Speeding up Waveform Relaxation Algorithms

International Workshop Computational Problems of Electrical Engineering
Zakopane 2004
Speeding up Waveform Relaxation
Algorithms
Marek Ossowski, Andrzej Kuczy«ski
I. Introduction
T
Authors are with the Institute of the Electrotechnics, Measuremants and Material Science, Technical University of
Lodz, ul.Stefanowskiego 18/22, 00-000 Lodz, Poland, e-mail:
[email protected], [email protected].
II. Fundamentls of the method
A. Circuit equations
...
We will investigate the circuit decomposed into
N mutually connected subcircuits. An exemplary
splitting for N=4 is shown in gure 1.
N1
N3
...
...
HE waveform relaxation (WR) methods, introduced by Lelarasmee [7] and developed during
last ten years by many authors, are still actractive
alternative to the incremental-time (IT) direct methods. They enable us to signicantly decrease the
size of analysed circuits and are suitable, in natural
way, to be implemented in parallel-processor computer systems (parallel programming). The main
advantage of this method is the possibility of using
dierent step size for dierent subsystems: subcircuits only with slow components can be integrated
with larger step size than those with fast components. Slow convergence, a result of a strong coupling between subcircuits, seems to be a major disadvantage of the investigated methods. Moreover,
depending on the size of time window, much more
memory is needed to store discrete solutions of the
previous iteration for every subsystem. WR algorithms become worth attention, if the number of
iterations, ensuring the predened accuracy of the
solution, is small enough. The problem with improving eciency without loosing exactness seems to
be hard to solve for general circuits. Some authors
[14], [15], used the Succesive Overrelaxation Algotithms with autoadaptive choice of the relaxation
factor. Today, multisplitting techniques together
with overlapping methods [11], [12], [13] are more
aplicable.
The main purpose of this paper is to present WR
algorithm for solving nonlinear dierential circuits that preserves convergence of iterations and guarantees much better eciency than standard techniques. Considerations related to convergence and
eciency of the proposed method are illustrated by
the representative example. Its comparison to stan-
dard algorithms is also included.
...
N2
N4
...
Abstract The paper deals with a problem of improving
eciency
of and
the waveform
relaxationcircuits.
(WR)
algorithms
for
linear
nonlinear
dynamic
An
algorithm forconcept
solvinganddierential
equations
using
multisplitting
overlapping
procedures,
was
formulated
inprocess.
a way that
provides
the the
convergence
of
iteration
In
order
to
split
circuit
into
independent
parts,
the
chosen
branches
are
cut
o
and
appropriate
sources,
voltage
or
current,
are
inserted
instead
ofsources
eliminated
links.
Voltages
and
currents
of
these
form
an
additional
set
ofseparated
algebraicparts.
variables,
carrying
information
between
Theiteration
conditions
that were
are toconsideensure
the
convergence
of
process
red.
The fullment
special circuit
and overlapping,
guarantee
of thesplitting
convergence
conditions.
Fig. 1. Circuit decomposed into 4 subcircuits
To simplify further explanations, let us consider
the circuits splitted into two subcircuits, labeled
N 1 and N 2 respectively, connected by three wires
(see gure 2).
2
i2
i1
u1
u2
1
N1
1
2
N2
Fig. 2. Connections between two subcircuits
Introducing additional sources, instead of the removed connections, we have two systems presented
in gure 3. Circuits from gures 2 and 3 are equivalent, if the following
relations are fullled:
1
1
2
b
bi11 =N
i21
1 ,
b
bi22 = −i12 ,u 12
i11
u21 i 22
2
b
b 12 = u22 ,
u
N2
b
b 21 = −u11 .
u
(1)
(2)
b
b
Let b
s1 and b
s2 be the vectors of additional sources
62
1
2
i12
u
^
i11 11
i 21
^
i 22 u22
1
2
International Workshop Computational Problems of Electrical Engineering
1
2
N1
i12
u^12
i 21
u^21
^
i 22 u22
relations is of the form:
^
i11

M1
 0
 0
0
^11
u
1
2
# "
b
bi11 (t)
i21 (t)
b
b
=
s1 (t) =
u22 (t) ,
b
b 12 (t)
u
"
# b
b 21 (t)
u
−u11 (t)
b
b
.
s2 (t) = b
=
−i12 (t)
bi22 (t)
and N
2
(3)
(4)

x1
x = · · · 
xN

b
s1
b
s = · · · 
b
sN
M1 ×1
b
M1 x
1 (t) + A1 x1 (t) = D1 s1 (t) − G01 b
s1 (t) , (5)
b
M2 x
2 (t) + A2 x2 (t) = D2 s2 (t) − G02 b
s2 (t) , (6)
where M1 ∈ RN1 ×N1 , M2 ∈ RN2 ×N2 (matrices M1 and M2 are nonsingular by assumption),
A1 ∈ RN1 ×N1 , A2 ∈ RN2 ×N2 , D1 ∈ RN1 ×Z1 ,
b
b
D2 ∈ RN2 ×Z2 , G01 ∈ RN1 ×Z1 , G02 ∈ RN2 ×Z2 ,
N2
N1
i x2 ∈ R
denote state variavectors x1 ∈ R
bles, vectors s1 ∈ RZ1 and s2 ∈ RZ2 contain independent sources placed inside individual circuits,
b
b
b
b
s2 ∈ RZ2 contain additional
vectors b
s1 ∈ RZ1 and b
sources.
It follows from the equations (3) and (4) that
the values of additional independent sources can be
dened by the relations:
(7)
(8)
where B2 ∈ RZ1 ×N2 , B1 ∈ RZ2 ×N1 , C2 ∈
b
b
RZ1 ×Z2 , C1 ∈ RZ2 ×Z1 N2 = diag(1, . . . , 1) ∈
b
b
b
b
RZ2 ×Z2 N1 = diag(1, . . . , 1) ∈ RZ1 ×Z1 .
b
where

are described by equations:
b
b
s1 (t) = −B2 x2 (t) + C2 s2 (t) = N2 b
s2 (t) ,
b
b
s2 (t) = −B1 x1 (t) + C1 s1 (t) = N1 b
s1 (t) ,
 x

1
0 0
x
2 
0 0 
· 

+
0 0 b
s1 
0 0
b
s
2
  
0 G1 0
x1
A2 0 G2  x2 
·
=
0 N1 0   b
s1 
b
B2 0 N2
s2


D1 0
 0 D2  s1
=
·
(9)
C1 0  s2
0 C2
Now, let us consider circuit, which consists of any
N subcircuits. This case can be described by general matrix equation:
x
M 0
A G
x
D
· +
·
=
s (10)
b
0 0
B N
s
C
b
s
for circuits N 1 and N 2 respectively:
1
0
M2
0
0

A1
0
+
B1
0
N2
Fig. 3. Circuits with additional sources
Circuits N
Zakopane 2004
b
Matrix description which results from the above


s1
s = · · · 
sN
M2 ×1

(11)
M3 ×1
moreover, xn ∈ RNn , sn ∈ RZn , b
sn ∈ RZn ,
M1 ×M1
M = diag(M1 , . . . , MN ) ∈ R
a Mn ∈
RNn ×Nn (n = 1 . . . N), A = diag(A1 , . . . , AN ) ∈
RM1 ×M1 a An ∈ RNn ×Nn (n = 1 . . . N),
G = diag(G1 , . . . , GN ) ∈ RM1 ×M2 a D =
diag(D1 , . . . , DN ) ∈ RM1 ×M3 ), B ∈ RM2 ×M1 ,
C ∈ RM2 ×M3 , N = diag(1, . . . , 1) ∈ RM2 ×M2 .
A type of the chosen WR procedures results in a
type of splittings of the structural matrices into two
parts: M = Ma −Mb , A = Aa −Ab , B = Ba −Bb ,
C = Ca − Cb , D = Da − Db , G = Ga − Gb ,
N = Na − Nb .
b
(j+1) (j+1)
x
Ma 0
A a Ga
x
· +
=
·
b
0
0
Ba Na
s
b
s
(j) (j) Mb 0 x
A
Gb x
D
=
· + b
·
+
s
b
0
0 b
Bb Nb
s
C
s
(12)
For the block Jacobi WR method we take as follows: Mb = 0 i Ab = 0, Mb = 0, Ga = 0, Bb = 0,
Nb = 0.
(j+1) (j+1)
x
M 0
A 0
x
· +
·
=
b
0 0
B N
s
b
s
(j) (j) x
0 0
0 G
x
D
=
· +
·
+
s
b
0 0
0 0
s
C
b
s
(13)
B. Multisplittings
Let L denote the number of dierent splittings
arisen from the process of circuit decomposition.
63
International Workshop Computational Problems of Electrical Engineering
Splitting denoted by k is dened as well by the
structural matrices decompositions
as by the addi
tional weight matrix E ∈ R(M1 +M2 )×(M1 +M2 ) .
k
Zakopane 2004
D. Convergence condicions
Convergence condition for the algorithm described by formula (14) is of the form [13]:
!
L
X
<1,
(18)
E(x) M−1
ρ
a,k Mb,k
(j+1) (j+1)
0 x
Aa,k Ga,k x
· +
·
=
k
0 b
Ba,k Na,k b
s k
s k
k=1
!
(j)
L
X
x
Mb,k 0
−1
(x) (19)
ρ
E Na,k Nb,k < 1 ,
=
· +
0
0
k
b
s
k=1
(j) Ab,k Gb,k
x
D
+
·
+
s (14) where ρ(∗) is an operator of nding spectral radius
b
Bb,k Nb,k
s
C
of a given matrix. Because Mb,k = 0 and Nb,k =
0 hold for k = 1, . . . , N + 1, convergence of the
The nal result of the j+1 iteration is
investigated algorithm is always ensured.
(j+1) k=L
(j+1)
X E. Nonlinear circuit analysis
x
x
=
E
(15)
b
b
s
s
k
Let us consider description of the splitted system:
k
k=1
x, b
fx (x,
s, s, t) = 0 ,
x(0) = x
and the relation E + E + · · · + s E =
L
2
1
fbs (x, b
s, s, t) = 0 ,
(20)
diag(1, . . . , 1) must be fullled.
Let us describe Jacobi WR method used with the
T
T
] , fbs = [fbs,1 . . . fbs,N ] and
concept of multisplittings. We assume that L = N where fx M=1,n[fx,1 . . . fx,NM
2,n
, fbs,n ∈ R
, wheras n = 1, . . . , N,
and that the splitting of circuit matrices doesn't fx,n ∈ R
M
+
·
·
·
+
M
=
M
,
M
+ · · · + M2,N = M2 .
1,1
1,N
1
2,1
vary at every stage k (k = 1, . . . , N). It means, the
We
are
looking
for
a
solution
vector x(t), which
splitting description given by (13) stays the same.
satises
equations
(20)
over
a
given
nite interval
We will change weight matrices E in such a man- [0, T ].
k
Thus, the equations describing subcircuit number
ner, that, at every stage k, we need to know only
n
are
of the form:
the values of vector variables xk , b
sk , which can be
calculated directly from the subcircuit k.
n
b
fx,n x
n , xn , b
sn , sn , t = 0 ,
xn (0) = x
Let us explain the weight matrix construction.
For that purpose we introduce
the
notations:
fbs,n (xn , b
sn , sn , t) = 0 ,
(21)
E(x) ∈ RM1 ×M1 i E(bs) ∈ RM2 ×M2 .
k
k
b
where b
sn is a vector containing sources acting in


(x) side
subcircuit
n, and b
sn contains sources supplying
E 0
k

other
subcircuits,
but
calculated in relation with
(16)
E = 
k
0
E(bs) xn .
Ma,k
0
k
For k = 1, . . . , N we set zeros into matrices E(x) k
and E(bs) , except the diagonal elements correk
sponding to the xk and b
sk variables respectively,
where we set 1.
C. Subcircuits overlapping
The basic idea of overlapping techniques is realized here in such a manner, that two subcircuits will
be analyzed simultaneously and the weight matrices are of the form:


(x) E
0
m
+
E = α 
k
0
E(bs)  m 
E(x) 0
n
 (17)
+ (1 − α) 
0
E(bs) F. Algorithm
1. Iterate: For j = 1, 2, . . . to satised do:
.
2. Set: x(j) (0) = x
3. Solve: for k = 1, . . . , N

(j)
(j+1) 
b
fx,n x
(j+1)
b
,
s
,
s
,
t
=0,
,
x

n
n
n
n


k
k



fbs,n x(j+1)
snj+1 , sn , t = 0
,b
n
k k (j)

(j+1) (j+1) b

sn+1 , sn+1 , t = 0 ,

fx,n+1 x
n+1 k , xn+1 k , b




(j+1) j+1

fbs,n+1 xn+1 , b
sn+1
, sn+1 , t = 0
k
4.
k
Calculate:
64
j+1 .
k
x(j+1) by the relationship
(j+1) X
N
(j+1)
x
x
=
E
b
s
s k
k b
n
where α ∈< 0, 1 > and for k = 1, 2, . . . , N pairs of numbers m,n, have the following values:
(1, 2) (2, 3) . . . (N, 1).
(22)
to obtain x(j+1) and b
s
k=1
where E is expressed by equation (17).
k
(23)
International Workshop Computational Problems of Electrical Engineering
To study convergence of the presented algorithm we
apply Newton's method in functional space [16]. As
a result we approximate nonlinear equations (20) by
the linear system with time-varying coecients
Ma (t)
0
Mb (t)
=
0
(j+1)
(j+1)
Aa (t) Ga (t) x
0 x
=
+
·
· Ba (t) Na (t) b
s
0 b
s
(j)
(j) A (t) Gb (t) x
0 x
+
+ b
·
· b
Bb (t) Nb (t)
s
0
b
s
D(t)
+
s , (24)
C(t)
where M(t) = Ma (t) − Mb (t) and N(t) = Na (t) −
Nb (t) are nonsingular matrices for all t ∈ [0, T ].
Waveform method is convergent [2] if
max ρ M−1
and
a (t) Mb (t) < 1
t∈[0,T ]
max ρ N−1
(25)
a (t) Nb (t) < 1 .
t∈[0,T ]
We assume Mb (t) = 0 and Nb (t) for every t ∈
[0, T ], hence the condition (25) is fullled for the
given iteration procedure.
III. Numerical example
To investigate eciency of the proposed algorithm we have analysed a simple testing circuit (the
N-segment model of transmission line with nonlinear conductance) presented in Fig.4. We compared standard block Jacobi WR algorithm with
the proposed multisplitting procedure for the wide
range of overlapping coecients. Calculating two
neighbouring segments and using the factor α = 0.6
we have speeded up the iteration process by 30%
(average result for dierent values of model elements). We have used zero initial vector, error tolerance ε = 10−4 and trapezoidal rule as basic code
of ODEs. As a stopping criteria we require
(j+1)
(t) − x(j) (t) ≤ ε ,
x
for all t ∈ [0, T ].
Fig. 4. Transmission line model for numerical test
IV. Conclusion
The concept of waveform relaxation algorithm
presented above is based on the special multisplitting of the nonlinear circuits. Partial results are
relaxed with overlapping techniques. Applying this
conncept we can ensure numerical stability of the
WR iterations performed for the nonlinear circuit
having nonsingular structural matrices fullling the
Zakopane 2004
convergence criteria (25). Hence, there is no need
to check convergence of the procedure in every step
of the discrete numerical iteration. The numerical experiments showed that applying proposed method we can improve the eciency of the WR algorithm by more than 30%. The proper choice of decomposition and optimisation of the linking structure may speed up the method signicantly, but
the general rules are very hard to be uniquely dened. The use of dierent overlapping coecient a
for dierent subcircuits is still investigated by authors and today results looks promising.
References
[1] J.Janssen, S.Vandewalle: Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The continuoustime case, SIAM J. NUMER. ANAL., vol. 33, no.2 , pp.
456-474, April 1996.
[2] Y.Jiang, W.Luk, O.Wing: Convergence-Theoretics of
Classical and Krylow Waveform Relaxation Methods for
Dierential-Algebraic Equations, IEICE , JEICE Trans.
Fundamentals, vol. E-80-A, no.10, pp.1969-1971, October 1997.
[3] Y.Jiang, O.Wing: On the spectra of waveform relaxation
operators for circuit equations, IEICE Trans. Fundamentals, vol. E-81-A, no.4, April 1998.
[4] Y.Jiang, O.Wing: A note on convergence conditions of
waveform relaxation algorithms for nonlinear-algebraic
equations, Applied Numerical Mathematics, vol.36 no.
2-3, Feb. 2001, pp. 281-297.
[5] A.Kuczy«ski, M.Ossowski: The two-level waveform
relaxation algorithm, Proceedings of IX International Symposium on Theoretical Electrical Engineering,
ISTET'2001, Linz, Austria, stron 4, 2001.
[6] A.Kuczy«ski, M.Ossowski: Convergence of the two-level
waveform relaxation algorithm, Proceedings of XXV International Symposium on Theoretical Electrical Engineering, IC-SPETO-2002, Gliwice-Ustro«, pp.337-340.
[7] E.Lelarasmee, A.E.Ruehli, A.L.Sangiovanni-Vincentelli:
The Waveform relaxation methods for time-domain analysis of large-scale integrated circuits, IEEE Trans. On
Computr-Aided Design, vol. CAD 1, no.6 , pp. 131-145,
July 1982.
[8] M.Ossowski, A.Kuczy«ski: Zachowanie algorytmów relaksacyjnych w analizie ukªadów nieliniowych o wielu
punktach równowagi, Proceedings of XXV International
Symposium on Theoretical Electrical Engineering, ICSPETO-2002, Gliwice-Ustro«, pp.212-222.
[9] M.Ossowski, A.Kuczy«ski: Waveform relaxation methods for solving dynamic circuits having multiple equilibrium points, Proceedings of XXVI International
Symposium on Theoretical Electrical Engineering, ICSPETO-2003, Gliwice-Niedzica,
[10] P.Saviz, O.Wing: Circuit Simulations by Hierarchical
Waveform Relaxation, IEEE Trans. On Computr-Aided
Design on Int. Cir. and Systems, vol. 12, no.3 , pp. 131145, July 1993.
[11] D.P.O'Leary, R.E.White, Multisplittings of matrices
and parallel solution of linear systems, SIAM J.Alg. Disc.
Meth. 6 , pp.630-640, 1985.
[12] G.Alefeld, I.Lenhrdt, G.Mayer, On multisplitting methods fo band matrices, Numer. Math. 75, pp.267-292,
1997.
[13] Y. Jiang, R.M.M. Chen, Multisplitting waveform relaxation for systems of linear integral-dierential-algebraic
equations in circuit simulation, Journal of Circuits, Systems and Computers, vol.10, No. 3&4, pp.205-218, 2000.
[14] M.J.Gardner, A.Ruheli, Optimaized waveform relaxation methods for RC type circuits, IEEE Trans. On Circuits and Systems, vol. 51, No. 4, April 2004.
[15] A.Lumsdaine, M.W.Reichelt, J.M.Squyres, J.K.White,
Accelerated waveform methods for parallel transient simulation of semiconductor devices, IEEE Trans. On
Computr-Aided Design on Int. Cir. and Sys. vol. 15 No
7 July 1996. Systems,
[16] D.J.Erdman, D.J.Rose, Newton waveform relaxation
techkiques for tightly coupled systems, IEEE Trans. On
Computr-Aided Design on Int. Cir. and Sys. vol. 11, No
3, pp.589-606, May 1992.
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