CC BY
19
connections where the pre-injected electrons are removed. Since a positive impulse represents an activity, the function of a negative impulse is given by means of the circuit that absorbs the positive impulse.
6 SUMMARY
The characteristics on the activity driven circuits and the design tools are reported in this paper. We made several activity driven circuits by using digital IC's. The manufactured circuits functioned well. But, now, those circuits are not much superior than the digital circuits. In principle, the activity driven circuit can be designed to form through the experience. The function of flowchart that is designed by the concept of activity will be transferred into a charge transfer circuits. It is possible to fabricate activity driven circuits as a semiconductor integrated circuit. Such activity driven circuit will be superior to today's digital system.
The trial manufacturing of activity driven circuit without automatic implementation is the step to realize the activity driven system. The manufactured circuit operates the designed function. These experiences teach us the importance of timing on the operation.
As another fruit of this research, the concept of activity helps to manufacture the circuit model for the brain mechanism. The impulsive actions are combined for a sophisticated activity. The group of activities is decoded by means of a working memory. A loop is able to keep an activity. The consciousness is intermittently interlocked to the other activity. These linkages of the short-term memories make possible to explain the activities of a brain such as linguistic activity. The intermittent transmission of activity will provide the deeper understanding of the brain mechanism that has relations to the artificial intelligence.
Although the concept of activity driven circuit is based on the activity in the real world, the circuit is a model. The author hopes that this new concept of hardware artificial intelligence will contribute to developments of the technology of intelligence.
ACKNOWLEDGMENT
The author is grateful to them who manufactured the impulse driven circuits as graduation studies. Mr. Y. Uchi-umi made a control circuit for automatic vending machine [13], Mr. Y. Fujiwara made a control circuit for automatic
washing machine [14], and Mr. Y. Monma made a sheet of short-term memories. [15]. The author would like to thank Prof. Takeshi Kurobane and Prof. Kimio Shibayama at Tohoku Univ. for their advices and encouragements as author's former teachers.
REFERENCES
1. Proc. IEEE, Vol.89, No7, Edited by M. Akay, Special issue on Neural Engineering: Merging Engineering and Neuroscience pp. 991.
2. Schalkoff R.J., Artificial Neural Networks, The McGraw-Hill Co. Inc. 1997.
3. Levy J.P., Bairaktaris D, Bullinaria J.A., Cairns P., Connection-ist Models of Memory and Language, UCL Press Limited, 1995.
4. Karasawa S., The strategy of impulse driven working memory for visual perception, Proc. of Inter. Conf. on Imaging Science, Systems, and Technology, June 24-27, Las Vegas, Vol.2, pp. 729-735, 2002.
5. Karasawa S., Impulse recurrent loops for short-term memory which merges with experience and long-term memory, Proc. of 3rd Int. Conf. on Cognitive and Neural systems, pp. 36, Boston Univ., May 26-29, 1999.
6. Nicholls J.G., Martin A.R., Wallace B.G., From Neuron to Brain, pp. 12, Sinauer Associates, Inc. Publishers, 1992.
7. Karasawa S., Decoder-for-neuron translation models for brain mechanism, Research report, Miyagi National College of Technology, pp. 19-25, 2000.
8. Carter J.W., Digital Designing with Programmable Logic Devices, Prentice-Hall, Inc, 1997.
9. Honjo I., Language Viewed from the Brain, Karger, ISBN 38055-6789-8, 1999.
10. Karasawa S., Oomori J., Impulse circuits for a distributed control inspired by the neuroanatomical structure of a cerebellum, Intelligent engineering systems through artificial neural networks, Vol.10, pp. 171-190, ASME press, New York, 2000.
11. Karasawa S., Monma Y., Fujiwara Y., and Uchiumi Y., Semiconductor devices for intelligent circuit that is operated by means of transferring impulses, Academics lecture meeting, Northeast branch of Japan Society of Applied Physics, pp.2627. Dec. 5, 2002.
12. Uchiumi Y., Manufacturing of an impulse driven circuit for the control of vending machine, Graduation Thesis at Department of Electrical Engineering, Miyagi National College of Technology, March 4, 2003.
13. Fujiwara Y., Manufacturing of an impulse driven circuit for the control of automatic washing machine, Graduation Thesis at Department of Electrical Engineering, Miyagi National College of Technology, March 4, 2003.
14. Monma Y., Manufacturing of a sheet of impulse driven short-term memories, Graduation Thesis at Department of Electrical Engineering, Miyagi National College of Technology, March 4, 2003.
15. Zimmerman T.A., Allen R.A., and Jacobs R.W., Digital chargecoupled logic, IEEE Journal of solid-state circuits, Vol.SC-12, No. 5, pp. 473-485, Oct. 1977.
16. Handy R.J., Use of CCD in the Development of digital logic, IEEE Trans. ED, Vol.24, No.8, pp. 1049-1061, 1977.
17. Geiger R.L., Allen P.E., and Strader N.R., VLSI design techniques for analog and digital circuits, pp. 826, McGraw Hill Publishing Co. 1990.
YAK 681.32:007.52
SOLVING OF BOUNDARY VALUE PROBLEMS FOR MATHEMATICAL PHYSICS EQUATIONS IN CELLULAR NEURAL NETWORKS
M.A.Novotarskiy
Cmamma npuceanena eueneunn çacmocyeaHHH MOKaMbHO-acunxpoHHux Memodie do eupiweHHH Kpaûoeux çadan MameMa-munHOÏ ôiçuêu Ha KëimKoeux HeûpoHHux Mepeœax. PoçiMMHymi apximeêmypu mpaduuiÛHoï aHaëoioeoï ma ducêpemHoï HeupoH-
Hux Mepeœ i apximeKmypa ôaïamomapoeoï KëimKoeoï HeûpoHHoï Mepeœi 3 ôaïamocimKoeuM çiëadxyeaHHMM. npedcmaeëeHuû aëzopumM acuHxpoHHoio ôyHKuioHyeaHHH dëa ôazamowapoeoï KëimKoeoï HeûpoHHoï Mepeœi.
Статья посвящена изучению применения локально-асинхронных методов для решения краевых задач математической физики на клеточных нейронных сетях. Рассмотрены архитектуры традиционной аналоговой и дискретной клеточных нейронных сетей, а также архитектура многослойной клеточной нейронной сети с мультисеточным сглаживанием. Представлен алгоритм асинхронного функционирования для многослойной клеточной нейронной сети.
This paper is devoted to studies of application of locally -asynchronous methods to the boundary value problems for mathematical physics equations solving in cellular neural networks. The following approaches are considered: traditional analog cellular network architecture, digital cellular network architecture, and multilayer cellular network with multigrid smoothing. The algorithm of asynchronous operation for the multilayer cellular neural network represented.
INTRODUCTION
The locally - asynchronous calculation concept is based on representation of a global task by a set of local tasks that are realized as independent local processes cooperating among themselves with the purpose of an interchanging by intermediate information. The interacting processes are implemented in cells of neural network [1]. We call them asynchronous, if no restrictions on interchange time exist. Localization is determined as the fixed amount of connections between neurons, which is much less than a possible amount of neurons in the neural network. Such organization of processes is characteristic for a living nature, therefore is rather actual to use it in an artificial neural network for realization of such concept.
1 MATHEMATICAL MODEL OF LOCALLY -ASYNCHRONOUS COMPUTATIONS
We will examine mathematical locally - asynchronous model of the form:
Uh - difference solution.
AU = F,
(1)
U(p1 ) = b, i = 1, 2,..., M.
(2)
Arbitrary iterative one-step method for deriving an approximate solution of the equation (1) can be noted as:
B„Uh(n + 1) = CUJn) + F , n _ 0, 1, 2,..., (4)
where Bn, Cn, Fn : H ® H.
B — C
Having made replacements D = —-n, Fn = jtn + i
Ln + 1
we shall receive two-layer iterative scheme Uh(n + 1) - Uh(n)
B
+ DUh(n) = j, n = 0, 1, 2,... .(5)
Ln + 1
From here
Uh(n + 1) = Uh(n) - tn + 1B-1 (DUh(n) - j). (6) Using (6) the iterative scheme is obtained
Uh(n + 1) = Uh(n) - p[Ah(DUh(n) - j)]. (7)
Locally - asynchronous method [3] defines the order of application of the iterative scheme (7) to nodes of grid area, on which the solution of a problem (1) exists. Such order is described with usage of a concept of a random sequence
{Jn};= 1 of nonempty subsets { 1, 2,...,h}. Using this
sequence, we create a sequence of iterations { Uh(n)}^_ 1 according to a rule:
Uh ( n ) =
[Щ(n - 1 ), i Ï Jn,
/[U1 (*1 (n)), U2(*2(n)),..., Uhl(Wn))], (8)
i î J„
where A - the differential operator,
U = U(x)|(x G D cRr) with Dirichlet boundary conditions of the arbitrary geometrical form.
The boundary is considered as a point set with an amount sufficient for a faithful representation of boundary. Let us select M points p^, p2,...pM G B for describing boundary B . Then the boundary conditions of Dirichlet look like:
We approximate the equation (1) on a homogeneous grid W with step h . Then Uh _ U(Xh)|(^h £ Dh) . The resulting system of the linear algebraic equations will look like:
AhUh _ Fh , (3) where Ah - difference operator; Fh - some grid function;
where Uh - is component of a vector Uh, and {si(n)}¥_ 1 _ 1, 2,...,h - is a sequence of non-negative numbers, which meet conditions: < (n - 1 )p, s_(n)® ¥ .
The delay allows using components of a vector of the previous iterations at an evaluation of a vector of the flowing iteration. This fact strictly justifies a possibility of usage of the asynchronous conditions for computations on parallel structures.
2 ANALOG CELLULAR NEURAL NETWORK ARCHITECTURE
The cellular neural networks [1] today are widely used for solving systems (3). Their structures are very similar to the structure of cellular automata, as they contain neurons, which are connected only to the nearest neighbors. An example of a two-dimensional cellular neural network is shown in fig. 1.
Nr ( i, j ) =
C(k, l)\max{|k- i|, \l-j\ }< r f ((k, l) î Dhl)
1 < k < n, 1 < l <
Ui,j = °.5(|Vi,j + 1 -Vj - 1 ) ■
(11)
\ RIO
U(U) ; F;g
Figure 1 - A two-dimensional cellular neural network
It consists of n X m nodes arranged in n rows and m columns. The nodes of a regular graph are neurons with their neighbor connections. The r -neighborhood for the neuron C ( i, j ) is determined in [1] as a subset of the cellular neural network that meets conditions:
f, (9)
where r is a positive integer number.
For the two-dimensional cellular neural network shown in fig. 1, we have
N (i, j) = { C(i - 1, j), C(i, j), C(i + 1, j),
C(i - 1, j - 1), C(i, j - 1), C(i + 1, j - 1),
C(i - 1, j + 1), C(i, j - 1), C(i + 1, j + 1)}, r = 1. (10)
The r -neighborhood Nr(i, j) always includes a central neuron C (i, j) that provides a symmetry property. If C(i, j) e N(k, l), then C (k, l) e N(i, j), for all C (i, j) and C(k, l) in a cellular neural network.
There are many equivalent circuits of neuron C (i, j) now [1,2]. They depend on a type of a problem. One of them, useful for obtaining solution of system (3), was offered in [1]. It can be implemented with usage of operational amplifiers, as shown in fig. 2.
The voltages Uk i, Ui j simulate output signals from neighbor neurons and feedback from the output signal. The node voltage Vi j is called the state of neuron C(i, j) . The
circuit consists of linear capacitor Cv , linear resistor Rv, linear voltage controlled current source with the characteristic Iv(i, j ;k, l) = A (i, j ;k, l) U j, Iu = B (i, j ;k, l) Uk l and a subcircuit with piece-wise-linear function
Figure 2 - Implementation of neuron C(i, j) for the cellular neural network with r = 1
The internal state of the neuron is set by the equation:
cvl vll{ t ) = j +
vdt '>jW Rv
A ( i, j;k, l) Uh j ( t) +
k, l
C (k, l) î N( i, j )
B ( i, j;k, l) U(t) +1.
(12)
k, l
C(k, l) î N(i, j)
Let's consider a two-dimensional heat equation:
d2 u (x, y, t) + d2 u (x, y, t) _ 1 du (x, y , t) dx2 dy2 k dt .
(13)
The heat equation (13) can be approximated by a set of equations:
Ui j(t)j
8 dui i(t)
H_ =
1 ■ ( t) + ut + 1 j ( t) + ui j ( t) +
i,jK 'k dt
+ uh j - 1( t ) + ut - 1, j - 1( t ) + u, j - 1( t ) + ut + 1, j - 1 ( t ) + + ui -1, j +1( t ) + ui, j +1( t ) + ui +1, j +1( t ).
(14)
Using cellular neural networks for solving differential partial equations is grounded on physical analogy between the equation (12) and equation (14). The solving process is combined with learning process. It consists of searching a stationary state of the cellular neural network that is a solution of a set of equations (14).
Analog neural networks provide the high rate of solution but relative low accuracy. The alternate way is usage of digital neurons in cells of the cellular neural network. It allows applying the numerical methods to solution of a set of equations (3).
3 DIGITAL CELLULAR NETWORK ARCHITECTURE
The digital cellular neural network structure is similar to analog structure but it consists of two layers, because iterations are carried out in two steps. The two-dimensional network is shown in fig. 3.
Figure 3 - A two-dimensional digital cellular neural network
Layer 1. This layer forms new approximation of the solution vector by the iteration expression (7). A layer 1 neuron model is shown in fig.5. One can describe it as a mathematical processor that receives input signal in the form of r -dimensional vector U ( n ) and produces scalar output Uj ( n + 1 ) .
Figure 5 - A layer 1 neuron model
It realizes the mapping operation:
Ne: U(t)® Uu}(t + 1 ),
(16)
where U = {Uk }, "D(k, l) e Nr(i,j), 1 < k < n , 1 < l < m.
The neuron can be a digital controller or computer program. In both cases, it should contain operational units and f - unit. At - unit is used for simulation of computation time delay. Feedback connection provides execution of internal iterations according to locally - asynchronous algorithm.
In common case one can describe Ne -mapping as
Each layer of the neural network performs specific calculations; therefore neurons in layers have different structures. Let us consider two steps of iterations implemented by the two-layer cellular neural network.
Layer 0. In this layer neurons calculate the discrepancy for points of discrete space:
dh( n ) = F h - AhUh ( n ).
(15)
For calculations, we use a neuron model, sown in fig.4. It contains a sigma-pi unit for product AhUh (n) calculations and bias Fh subtraction. In both layers, we use the piece-wise-linear activation function, which is typical of computing neural networks.
Figure 4 - A layer 0 neuron model
U,j(t + 1) _ Uuj(t) + hf(WU(t) + C) _ (17)
_ U/j(t) + hf(f tUk, l(t) + %[) ,
k, l
where wkl £ W, ck l £ C , "D(k, l) £ Nr(k, l) , 1 < k < n , 1 < l < m.
Ne -mapping (17) is a physical analog of the iterative formula (7). Being grounded on this analogy, we shall set a way of definition of weight coefficients W. Generally W is a matrix interpretation of differential operator Ah . Therefore, the process of creation of a matrix of weight coefficients W is considered as a stage of synthesis of the cellular neural network by cloning a template.
One more function of this layer consists in detection of stopping moment at obtaining convergence of asynchronous algorithm. For this purpose, the neurons evaluate expression:
I Uh (n + 1 ) - Uh ( n ) j
I U h ( n ) |
< e
(18)
If convergence has been obtained, the vector Uh is the problem solution; otherwise, we go to the layer 0 again.
4 MULTILAYER CELLULAR NETWORK WITH MULTIGRID SMOOTHING
The asynchronous way of operation of the digital cellular neural network can cause a mismatch in speed of operation of neurons. Such mismatch reduces convergence speed of the iterative method. Usage of multigrid smoothing [4, 5] allows achieving essential improvement of convergence speed. Some accelerating schemes have been proposed to modify standard multigrid method under different circumstances [6, 7, 8]. Here we present multilayer cellular neural network paradigm with multigrid smoothing.
Let Uh( n) - some approximating of the equation (1) solution Uh , then the absolute error and discrepancy of the given iterative step are determined by expressions:
Vh(n) _ Uh - Uh(n), dh _ Fh - AhUh(n). (19) Outgoing from (19) equations of a discrepancy looks like:
AhVh ( n ) = dh ( n )
h(l) = h(L)2l -l,l < L ,
(21)
dh(i-1)(n) = P(l)dh(0(n) ,
Vh(i)(n) = I(l- 1} Vh(i-1 )(n) .
h (l- 1A
Uh ( n ) = Uh ( n ) + Vh ( l)( n )
(20)
According to [4] we receive solutions of the equation (20) on a subset of grids W(L), W(L - 1),..., W( 1), W(0), with steps
where l - number of a flowing coarse grid.
Linear projective operator P carries out the transition from fine grid to rougher grid
(22)
and the transition in an opposite direction is ensured with an interpolating operator
(23)
Computation of the absolute error at the given grid level is performed with usage of the difference scheme
Figure 6 - A multilayer cellular neural network
It consists of the input layer and 2 L + 1 computing layers. The neurons of an input layer are intended for the grid function Uh (n) correction according to expression (25). All computing layers are doubled except for the layer 0. These layers are intended for implementation of a multigrid smoothing algorithm. For simplicity, the one-dimensional variant of the neural network is represented in fig 6.
In the layer L . each neuron corresponds to the point of a
fine grid with step h _ h (L) . According to expression (21) step h (L - 1) is magnified twice in comparison with step
h (L) . Such relation of steps is accepted as a matter of convenience but this is not necessary.
The implementation of a multigrid locally - asynchronous method is determined by the following algorithm. For 1 < i < N do Choose initial value U^ ( 0) ;
1: For 0 < l <L & Jo < j < N-Pi do
(0 £ j £ ""ho;
Vh(l)(n + 1 ) = Vhl)(n) - p[Ah(l) Vh(l)(n) - dhl)(n)]. (24) begin
The cycle of computations on coarse grids is completed by correction of the grid function:
(25)
There is a set of multigrid algorithms ( V -cycle, W -cycle etc.), which use in various sequences base operations of smoothing, restricting and interpolating. Each of variants of multigrid algorithm can appear effective for the separate problem.
Therefore, we speak about the multigrid technique, which defines some ideology of solution of boundary value problems.
The multilayer cellular neural network shown in fig. 6 performs computations to solve (3) and (20).
Choose initial value Vj ( 0 )
d( l ) :=F(l) - A(l ) U (l ) ; d(l -1 ) := P (l ) d(l ) ;
Vjl)(n + 1 ) := Vjl)(n) - p[A(l) V(l)(n) - d(l)(n)] ;
V(l):=/(l- 1) V(l- 1) ;
end;
For 0 < i < N do begin
UJ: = Ut + V(L) ;
U(n + 1 ) : = U(n) - p[A(L) U(L) - j(L)] ; Ut ( n + 1 ) - Ut ( n )
if then end;
U, ( n )
■ < E then exist else goto 1;
А.Ю.Балыкова, В.М.Жигачев, Б.В.Чувыкин: АППАРАТНО-ПРОГРАММНАЯ РЕАЛИЗАЦИЯ АЛГОРИТМА ВОССТАНОВЛЕНИЯ ИНТЕРПОЛЯЦИОННЫМИ СПЛАЙНАМИ
This algorithm sets a sequence of data conversions that mainly are the characteristics of one neuron. We do not circumscribe an exact sequence of interneuron data exchange, as it cannot be defined because of asynchronous character of interaction.
The algorithm of asynchronous iterations (8) has allowed applying the numerical methods for PDE solving on cellular neural networks. We used multigrid smoothing to improve convergence speed. Our numerical results showed that remarkable accelerations rates ware achieved by these techniques.
REFERENCES
1. Chua L. O., Yang L. Cellular Neural Networks Theory// IEEE Transactions on Circuits and Systems, 1988, vol. 35, No10,
pp. 1257 - 1272.
2. Lijuan S., Panzhen W., Xinyu W. The Application of Cellular Neural Networks for Solving Partial Differential Equations// The Journal of China Universities of Posts and Telecommuni-cations,1997, vol. 4, No.1, pp.45-50.
3. Baudet G.M. The Design and Analysis of the Algorithms for Asynchronous Multiprocessors. Ph.D. Diss.-Pittsburg: Carnegie-Mellon Univ.,1978, PA, 182 p.
4. McCormic S., Rodrigue G.H. Multigrid Methods for Multiprocessor Computers. Tecn. Rept. - Lawrence Livermore: Lab. Livermore CA, 1979, 29 p.
5. Wesseling P. An Introduction to Multigrid Methods. John Wil-ley & Sons, Chichester, 1992, 294 p.
6. Brandt A., Yavneh I. Accelerated multigrid convergence and high-Reynolds recirculating flows// SIAM Journal on Scientific. Computing., 1993, vol. 14, pp. 607-626.
7. Reusken A. Steplength optimization and linear multigrid methods// Numer. Math, 1991, vol. 58, pp. 819-838.
8. Vanek P. Fast multigrid solver// Appl. Of Math., 1995, vol.40, pp. 1-20.
УДК 681.2.06
АППАРАТНО-ПРОГРАММНАЯ РЕАЛИЗАЦИЯ АЛГОРИТМА ВОССТАНОВЛЕНИЯ ИНТЕРПОЛЯЦИОННЫМИ СПЛАЙНАМИ
А.Ю.Балыкова, В.М.Жигачев, Б.В.Чувыкин
Розглянуто питання проектування гнтерполюючих анало-го-цифрових ф1льтр1в, що реал1зують алгоритми сплайн-iнтерполяцп. Приведено приклади розрахунку вар1ант1в структур АЦФ i результати iмiтацiйного моделювання.
Рассмотрены вопросы проектирования интерполирующих аналого-цифровых фильтров, реализующих алгоритмы сплайн-интерполяции. Приведены примеры расчета вариантов структур АЦФ и результаты имитационного моделирования.
The questions of designing of an interpolation analog-digital filters (ADF) realizing algorithms of spline-interpolation are considered. The examples of realization of variants of structures ADF and results of imitating modeling are given.
Использование сплайнов в вычислительной математике началось в 70-е годы. Они оказались эффективным инструментом в решении задач приближения функций, восстановления функций по неполной информации, сглаживания экспериментальных данных, цифровой фильтрации. В измерительной технике сплайны нашли широкое применение при решении задач синтеза весовых функций помехоустойчивых аналого-цифровых преобразователей (АЦП) и интерполирующих фильтров в первую очередь благодаря возможности их аппаратной реализации в аналого-цифровой форме с высокой точностью.
Рассмотрим наиболее важные свойства сплайнов.
Сплайны - это финитные функции, составленные из "кусочков" многочленов данной степени, которые состыкованы так, чтобы получившаяся функция была непрерывной и обладала несколькими непрерывными производными, т.е. была достаточно "гладкой".
Полиномиальный сплайн степени n для равномерных отсчетов (сплайн Шенберга) имеет вид [1]:
n + 1
B
(t) = f (-1 )kCkn + i (t - k)
(1)
k = 0
где знак "+" обозначает одностороннее ограничение:
tn =
tn, t > 0; 0, t< 0.
Если рассматривать полиномиальный сплайн (1) как импульсную характеристику (ИХ) некоторого аналого-цифрового фильтра (АЦФ), то можно найти его передаточную функцию через преобразование Лапласа:
Hn(p) = I (Bn (t)) = { 1 - 7/ * \'
(2)
где L - оператор преобразования Лапласа, p - оператор дифференцирования, h - шаг дискретизации. Поскольку оператору аналоговой задержки exp (-phh) на шаг дискретизации h для цифрового входного сигнала соответствует оператор дискретной задержки z-1, то передаточная функция (2) АЦФ можно представить в виде функции двух переменных, что соответствует аналоговой и цифровой представлении информации:
Hn(z, p) = (1 - z-1)n -(ph)-
(3)
В соответствии с формулой (3) реализация алгоритма восстановления цифрового сигнала интерполяционным сплайном состоит из процедур формирования n -ой конеч-
