Error control in Krylov subspace methods for Model Order Reduction

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Error control in Krylov subspace methods for
Model Order Reduction

• Pieter Heres
• June 21, 2005
• Eindhoven

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Overview

• Application
• Krylov subspace methods
• Error control

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Interconnect structures
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Coupled simulation

• Incorporate passive layout effects in full chip
  simulation

Passive circuit
Active circuit
Maxwells equations
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Model Order Reduction

• To quickly capture the essential features of
  passive structure
• Implementation in Philips layout simulator
  Fasterix
• Preservation of stability (and passivity)
• Example

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RF Transformer

• Courtesy to Jos Bergervoet,
• Philips Research

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System equations

• Circuit equations
• Matrices 5202 x 5202, partly full
• 4 ports
• Simulated for frequencies up to 30 GHz
• Defined such that afterward components can be
  added

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Frequency domain

• Laplace transform to frequency domain
• Transfer function
• Approximation for frequency behavior

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Overview

• Application
• Krylov subspace methods
• Error control

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Krylov subspace methods

• Expand X(s)
• Collect the terms for different powers of s
• In general

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Krylov subspace methods (2)

• Collecting the moments in one space
• gives a Krylov space
• In general
• Orthonormal basis of space
• Projecting the space onto the space VTGV,
  preserves the first moments of X(s)

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Algorithm

• Solve G W B
• V1 R W (QR step)
• for j 1,2,
• Solve G W C Vj
• for i 1,2,,j
• Hi,j ViT W
• W W Vi Hi,j
• end
• Vj1 Hj1,j W (QR step)
• end
• Project system matrices

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Reduced system

• Projected system matrices
• Reduced system
• Transfer function of reduced system

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Reduced system (2)

• Consider the shift-and-inverted system
• Or
• And the reduced form
• where

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Frequency domain simulation
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Time domain simulation
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Overview

• Application
• Krylov subspace methods
• Error control

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When to stop?

• Approximate
• There is a closed expression for error function
• In-practical
• Closed expression (eventually a bound) becomes
  approximation

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Error

• We state

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Updating reduced matrices

• Projection
• In every iteration a block is added
• The matrices can be cheaply updated in every step

Solve G W B V1 R W Calculate 1st matrix for j
1,2, Solve G W C Vj Orthogonalize Vj1
Hj1,j W Update matrices end
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Error definition

• For practical reasons we define the error
• Officially we should take

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Results
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Results (2)
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Sequence argument

• The following bounds can easily be derived

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Sequence argument (2)

• In terms of errors
• This all can be formulated as

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Sequence argument (3)

• Finally
• Choose the value in between
• Finally

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Conclusion

• Model Order Reduction techniques have shown to be
  useful for passive electronic applications
• Krylov subspace methods for Model Order Reduction
  can be fully automatic