and PMOR
• A rigorous mathematical definition of projection
  subspaces is given for the design of Pade-like
  approximation of the transfer function
• The orthonormal basis of the projection subspace
  is computed adaptively via a novel 2D Arnoldi
  process

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Moment tree Zhu et al. DATE07

• Rotate the lattice by 45 degree

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Recursion on T