Redundancy reduction of IC models · 11/13/2007 · 3 Model order reduction 4 Applications 5 Conclusions 15. GF NPY_]PQZ]LYLWd^T^ ^NTPY_TQTNNZX[`_TYRLYOL[[WTNL_TZY^ Introduction

/centre for analysis, scientific computing and applications

Introduction Multirate time-integration Model order reduction Applications Conclusions

Redundancy reduction of IC modelsby Multirate time-integration and

Model order reduction

A. Verhoeven1,2 E.J.W. ter Maten1,2 R.M.M. Mattheij1

[email protected]

1Eindhoven University of Technology (CASA)

2NXP Semiconductors (Design Methods)

CASA AIO-day, Dorint Hotel, Eindhoven, November 13 2007



Overview

1 Introduction

2 Multirate time-integration

3 Model order reduction

4 Applications

5 Conclusions

2



Outline

1 Introduction



4 Applications

5 Conclusions

3



Design of Integrated Circuits

Applications of electricalcircuits

analogousdigital

Circuit simulation is usedfor optimisation andverification.

Design Methods (NXP)provides circuit simulationsoftware (Pstar).

Fast but accurate methodsare needed.

4



Modified Nodal Analysis

Network models consisting of nodes and branches

Kirchhoff’s Laws (KCL,KVL) and constitutive relations (CR)

Dynamics of a circuit model can be described by vn, ib2 .

Hierarchical structure because of the modular design.

Unknowns and equations

branch currents : ib =

[ib1

ib2

],

branch voltages : vb =

[vb1

vb2

],

nodal voltages : vn,KCL : Aib = 0,KVL : AT vn = vb,Current-defined CR : ib1 = d

dt q(t , vb, ib2) + j(t , vb, ib2),Voltage-defined CR : vb2 = d

dt q(t , vb, ib2) + j(t , vb, ib2). 5



Circuit simulation

The derived equations can be written as a system ofdifferential-algebraic equations (DAE):

ddt

[q(t , x)] + j(t , x) = 0. (1)

A transient analysis computes the solution x : [0, T ] → Rd

for a given initial solution x(0) = x0.

In Pstar this DAE is discretised by the Backward DifferenceFormula (BDF). The linear systems at each Newtoniteration are solved by an hierarchical type of Gausselimination.

6



Limits of default transient analysis

The default single-rate time-integration algorithm for thetransient analysis is not efficient if the continuous ornumeric model contains redundancy, i.e. if large parts havea low activity level or even stay constant.

Then it is possible to decrease the simulation costs whilethe accuracy is maintained. This can be done by

Efficient simulation of theoriginal model: Multiratetime-integration, dynamicalpartitioning, bypassing,etc.

Model order reduction: anew model of smaller size(and complexity) iscreated. 7



Outline

1 Introduction



4 Applications

5 Conclusions

8



Multirate time-integrationThe circuit model is partitionedin an active (A) and a latent (L)part that are integrated at thefine and coarse time-grids,respectively.

ddt

[qA(t , xA, xL)] + jA(t , xA, xL) = 0, (2)

ddt

[qL(t , xA, xL)] + jL(t , xA, xL) = 0. (3)

s

s

ssss

ss

L A

xL ∈ RdL xA ∈ RdA

�

�

-

Hn

Tn

Tn+1

6Interface

��hn,1tn,0

tn,1

tn+1,0

9



Multirate for hierarchical circuit models

For circuit models the partitioning should coincide with theexisting hierarchical structure.

We can only refine the internal part x(A) for given terminalpart x(A) = v(t) (interpolation-based voltage source).

ddt

[q(A)(t , x(A))] + j(A)

(t , x(A)) = 0, x(A) =

[v(t)x(A)

]. (4)

It is preferable to refine the complete xA for given terminalcurrents iL→A = j(t) (interpolation=based current source).

ddt

[q(A)(t , x(A))] + j(A)(t , x(A)) = j(t). (5)

New trend is to use controlled current sources instead, e.g.

iL→A = j(t) + Gx(A) or iL→A =ddt

[q(A)(t , x(A))] + j(A)

(t , x(A)). 10



Error analysis of multirate BDF method

Weighted error norm at the coarse time-grid (compoundphase), where 0 ≤ τ � 1

rnC = ‖d

nL‖+ τ‖d

nA‖.

Error analysis of xA at the fine time-grid

dn−1,mA

.= αqA(t , xA(t), xL(t)) + hjA(t , xA(t), xL(t)) + b.= d

n−1,mA + hKn−1,mrn−1,m

L .

We obtain the local error bound

‖dn−1,mA ‖ ≤ ‖ˆdn−1,m

A ‖+ h‖Kn rnL‖,

= ˆrn−1,mA + hrn

I =: rn−1,mA .

11



Multirate stepsize control

Error constraints

{rnC ≤ TOL,

rn−1,mA ≤ TOL.

Error model

rnC = φn

CHK+1n

rn−1,mA = ˆrn−1,m

A + rnI

ˆrn−1,mA = φn−1,m

A hk+1n−1,m

rnI = φn

I HK+1n

Coupled tolerance levels forw ∈ (0, 1):{

TOLA = (1− w)TOL,TOLI = wTOL.

This parameter w can be chosensuch that the expected workload isminimised.Independent stepsize control of Hnand hn−1,m such that{

ˆrn−1,mA ≤ TOLA,

hmax rnI ≤ TOLI

⇒ rn−1,mA ≤ TOL.

12



Adaptive partitioning control

For (digital) circuits with moving active areas a staticpartitioning is not sufficient. Therefore the multiratepartitioning is adaptively controlled such that the expectedlocal speed-up factor is optimised.

The partitioning is controlled after each refinement phasebased on a local efficiency analysis at t = Tn.

It is also allowed to modify the partitioning just after thecompound step if it was not accepted. 13



Semi-optimal partitioning techniques

1 By means of d all possible transitions of the most activelatent element and the most latent active element arecompared and optimised, iteratively.

2 Tolerance level εrel < 1 for relative local error per element:

|di | > εrel‖d‖. (6)

3 Tolerance level εabs>TOL for absolute local error perelement:

|di | > εabs. (7)

4 From all needed stepsizes per element the largest gap isdetected to separate the system in a fast and a slow part.

14



Outline

1 Introduction



4 Applications

5 Conclusions

15



Model order reduction (MOR)

Consider the following nonlinear DAE system{ ddt [q(x)] + j(x) = Bu , x(0) = x0,

y = h(x),

where x ∈ Rd , u ∈ Rm, y ∈ Rp with d � m, p. We are onlyinterested in the relationship between u and y in thetime-domain. With offline MOR the model is replaced by alow-order model for z ∈ Rr{ d

dt [q(z)] + j(z) = Bu , z(0) = z0,

y = h(z).

Onesided methods only consider the function u → z, whiletwosided methods really consider u → y. 16



Linear Time-Invariant (LTI) models

{x = Ax + Bu , x(0) = x0,

y = Cx.(8)

The observability and controllability functions are

Lc(x0) = min{12

∫ 0

−∞‖u(t)‖2dt : u ∈ L2(−∞, 0), x(−∞) = 0},

Lo(x0) =12

∫ ∞

0‖y(t)‖2dt ,∀τ∈[0,∞)u(τ) = 0.

For linear systems we have Lc(x0) = 12xT

0 W−1x0 andLo(x0) = 1

2xT0 Mx0, where W, M ∈ Rd×d are the

controllability and observability Gramians.

The (energy) ratio Lo(x)Lc(x) should be balanced.

17



Truncated balanced realization (TBR)

The Gramians W, M satisfy the Lyapunov equationsAW + WAT = −BBT , (9)

AT M + MA = −CT C. (10)

The system is balanced w.r.t. basis V if W = VΣVT andM = V−T ΣV−1 are simultaneously diagonalised. such that

Lo(x)

Lc(x)=

xT Mx

xT W−1x=

xT V−T ΣV−1x

xT V−1Σ−1V−T x=

zT Σ2zzT z

.

The singular values of Σ often converge rapidly to zero. Areduced model can be derived by truncation.{

z = V−1AVz + V−1Bu , z(0) = z0,y = CVz.

(11)

There exist many other MOR techniques for LTI systems,like PRIMA, PVL, PMTBR, etc. 18



Trajectory PieceWise Linear (TPWL)

−1 0 1 2 3 4−1

−0.5

0

0.5

1

1.5

2

2.5

3

x1

x 2

B

A

C

D

E

The nonlinear system is linearisedat {(t1, x(ts)), . . . , (ts, x(ts))} along agiven trajectory. Each linearisedsystem is reduced by LTI modelreduction techniques. A globalbasis V is computed by a svd of alllocal reduced basisvectors. Finallythe global model is constructed bya weighted sum of all locallyreduced linearised systems.

{ ∑si=1 wi(z)

[VT CiVz + VT GiVz− VT B iu(t)

]= 0,

y = h(z).(12)

19



Empirical Gramians

The observability and controllability Gramians satisfy

W =

∫ ∞

0eAtBBT eAT

tdt , M =

∫ ∞

0eAT

tCT CeAtdt . (13)

Consider [x1, . . . , xm] and [y1, . . . , yn], where x i and y j satisfy{ddt [q(x i)] + j(x i) = b iδ(t),

x i(0) = 0.

{ddt [q(t , x j)] + j(t , x j) = 0, x j(0) = ej ,

y j = h(x j).

Then W, M can be numerically integrated as follows

W =m∑

i=1

1N

N∑k=1

x i(tk )x i(tk )T , M =n∑

i=1

1N

N∑k=1

y i(tk )T y i(tk ). (14)

For LTI systems we have that W → W, M → M if N →∞. 20



Galerkin projection

Empirical balanced truncation (EBT) uses these formulasfor W, M with a larger sets of inputs and initial values fornonlinear systems.

TBR is used to balance W, M by solving a system ofLyapunov equations. Thus a basis V can be constructedby truncation.Proper Orthogonal Decomposition (POD) approximatesW, M by using only one trajectory.

W = M =1N

N∑k=1

x1(tk )x1(tk )T = VΣVT . (15)

The reduced model for z ∈ Rr is constructed by Galerkinprojection.{

ddt

[VT q(t , Vz)

]+ VT j(t , Vz) = VT Bu , z(0) = z0,

y = h(Vz).(16)

21



Missing Point Estimation (MPE)For Galerkin projection the reduced functions, e.g.VT q(t , Vz), need the complete evaluation of q, etc.

Let V ∈ Rd×r be a given basis and P ∈ {0, 1}g×d aselection matrix with PPT = Ig , then V, VT areapproximated by

V ≈ TPV, VT ≈ VT TP.Here T ∈ Rd×g is an interpolation matrix that can beoptimised in a Least Squares sense.

Define V = PV ∈ Rg×r , W = TT V ∈ Rg×r , then we canapproximate x ≈ TVz, VT q(Vz) ≈ WT Pq(TVz), etc.The original reduced model is replaced by{

ddt

[W

TPq(t , TVz)

]+ W

TPj(t , TVz) = W

TPBu , z(0) = z0,

y = h(TVz).(17) 22



Outline

1 Introduction



4 Applications

5 Conclusions

23



Inverter chain model

Equations

u1 = Uop− u1 −Υf (uin, u1, 0)

uk = Uop− uk −Υf (uk−1, uk , 0)

for k = 2, . . . , n

uin =

t − 5 5 ≤ t ≤ 105 10 ≤ t ≤ 15(17− t)5/2 15 ≤ t ≤ 170 otherwise

24



Numerical results

Method α εrel nC nR kC kR av( dAd )(%) time (s) S

Single-rate 1340 0 5440 0 0 2661 2 82 1651 1008 3415 16 87 3.11 3

2 94 1663 996 3429 15 86 3.12 10−1 166 1953 1313 4034 9 100 2.72 10−2 97 2001 1225 4105 16 105 2.52 10−3 94 1992 1637 4093 22 133 2.0

We compared the first two presented partitioningalgorithms using iterative optimization and the relativetolerance level, respectively.

All multirate algorithms are at least two times faster thansingle-rate because of dynamical partition.

25



Hierarchical scalable test circuit HSTC

Test circuit with 5× 10subcircuits.

The circuit is driven byvoltage sources ofdifferent frequencies.

The active part consistsof the subcircuitsS11, S12, S13.

BDF2 multiratesimulation on [0, 10−8].

26



Results for hierarchical scalable test circuitHSTC

method nC nR comp. time (s) Smax.error

single-rate 2937 7330 5.8 · 10−2

multirate 111 3765 668 11 1.8 · 10−1

27



Multirate results in Pstar

Figure: The high-speed operationaltransconductance amplifier.

Figure: The temperature-independentoscillator.

Table: Multirate results with dynamical partitioning. Notation: d- number of

unknowns, NC - number of compound steps, NR - number of refinement steps, NS - number of single-rate steps, dA-

number of active unknowns, Rp - number of repartitionings, S- speed-up factor.

Circuit name d NC NR NS q dA/d Rp SHSOTA 66 120 13983 13963 117 55% 1 1.6Temp.ind.osc. 245 172 9408 80 55 11% 8 4.2 28



MOR results [inverter chain]

TPWL

Proper numerical results forr = 35, . . . , 50 (n = 104).

The higher accuracy ofPMTBR can be used to getsmaller models than PRIMAgets.

Reduced models are stillvalid for different inputs.

Linearisation points aredirectly computed during arough transient simulationbefore.

POD

The POD basis V is found bysolving the eigenvalueproblem for the correlationmatrix.

The POD basis V comprisesthe eigenvectorscorresponding to 20 largesteigenvalues and captures thedynamics on [0, 20ns] verywell.

With MPE it is possible toremove 70% of the equationswhich reduces the evaluationcosts. 29



Outline

1 Introduction



4 Applications

5 Conclusions

30



Conclusions for multirate

Normal transient simulation can be inefficient for circuitswith large slow part. Multirate algorithms can be muchmore efficient for these applications while the accuracy ispreserved.

A multirate error control mechanism has been developedwhich allows much larger steps at the coarse time-grid.Also several partitioning algorithms have been investigatedand implemented.

Current prototype in Pstar shows good results and apotential to improve. Search for the realistic designexamples to determine in which areas the multiratepotential is largest.

31



Conclusions for MOR

TPWL: behaves very well. Small balls may need longerextraction time.

Empirical Balanced Truncation: approximates the systemu → y but needs a very high extraction time.

POD: cheap version that is only accurate if x stays close tosnapshot-subspace.

Galerkin projection methods based on empirical Gramianswork nicely but have high evaluation costs. Theseevaluation costs can be reduced by techniques like MissingPoint Estimation.

Promising results of nonlinear MOR for inverter chain anddiode chain models.

32



Future plans

Publication of paper about MOR in IFIP proceedings

PhD thesis will be sent to press

PhD defence at January 8th, 2008

Next year I will start as software engineer at VORtech BV(Delft).

Thank [email protected]

33

Documents

Redundancy reduction of IC models · 11/13/2007 · 3 Model order reduction 4 Applications 5 Conclusions 15. GF NPY_]PQZ]LYLWd^T^ ^NTPY_TQTNNZX[`_TYRLYOL[[WTNL_TZY^ Introduction