General Linear Methods for Integrated Circuit Design

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General Linear Methods for Integrated Circuit Design

Steffen Voigtmann

Oberwolfach, April 2006

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

Integrated circuit design

design constraints

design specifications

schematic generation

simulation and characterisation

specification met? release to

manufacturing

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General linear methods for integrated circuit design – St. Voigtmann, p. 2

Classical methods

I BDF

. artificial damping

-1 0 1

20 time 10 0

I Trapezoidal rule

. undesired oscillations

−2

−1 0 1 2

2π time 3π π 2 π 0 2

I Runge-Kutta methods . high computational costs

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

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Classical methods

I BDF

-1 0 1

20 time 10 0

I Trapezoidal rule

−2

−1 0 1 2

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

(5)

Classical methods

I BDF

-1 0 1

20 time 10 0

I Trapezoidal rule

−2

−1 0 1 2

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

(6)

Classification of methods

I Linear multistep methods . low costs

. very successful (BDF) . not A-stable forp>2 I Runge-Kutta methods

. very good stability properties . stepsize change is easy . high costs

I General linear methods (GLM) . combine advantages

of both classes

. make new methods possible . provide unifying framework

for known methods

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DAEs of increasing complexity

A[Dx]⁰+Bx=q A[Dx]⁰+b(x,·) =0 A d⁰(x,·)+b(x,·) =0

Index2 Index1 Index0

ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

y⁰ =f(y,t) ordinary differential equations

I well understood (theoretically, numerically)

I Butcher, Dahlquist, Gear, Hairer, Petzold, . . .

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

E x⁰+Fx=q A[Dx]⁰+Bx=q

linear DAEs

I standard form (Hairer/Wanner, Kunkel/Mehrmann)

I prop. stated (M ¨arz, Balla, Kurina)

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

y⁰=f(y,z) 0=g(z)

DAEs in Hessenberg form

I Runge-Kutta (Hairer/Wanner, Kværnø)

I lin. multistep (Campbell, Gear, Petzold)

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs

circuit simulation

A d⁰(x,·)+b(x,·) =0 nonlinear index-1 DAEs (prop. stated)

I extension of decoupling procedure

I M ¨arz, Higueras

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

A d⁰(x,·)+b(Ux,·) +BTx=0

DAEs appearing in electrical circuit simulation

I index-2, no Hessenberg form

I Tischendorf, Est ´evez Schwarz (initialisation)

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DAEs of increasing complexity

Mx⁰=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

prop. stated index-2DAEs A d⁰(x,·)+b(x,·) =0 nonlinear DAEs with properly stated leading terms

I existence and uniqueness of solutions

I convergence results for numerical methods

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DAEs in electrical circuit simulation

I Modified Nodal Analysis Aq(x,˙ ·)+b(x,·) =0

I analysis: tractability index . low smoothness requirements . use projectors (Pi,Qi,U,T, . . . )

and matrix sequences

I index can be determined topologically . look for CV loops and LI cutsets I index-2componentsTxare given by

. currents ofV-sources inCVloops . voltages of inductors andI-sources

inLIcutsets

I index-2componentsTxenter linearly Aq(x,˙ ·)+b(Ux,·) +BTx=0

V₁

V₂

V_BB

V_DD u₄

u₁₂ u₁

u₃

u₂

u7

u₆ u₅

u₁₁

u₁₀

u₉ u₈

C

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DAEs in electrical circuit simulation

inLIcutsets

V₁

V₂

V_BB

V_DD u₄

u₁₂ u₁

u₃

u₂

u7

u₆ u₅

u₁₁

u₁₀

u₉ u₈

C

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DAEs in electrical circuit simulation

Est ´evez Schwarz (2000) Cons. initialization for index-2 DAEs and it’s application to circuit simulation, PhD thesis

inLIcutsets

V₁

V₂

V_BB

V_DD u₄

u₁₂ u₁

u₃

u₂

u7

u₆ u₅

u₁₁

u₁₀

u₉ u₈

C

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DAEs in electrical circuit simulation (cont.)

A[Dx]⁰+b(Ux,·) +BTx=0 ➪ index-2 components enter linearly

I Idea: Introduce new variables

u=DP1x, w=P1D⁻(Dx)⁰+ (Q0+Q1)x.

For a solutionxthis implies

x=D⁻u+ (P₀Q₁+Q₀P₁)w+Q₀Q₁D⁻(Dx)⁰.

I Consequences:

Ux=D⁻u+ (P₀Q₁+UQ₀)w, A[Dx]⁰+BTx= (AD+BT)w

➪ F(u,w,·) =A[Dx]⁰+b(Ux,·) +BTx=0.

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DAEs in electrical circuit simulation (cont.)

I Consequences:

➪ F(u,w,·) =A[Dx]⁰+b(Ux,·) +BTx=0.

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DAEs in electrical circuit simulation (cont.)

I Consequences:

➪ F(u,w,·) =A[Dx]⁰+b(Ux,·) +BTx=0.

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Local existence and uniqueness of solutions

V. (2004) General linear methods for nonlinear DAEs in circuit simulation, SCEE

Theorem.

I The properly stated index-2DAE

F(u,w,·) =A[Dx]⁰+b(Ux,·) +BTx=0, F(u₀,w₀,t₀) =0,

is locally equivalent to w(u0,t0) =w0, F u,w(u,t),t

=0.

I For everyx⁰∈IR^m, the initial value problem

A[Dx]⁰+b(Ux,·) +BTx=0, DP1x(t0) =DP1x⁰.

is uniquely solvable. The solutionx=D⁻u+z0+z1satisfies u⁰ =f u,w(u,t),t

, z1=P0Q1w(u,t), z0 =g u,(Dz1)⁰,t .

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Local existence and uniqueness of solutions

Theorem.

=0.

, z1=P0Q1w(u,t), z0 =g u,(Dz1)⁰,t .

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Local existence and uniqueness of solutions

V. (2004) General linear methods for nonlinear DAEs in circuit simulation, SCEE

Theorem.

=0.

, z1=P0Q1w(u,t), z0 =g u,(Dz1)⁰,t

➩ .

inherent ordinary differential equation

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Properly stated index-2 DAEs

A q(x, ˙ ·)+ b(x, ·) = 0

I split solution into characteristic parts x=D⁻u+z+w

x_∗

P0x_∗ z0∗=Q0x_∗

D⁻DP¯1x∗ z1∗=P0Q¯1x∗ U Q0x∗ T Q0x∗ D⁻u z=z1∗+U z0∗ w=T z0∗

I split equations similarly u⁰=f(u,v⁰,t)

v=g(u,t)

. x =D⁻u+z(u,·) +w(u,v⁰,·) z=z(u,t) w=w(u,v⁰,t)

F(u, w, z, η, ζ, t) = 0

Fˆ₁(u, z, t) = 0 u⁰= (u, w, t) Fˆ₂(u, w, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u⁰=f(u, v⁰, t) v=g(u, t)

¯ ZG¯−1

2 D¯P1G¯−1

2 T¯G−1

2

I I−fv⁰gu remains non-singular (locally) Implicit Index-1 System

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Properly stated index-2 DAEs

V. (2004) Accessible criteria for the local existence and uniqueness of DAE solutions, MATHEON

A q(x, ˙ ·)+ b(x, ·) = 0

x_∗

P0x_∗ z0∗=Q0x_∗

D⁻DP¯1x∗ z1∗=P0Q¯1x∗ U Q0x∗ T Q0x∗

D⁻u z=z1∗+U z0∗ w=T z0∗

v=g(u,t)

F(u, w, z, η, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u⁰=f(u, v⁰, t) v=g(u, t)

¯ ZG¯−1

2 D¯P1G¯−1

2 T¯G−1

2

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Properly stated index-2 DAEs

A q(x, ˙ ·)+ b(x, ·) = 0

x_∗

P0x_∗ z0∗=Q0x_∗

D⁻DP¯1x∗ z1∗=P0Q¯1x∗ U Q0x∗ T Q0x∗

D⁻u z=z1∗+U z0∗ w=T z0∗

v=g(u,t)

F(u, w, z, η, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u⁰=f(u, v⁰, t) v=g(u, t)

¯ ZG¯−1

2 D¯P1G¯−1

2 T¯G−1

2

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Properly stated index-2 DAEs (cont.)

V. (2006) General linear methods for integrated circuit design, PhD thesis

A q(x, ˙ ·)+ b(x, ·) = 0

⇔ u⁰=f(u,v⁰,t) v=g(u,t)

x=D⁻u+z(u,·) +w(u,v⁰,·) z=z(u,t) w=w(u,v⁰,t)

I new decoupling procedure

I existence and uniqueness results

I only mild smoothness assumptions

I covers/extends results on

. linear DAEs (Balla, M ¨arz, Kurina) . nonlinear index-1 DAEs (Higueras, M ¨arz)

. DAEsA[Dx]⁰+b(Ux,·) +BTx=0 (Tischendorf, Est ´evez Schwarz) . Hessenberg DAEs (Hairer, Lubich, Roche, Wanner)

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Properly stated index-2 DAEs (cont.)

A q(x, ˙ ·)+ b(x, ·) = 0

⇔ u⁰=f(u,v⁰,t) v=g(u,t)

x=D⁻u+z(u,·) +w(u,v⁰,·) z=z(u,t) w=w(u,v⁰,t)

I new decoupling procedure

I existence and uniqueness results

I only mild smoothness assumptions

I covers/extends results on

. linear DAEs (Balla, M ¨arz, Kurina) . nonlinear index-1 DAEs (Higueras, M ¨arz)

. DAEsA[Dx]⁰+b(Ux,·) +BTx=0 (Tischendorf, Est ´evez Schwarz) . Hessenberg DAEs (Hairer, Lubich, Roche, Wanner)

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Decoupling and discretisation

Aq(x,˙ t) +b(x,t) =0 index-2 DAE

decoupling u⁰=f(u,v⁰,t) v=g(u,t) +constraints

discretisation discretisation

discretised index-2 DAE

decoupling discretised index-1 DAE +discretised constraints

I If two subspaces associated with the DAE,DN1andDS1are constant, then this diagram commutes.

I It is always assumed thatN₀∩S₀does not depend onx.

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Decoupling and discretisation

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Decoupling and discretisation

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GLMs for ODEs y

⁰

= f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i

I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y^[n−1], y^[n] = hPs

i=1 b_if(Y_i) + y^[n−1]

I General linear: Y_i = hPs

j=1a_ijf(Y_j) + Pr

j=1u_ijy^[n−1]_j , y^[n]_i = hPs

j=1 b_ijf(Y_j) + Pr

j=1v_ijy^[n−1]_j

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GLMs for ODEs y

⁰

= f (y)

i=1αiyn−i

I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y^[n−1], y^[n] = hPs

i=1 b_if(Y_i) + y^[n−1]

j=1u_ijy^[n−1]_j , y^[n]_i = hPs

j=1v_ijy^[n−1]_j

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GLMs for ODEs y

⁰

= f (y)

i=1αiyn−i I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y^[n−1], y^[n] = hPs

i=1 b_if(Y_i) + y^[n−1]

j=1u_ijy^[n−1]_j , y^[n]_i = hPs

j=1v_ijy^[n−1]_j

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GLMs for ODEs y

⁰

= f (y)

j=1aijf(Yj) + y^[n−1], y^[n] = hPs

i=1 b_if(Y_i) + y^[n−1]

j=1u_ijy^[n−1]_j , y^[n]_i = hPs

j=1v_ijy^[n−1]_j

(37)

GLMs for ODEs y

⁰

= f (y)

j=1aijf(Yj) + y^[n−1], y^[n] = hPs

i=1 b_if(Y_i) + y^[n−1]

j=1u_ijy^[n−1]_j , y^[n]_i = hPs

j=1v_ijy^[n−1]_j

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An example method

"

A U B V

#

=







1

4 0 0 0 1 0 −₃₂¹ −₁₉₂¹

49 25

1

4 0 0 1−¹⁷¹₁₀₀ −₁₀₀⁴⁹ −₆₀₀⁴³

123

1225 −²²⁵₃₉₂ ¹₄ 0 1 ¹³⁶³₁₄₀₀ ¹³⁹⁴¹₃₉₂₀₀ ₇₈₄₀₀⁵³⁷⁹

−⁹⁵₈₄ −⁵⁹₈₄ ₃₆⁷ ¹₄ 1 ⁴³₁₈ ³¹₄₂ ₃₃₆³⁷

0 0 0 1 0 0 0 0

−²⁶⁸₂₁ ⁸⁶₂₁ −²⁸₉ 4 0 ⁷⁰₉ ¹⁰₂₁ −₂₁⁵

−³²₂₁ ⁸⁸₂₁ −²²⁴₉ 16 0 ⁵⁶₉ ²⁰₂₁ −¹⁰₂₁







(Butcher, 2004) I diagonally implicit

I orderp=3and stage orderq=3

I stiffly accurate, A-stable, L-stable

I Nordsieck form, y^[n]_i+1 ≈hⁱy⁽ⁱ⁾(t_n)

(39)

An example method

"

A U B V

#

=







1

4 0 0 0 1 0 −₃₂¹ −₁₉₂¹

49 25

1

4 0 0 1−¹⁷¹₁₀₀ −₁₀₀⁴⁹ −₆₀₀⁴³

123

1225 −²²⁵₃₉₂ ¹₄ 0 1 ¹³⁶³₁₄₀₀ ¹³⁹⁴¹₃₉₂₀₀ ₇₈₄₀₀⁵³⁷⁹

−⁹⁵₈₄ −⁵⁹₈₄ ₃₆⁷ ¹₄ 1 ⁴³₁₈ ³¹₄₂ ₃₃₆³⁷

0 0 0 1 0 0 0 0

−²⁶⁸₂₁ ⁸⁶₂₁ −²⁸₉ 4 0 ⁷⁰₉ ¹⁰₂₁ −₂₁⁵

−³²₂₁ ⁸⁸₂₁ −²²⁴₉ 16 0 ⁵⁶₉ ²⁰₂₁ −¹⁰₂₁







(Butcher, 2004) I diagonally implicit

I orderp=3and stage orderq=3

I stiffly accurate, A-stable, L-stable

I Nordsieck form, y^[n]_i+1 ≈hⁱy⁽ⁱ⁾(t_n)

Diagonally implicit methods with high stage order

are possible!

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Why general linear methods?

"

Y y^[n]

#

=

"

A U B V

#

·

"

hf(Y) y^[n−1]

#

I improve stability . damping properties

similar to RK methods ^-1

0 1

20 time 10

0 −2

−1 0 1 2

I improve efficiency

. diagonally implicit schemes . solve stages sequentially

A=





λ 0

... . .. aij · · · λ





I benefit from high stage order . no order reduction

. cheap and reliable error estimates

1.0·10⁻² 1.5·10−2

2.0·10⁻²

0 5 10 15 time

p= 3 p= 2 p= 1

(41)

Why general linear methods?

"

Y y^[n]

#

=

"

A U B V

#

·

"

hf(Y) y^[n−1]

#

0 1

20 time 10

0 −2

−1 0 1 2

A=





λ 0

... . ..

aij · · · λ





1.0·10⁻² 1.5·10−2

2.0·10⁻²

0 5 10 15 time

p= 3 p= 2 p= 1

(42)

Why general linear methods?

"

Y y^[n]

#

=

"

A U B V

#

·

"

hf(Y) y^[n−1]

#

0 1

20 time 10

0 −2

−1 0 1 2

A=





λ 0

... . ..

aij · · · λ





1.0·10⁻² 1.5·10−2

2.0·10⁻²

0 5 10 15 time

p= 3 p= 2 p= 1

(43)

GLMs for index-2 DAEs

A q x(t),t ⁰+b x(t),t

=0

| {z }

↑ ↑ ↑

singular charges/ voltages/

fluxes currents

I input quantities

q^[n−1]_k+1 ≈h^{k d}_{d t}^kkq x(t),t

I

"

q(X_n,t_c) q^[n]

#

= A U

B V

·

" hQ⁰_n q^[n−1]

#

such that

AQ⁰_n+b(Xn,tc) =0

I solve for the stagesX_n Remark

I use implicit methods (Anon-singular)

I charge conservation is guaranteed

I only charges / fluxes are passed on from step to step

I analysis usesimplicit index-1 systemy⁰=f(y,z⁰), z=g(y)

(44)

GLMs for index-2 DAEs

A q x(t),t ⁰+b x(t),t

=0

| {z }

↑ ↑ ↑

fluxes currents

I input quantities

q^[n−1]_k+1 ≈h^{k d}_{d t}^kkq x(t),t

I

"

q(X_n,t_c) q^[n]

#

= A U

B V

·

"

hQ⁰_n q^[n−1]

#

such that

I solve for the stagesX_n

Remark

(45)

GLMs for index-2 DAEs

A q x(t),t ⁰+b x(t),t

=0

| {z }

↑ ↑ ↑

fluxes currents

I input quantities

q^[n−1]_k+1 ≈h^{k d}_{d t}^kkq x(t),t

I

"

q(X_n,t_c) q^[n]

#

= A U

B V

·

"

hQ⁰_n q^[n−1]

#

such that

I solve for the stagesX_n Remark

(46)

GLMs for index-1 DAEs

Apply M=

A U

B V

to implicit index-1 DAEs y⁰=f(y,z⁰), z=g(y) Y=hAf(Y,Z⁰) + Uy^[n] g(Y) =hAZ⁰+Uz^[n]

y^[n+1]=hBf(Y,Z⁰) +Vy^[n] z^[n+1]=hBZ⁰+ Vz^[n]

General Linear Methods for Integrated Circuit Design

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