Application of the theory of optimal set partitioning before building multiplicatively weighted Voronoi diagram with fuzzy parameters Текст научной статьи по специальности «Математика»

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APPLICATION OF THE THEORY OF OPTIMAL SET PARTITIONING BEFORE BUILDING MULTIPLICATIVELY WEIGHTED VORONOI DIAGRAM WITH FUZZY PARAMETERS

DOI: 10.31618/ESU.2413-9335.2020.6.71.615

Kiseleva E.M., Prytomanova O.M., Padalko V.H.

Oles Honchar Dnipro National University

ABSTARCT

An algorithm for constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with the optimal location of a finite number of generator points in a limited set of «-dimensional Euclidean space En has been suggested in the paper. The algorithm has been developed based on the synthesis of methods of solving the problems of optimal set partitioning theory involving neurofuzzy technologies modifications of N.Z. Shor r -algorithm for solving nonsmooth optimization problems.

Keywords: multiplicatively weighted Voronoi diagram, optimal set partitioning problem, optimal location of generator points, neurofuzzy technologies, N.Z. Shor r-algorithm, nonsmooth optimization problems.

Introduction. Currently, hundreds of literary sources on Voronoi diagrams and their application in various fields [1]. Voronoi diagrams for two- and three-dimensional spaces are used in many different areas of applied sciences. Quite apart from the fact that a large amount of known algorithms for constructing Voronoi diagrams of given finite Msets of a plane (space) points, called generator points, have 0(|M| log( |M|)) complexity, all these algorithms are quite complicated. In addition, in practical problems the parameters of the Voronoi diagrams often may be fuzzy.

In the papers [2, 3], the algorithms for constructing a standard (classical) Voronoi diagram with deterministic parameters and various generalizations are suggested, based on the application of methods of optimal set partition (OSP) theory and having several advantages over the known ones, which are described in the scientific literature [1, 4, 5]. Particularly, they do not depend on the En space dimension containing a limited set to be partitioned, which is independent of the geometry of the sets to be partitioned; the complexity of algorithms for plotting Voronoi diagrams based on the approach described above does not increase with the growing number of generator points; they can be used to construct not only Voronoi diagrams of a given number of generator points of fixed locations, but also with the optimal location of these

points in a limited amount of En space, and other advantages. The result of such a versatile approach is the ability to easily construct not only already known Voronoi diagrams but also the new ones.

The versatility of the approach proposed in studies [2, 3] in the construction of Voronoi diagrams is further supported by the fact that models and methods for solving continuous problems of optimal set partitioning can be generalized in the case of fuzzy initial parameters of the problem or a requirement of a fuzzy set partitioning, resulting in a fuzzy nature of Voronoi diagrams.

In this paper, there has been developed the algorithm of constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with optimal positioning of N finite number of generator points in n limited set in n-dimensional En(n > 2) Euclidean space. The algorithm has been developed based on the synthesis of methods of solving problems of the theory of OSP [7]) with neuro-fuzzy technologies [8]) and modifications of N.Z. Shor r -algorithm for solving non-smooth optimization problems [9, 10].

Problem setting. A classical Voronoi diagram of a finite set M = {t1,t2,...,tw} c En of generator points Tj = (T(1),T(2),...,T(n)), i = l,2,...,N in n-dimensional Euclidean space En (n > 2) is Voronoi polyhedra set

Vor(Ti) = [x e En: c(x,Ti) < c(x,Tj), j = 1,2,...,N,j &i},i = 1,2,...,N,

of seed points t1,t2,...,tn, where ci(x,Ti) are In a multiplicatively weighted Voronoi diagram

functions of the distance between x and Ti points which of the set M = {t1, t2,..., tn] c En are defined in E„ as a Euclidean metric.

MW

Vor(M) = ^ MW Vor(Ti)

each Voronoi polyhedron

MWVor(Ti) = [x e En: c(x,Ti)/wi < c(x,Tj)/wj, j = 1,2,..., N,j ± i}, i = 1,2,..., N, (1)

represents a set of space, the weighted distance from which to the generator point Ti £ M does not exceed the weighted distance to any other seed point (wi > 0,i = 1,2, ...,N,are given weight coefficients).

Growing crystals is one of the illustrative ways of obtaining a multiplicatively weighted Voronoi diagram [6]. When all crystals begin to grow at the same time but at different speeds, each Ti £ M point gets a weight coefficient, and when measuring the

distance to it, one needs to multiply the function that sets the distance by that weight coefficient.

Voronoi diagrams with fuzzy parameters appear, for example, when the weight coefficients of the two-point distance functions that determine the elements of the Voronoi diagram are fuzzy. Let us set for each ci(x,Ti) function of (1) a fuzzy weight wi, which depends on external factors, which may also be fuzzy, and at the same time the type of this dependency is unknown in advance.

Then (1) shall be put down as follows

MW Vor(Ti) = [x e En: c(x,xi)/WWi < c(x,Tj)/wj, j = 1,2,...,N,j ±i},i = 1,2,...,N,

(2)

For a mathematical formulation of a problem of constructing a multiplicatively weighted Voronoi diagram with an optimal positioning of a finite number N of generator points in n limited set of n -dimensional Euclidean space En (n > 2) on the basis of methods for solving OSP problems, let us eliminate the fuzziness in (2) by the neurolinguistic identification method of unknown complex nonlinear relationship of [8].

To apply the specified method in recovering values of fuzzy parameters iv1, ...,wN, without limiting the generalization of considerations, let us denote their restored values as w and consider the functional

dependency of the identification object output w from its inputs y1,...,yq as follows:

w = w(yi,...,yq),

(3)

and,

where y1,...,yq are the factors affecting w, as noted earlier, may also be fuzzy.

Having applied the method of neurolinguistic identification of unknown complex nonlinear relations developed in [8], we obtain the deterministic (clear) value of the original variable w, which is calculated by the following formulas:

w

_ T.k=idk'V*Dk(w) Zk=1^*Dk(w)

(4)

■cteM

M*Dk (w) =

sk

sk

S p *k (yf yq X if Z p *k (yf y 2' ■■■' yq) ^1

j=1

1 otherwise,

(5)

q

P *j (yi,y2,...,yq) = Vrf^ß *'[j (Уi),

1

ß *ii (yd =

i+i-

(yi-b

\ e'kj

,yq) = V *■ *i[j

i = 1

i = 1,...,q,j = 1,...,sk, k = 1, 2, ..., L,

(6)

(7)

where in formulas (4) - (7):

number of classes (linguistic terms) of the original

- n *Dk (w) is the membership function of the variable w, dk is the centre of the class Dk; original variable w of Dk class, k = 1,2,... ,L, Lis the

<

2

k

- P *f (y1,y2,...,Vq) are fuzzy production rules derived from expert and experimental information on dependency (6), j is the rule number in k -class, j = 1,2, ...,sk, sk - means the amount of rules in k -class; v *k - weight of j -rule in k -class of output w;

- №*kj (yd is the bell-shaped function of y variable membership in its linguistic term in -rule of k -class of the output variable w, b *k is the coordinate of the maximum and e *k is the concentration coefficient of this membership function.

It should be noted that the value v *k - of the weights of the rules in (6) and the parameters t *k, e *k of the membership function (7) are marked by an asterisk as optimal ones, that is, the values obtained as a result of the parametric identification of the method of neurolinguistic identification, for which the deviation of the experimental data from the artificial data obtained after adjusting the fuzzy model of the object (3) reaches its minimum. Values of n *Dk (w), and P *kj (yO in (4) - (7) are calculated at optimal values of v *k, b *k, e *kj.

Thus, after restoring fuzzy parameter values iv1, ...,wN in (2) using the described method of neurolinguistic identification, we obtain their recovered (crisp) values of w1, ...,wN.

We now formulate the mathematical setting of a problem of constructing a multiplicatively weighted Voronoi diagram with the optimal placement of a finite number of N generator points in П limited set of n -dimensional Euclidean space En (n > 2) on the basis of methods of solving OSP problems and restored values of its fuzzy parameters.

Let us assume that П is a given bounded set of En, Ti, ..., tn is a finite set of generator points in П. In cases where the location of points t1,t2,...,tn in П is unknown and these must be placed (selected) in П, another Voronoi diagram can be introduced on a set П с En, that is, the Voronoi diagram of points optimally located in a finite set.

We will consider the following total of Voronoi polyhedra as a multiplicatively weighted Voronoi diagram of a finite number N of generator points т1, т2,..., tn in П с En limited set

MW Vor(Ti) = {x £ En: c(x,Tj)/Wj < с (x,Tj)/Wj, j = 1,2 ,...,N,j Ф i }, i = 1,2,..., N,

(8)

of t1,t2,...,tn points optimally placed in a finite set, for which the total weighted distance from the

points of n set to the corresponding t1,t2,...,tn generator points is the smallest, that is, the functional

/({t1,...,tw}) £i=lfvor(t.)

(c(x,Ti)/Wi)dx

(9)

takes on the minimum value.

The method of solving the problem. Let's develop an approach to constructing a multiplicatively weighted Voronoi diagram of a finite number of generator points t1,t2,...,tn, optimally placed in П с En finite set, with fuzzy parameters, based on the application of the mathematical apparatus of neurofuzzy technologies [8] and the OSP theory [7]. To do this, we must first formulate the corresponding continuous problem of optimal partitioning of the set of

En to subsets with unknown coordinates of some points characteristic of each subset, which are called the centres of subsets, being a generalization of the problem of [7].

Let Q be a bounded Lebesgue set in n-dimensional Euclidean space En. Let a total of Lebesgue measurable subsets n1,...,nN of n c En be called the possible division of the Q set into its non-intersecting subsets H1,...,nN if

Uf=l fy = П, mes(p.i П П,) = 0, i,j = 1,2,..., N (i Ф j),

(10)

where mes(-)means Lebesgue measure.

Let denote the class of all possible partitions

of Q set into non-intersecting subsets, n1,...,nN that is = {(Di, ...,nN): Uf=1 A = n, mes(ni nDj) = 0, i,j = 1,2, ...,N (i * j)}. Let's introduce the functional

f(№......nw}, {ti.....tn}) = £f=i Jn.(c(x, Ti)/Wi) dx, (11)

where c(x, Tt) is a given real-valued limited Q*Q function measured by x = (x(1),..., x(n)) £ n for any fixed point Tj = (t-1),...,t-'i)) £ n for all i = 1,2, ...,N; Wj > 0 (i = 1,2, ...,N) are given weight coefficients.

Here, integrals are thought of as being Lebesgue. We assume that a measure of a set of the limiting points of the set , i = 1 ,..,N is zero.

Problem A. Find

min F({ü1,...,ün},{t1,...,tn}),

(«i.....

(T1,...,Tw}6 nN

where functional F({ß1,...,ßN}, {t1,...,tn}) is

(1) (n)

represented in the form of (11); the rt- ,...,t£ coordinates of centres of tj = (t(1), ..., r(n)) £ üt,i =

1,2,..., N, are unknown in advance and need to be identified.

To solve the Problem A for each fixed wi (i = 1,2,...,N) we introduce the characteristic

The ({ß*,..., ßß}, {t* ..., tN}) pair which functions of subsets ßt :

represents the minimum of functional (11) on ZN x ßN set may be called an optimal solution of the Problem A. In this case, let us call {ß*,... ,ßN} £ Zn partition an optimal partitioning of ß c En set into N subsets, and t* = (t*,...,tN) £ ßN total of t* £ ß*, i = 1,2 ,...,N centres - the optimal centres of ß* subsets in problem A.

Problem B. Find

1,x £ ßt,

0,x£ß\ßt, i = 1,...,N,

and restate the Problem A in terms of characteristic functions as follows.

min L £N=i№ ri)/wi) A(x) dx,

(X(-),x)erxQ

N

where

N

r = {k(x) = (\(x),...,kN(x)):^k(x) = 1 almost everywhere (a.e.) forxeQ

X;(x)=0vl a.e. forxeQ i = l,...,N}; t = t1,...,tn gQx...xQ = Q

N

N

For Problem B, in [7], we prove the following theorem 1, which establishes the optimal solution (A* (■), t„).

Theorem 1. C°mp°nents of a characteristic optimal solution (ß*1.....ß*t.....ß*N) of the

Vector , , , function A*(x) = Problem A for i = 1.....N and almost all x £ ß are as

(A*i(x), ...,A*t (x), ...,A*n (x)) that matches the follows:

k !( x):

1, if c( x, X*; ) / w ^ c( x, 1*j)/ w,

i ^ j a.e. for x e Q, j = 1,...,N, then x e Q*;, 0 otherwise,

the optimal solution to the problem, which is dual to problem B, is chosen as rt1,..., t,n :

G(t) = f min [c(x,Ti)/wl]dx ^ min,t £ nN.

" i=1,...,N

1

We now present Theorem 2, based on the results matches the optimal solution (nt1,...,nti,...,ntN) of of [7, 11] papers, which summarizes our considerations the Problem A for i = 1,..., N and almost allx £ n are and will be further used in the formulation of the as follows: algorithm for solving Problem A.

Theorem 2. Components of a characteristic vector function A,(x) = (At1(x),...,Atl(x),...,A,N(x)) that

k i( x)

1, if c(x,X*;) / w ^ c(x, Xy) / w .,

i ^ j a.e. for x e Q, j = 1,...,N, then x e Q»;, (12)

0 otherwise,

the optimal solution of the problem is chosen as ...,

g(t) = f min [c(x,Ti)/wi]dx ^ min,t £ D.N. (13)

" i=1,...,N

Here, each parameter Wi, i = 1,... ,N previously formulas (4) - (7).

specified as the output w that depends on y1,...,yq Problem A solution algorithm. To describe the

inputs as w = w(y1,...,yq), is calculated by algorithm, we define i -component, i = l,...,N, being

the component of the generalized gradient vector of S'gCO = (5c1(T),...,5ci(T),...,5GW(T)) functions (13) at point as follows:

el10") = /о9Тс т) Ài(x)dx, i = 1,...,N,

(14)

where gTcl(x,t) - i-component of N-dimensional vector of generalized gradient gTc(x,t) of c(x,Tj) In the formula (14) (x), i = 1,..., N, is defined as follows:

x) =

1, if c(x, zi ) / w < c(X, X ) / w ,

in other cases,

0 otherwise,

ie.n 1,...,N, thenx eQ , (15)

where Wj, i = 1,..., N, are calculated by formulas (4) - (7).

Algorithm. We place Q domain to n-dimensional parallelepiped P, the sides of which are parallel to the axes of the Cartesian coordinate system. We cover the parallelepiped P with a rectangular grid and set the initial approximation t = t(0). We calculate A(0)(x)

value at the nodes of the grid according to formulas (12), taking into account (4) - (7) formulas for calculating the parameters Wj, at t = t(0); gG(r) value - by the formula (14) at A(x) = A(0)(x),T = t(0); select the initial test step of r-algorithm h0 > 0 and find

т1 = Рп

ho

hi3g(c°)

(Н1дс(с°),дс(с0))

Pn is a projection operator on n. Proceed to the second step. Let the result of calculations after k (k = 1,2,...,) steps of the algorithm be the obtained values of T(k),A(k-1)(x) in the nodes of the grid. Let us describe (£+1) step. 1. We calculate A(k) (x) value at the nodes of

the grid according to formulas (15), taking into account (4) - (7) formulas to calculate the parameters of Wj, at

=

2.

(k)

We find gG(т) value by formulas (14) at

( k)

A(x) = A(k)(x),T =

3. We then take (k+1) step of r-algorithm in N-form, an iterative formula of which is as follows

rk + 1 _

Pn ( Tk - h

Hk+iac(ck)

(нк+1дс(тк),дс(тк)}

4. If the condition ||Tk - Tk+1|| < e, e > 0, (16) is not met, we go to (£+2) step of the

algorithm, otherwise - to cl. 5.

5. Let A„(x) = A®(x), t„ = T(i), where lis the iteration number at which condition (16) is satisfied.

6. We calculate the optimal value of the objective function g(t) of (13) by the formula

G(t) = / min [c(x,Ti)/wi]dx,

" i=1,...,N

at t = t„ and wj, i = 1,...,N calculated by formulas (14) - (17).

The described algorithm. Thus, as a result of the solution of Problem A by the algorithm described above, which is based on theorem 2 above, we obtain a total of Voronoi polyhedra (3) of generator points rhi = 1,...,N:

Vor(xi) = {x £ П с En: с (x, Ti)/wi < с (x, т¡)/Wj, i Ф j, j = 1,...,N},

but unlike the standard Voronoi diagram (1), where points t1,...,tn are fixed and parameters wj, i = 1,...,N are clear, in order to find the coordinates of the generator points t1, ..., tn, optimally

located in n c En, we need to solve a finite-dimensional optimization problem

g(t) = / min [c(x,Ti)/wi]dx ^ min,т £ nN = П Х...ХП,

" i=1,...,N '-■-'

0

with a non-differentiated objective function G( т) and Wj, i = l,...,N parameters recovered using the neurolinguistic identification method of unknown complex nonlinear relationship.

Summary. The method and algorithm has been suggested for constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with optimal location of a finite number of generator points in a limited set of w-dimensional Euclidean space En. The method is based on the formulation of an appropriate continuous problem of optimal set partitioning into non-intersecting subsets, where the centers of these subsets are located involving fuzzy parameters in the objective functional and with the criterion of the partition quality, which provides an appropriate Voronoi diagram with fuzzy parameters. The method of solving the above problem of optimal set partitioning is based on the application of the mathematical apparatus developed in [11], while the method of neurolinguistic identification, developed in [8], was used to eliminate the fuzziness in the OSP problem.

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3.Kiseleva Е.М., Koriashkina L.S. Theory of continuous optimal set partitioning problems as a universal mathematical formalism for constructing voronoi diagrams and their generalizations. II. Algorithms for constructing Voronoi diagrams based

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ЧИСЛЕННОЕ РЕШЕНИЕ ЗАДАЧИ КОШИ ДЛЯ ОБЫКНОВЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ С ПАРАМЕТРОМ ПРИ ПРОИЗВОДНОЙ

Абрамова Вера Викторовна

Канд. ф.-м. наук, доцент кафедры автоматизации и управления, Набережночелнинский институт (филиал Казанского (Приволжского) федерального университета,

Набережные Челны,

АННОТАЦИЯ

Данная работа посвящена методу коллокации решения обыкновенных дифференциальных уравнений первого порядка с параметром при производной, Основой исследований являются: общая теория приближённых методов анализа и конструктивная теория функций.

ABSTRACT

This paper is devoted to the method of collocation of the solution of first-order ordinary differential equations with the parameter for the derivative. The basis of the research is the general theory of approximate analysis methods and the constructive theory of functions.

Ключевые слова: дифференциальное уравнение, метод коллокации, обратный оператор, сходимость, скорость сходимости, оценка погрешности.