Circuit Simulation via Matrix Exponential Operators CK Cheng UC San Diego 1.

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Circuit Simulation via Matrix Exponential Operators

CK ChengUC San Diego

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Outline

• General Matrix Exponential• Krylov Space and Arnoldi Orthogonalization

• Matrix Exponential Method– Krylov Subspace Approximation– Invert Krylov Subspace Approximation– Rational Krylov Subspace Approximation

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General Matrix Exponential

• 𝝓0(At)= eAt• 𝝓1(At)= d𝞽=A-1(eAt-I)• 𝝓2(At)= 𝞽d𝞽=A-2(eAt-A-I)• 𝝓k(At)=Exercise: Expand the right hand side expression to remove the inverse operation.

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Krylov Space and Arnoldi Orthonormalization

Input A and v1=x0/|x0|

Output AV=VH+hm+1vm+1emT

For i=1, …, m• Ti+1=Avi

• For j=1, …, I– hji=<Ti+1,vj>

– Ti+1=Ti+1-hjivj

• End For• hi+1,i=|Ti+1|

• vi+1=1/hi+1 Ti+1

End For

In other words,Avi-=hi+1,ivi+1

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Standard Krylov Space

Generate: AV=VH+hm+1vm+1emT

Thus, we have eAhv1≈VeHhe1

Residual r=Cdx/dt-Gx=-hm+1Cvm+1emTeHhe1

Derivation:Cdx/dt-Gx=CVHeHhe1-GVeHhe1

=(CVH-GV)eHhe1 = C(VH-C-1GV)eHhe1

=C(VH-VH-hm+1vm+1emT)eHhe1

=-hm+1Cvm+1emTeHhe1

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Standard Mexp

Error trend

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hhError e e e mHAmv v V

sweep m and h

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Invert Krylov Space

Generate: A-1V=VH+hm+1vm+1emT

Let H=H-1, we have eAhv1≈VeHhe1

Residual r=Cdx/dt-Gx=hm+1Gvm+1emTHeHhe1

Derivation:Cdx/dt-Gx=CVHeHhe1-GVeHhe1

=(CVH-GV)eHhe1 = G(G-1CVH-V)eHhe1

=G(A-1VH-V)eHhe1

=hm+1Gvm+1emTHeHhe1

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• large step size with less dimensionInvert Matrix Exponential

sweep m and h1

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hherror e e e

mHA

mv v V

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Rational Krylov SpaceGenerate: (1-rA)-1V=VH+hm+1vm+1em

T

Let H=1/r (I-H-1) we have eAhv1≈VeHhe1

Residual r=Cdx/dt-Gx=-hm+1(C/r-G)vm+1emTH-1eHhe1

Derivation:Cdx/dt-Gx=CVHeHhe1-GVeHhe1

=(CVH-GV)eHhe1 = (1/r CV(I-H-1)-GV)eHhe1

=(1/rCV(H-1)-GVH)H-1eHhe1

=((1/rC-G)VH-1/rCV)H-1eHhe1

=-hm+1(C/r-G)vm+1emTH-1eHhe1

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• large step size with less dimensionRational Matrix Exponential

fix , sweep m and h 1

~

2eeeError

hh mH

mA Vvv

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Different

needs large m

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Different

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Spectral Transformation – = 10f• Small RC mesh, 100 by 100• Different h for Krylov subspace• Different for rational Krylov subspace

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Spectral Transformation– = 1p• Small RC mesh, 100 by 100• Different h for Krylov subspace• Different for rational Krylov subspace

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Spectral Transformation– = 100p• Small RC mesh, 100 by 100• Different h for Krylov subspace• Different for rational Krylov subspace

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Sweep for Large Range

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Sweep for Large Range

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Difference Between Inverted and Rational

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Fixed = 1p, sweep time step h

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Fixed = 1n, sweep time step h

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Fixed = 1u, sweep time step h

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Fixed = 1m, sweep time step h

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Fixed = 1, sweep time step h

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Fixed = 1k, sweep time step h

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Fixed = 1M, sweep time step h

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Krylov Space ResidualGenerate: AV=VH+hm+1vm+1em

T

Thus, we have eAhv1≈VeHhe1

Residual r=Cdx/dt-Gx=-hm+1Cvm+1emTeHhe1

Derivation:1. Set Y=[e1 He1 H2e1 … Hm-1e1]

2. We have YC=HY where C=zm+cm-1zm-1+…+c1z+c0=0

has roots 𝞴1, 𝞴2,… 𝞴m

0 -c0

1 0 -c1

1 0 -c2

… … …1 0 -cm-2

1 -cm-1

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Krylov Space ResidualResidual r=Cdx/dt-Gx=-hm+1Cvm+1em

TeHhe1

3. C=V-1DV (VC=DV), V= D=Diag(𝞴1, 𝞴2,… 𝞴m)

4. H=YCY-1=YV-1DVY-1

5. eH=YV-1eDVY-1

6. emTeHhe1

=emTYV-1eDhVY-1e1

=emTYV-1eDh1, 1=[1,1,…1]T

=emTYV-1[e𝞴1h, e𝞴2h,…,e𝞴mh]T ≈hm-1/(m-1)!

1 𝞴1 . 𝞴1m-11 𝞴2 . 𝞴2m-1. . . .. . . .. . .

1 𝞴m . 𝞴mm-1

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Invert Krylov Space ResidualGenerate: A-1V=VH+hm+1vm+1em

T

Thus, we have eAhv1≈VeHhe1

Residual r=Cdx/dt-Gx=-hm+1Cvm+1emTeHhe1

Derivation:6. Exercise to derive: em

TH-1eHhe1≈