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Vladikavkaz Mathematical Journal 2018, Volume 20, Issue 3, P. 37-47
УДК 517.9+541.124+541.126 DOI 10.23671 /VNC.2018.3.17981
BINARY CORRESPONDENCES AND THE INVERSE PROBLEM OF CHEMICAL KINETICS8
A. E. Gutman1,2, L. I. Kononenko1'2
1 Sobolev Institute of Mathematics, 4 Academician Koptyug av., Novosibirsk 630090, Russia; 2 Novosibirsk State University, 1 Pirogova st., Novosibirsk 630090, Russia
E-mail: gutman@math.nsc.ru; larakon2@gmail.ru, larak@math.nsc.ru
Abstract. We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters.
As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of the composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics.
Key words: binary correspondence, inverse problem, solvability, composition, stability, correctness, differential equation, chemical kinetics, linear independence. Mathematical Subject Classification (2000): 34A55.
We continue the study started in fl, 2] which is devoted to formalization of the notion of problem and solution of the inverse problem of chemical kinetics. In particular, we extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations.
1. Formalization of the notion of problem
In this section, we employ binary correspondences for formalizing the notion of problem, basic components of problems, their properties, and constructions: the condition of a problem, data and unknowns, solvability and unique solvability, inverse problem, composition and rest" The work was supported by the program of fundamental scientific researches of the SB RAS № 1.1.2., projects № 0314-2016-0005 and № 0314-2016-0007, and by the Russian Foundation for Basic Research, project № 18-01-00057.
© 2018 Gutman A. E., Kononenko L. I.
riction of problems. We also consider topological problems, the related notions of stability and correctness, and problems with parameters.
1.1. By a problem we mean an arbitrary correspondence between the elements of two sets, i. e., a triple P = (A, B, C), where ^d B are any sets and C C A x B. The sets A, B, and C (i. e., the set of departure, the set of destination, and the graph of the correspondence P) are denoted by Dom P, Im P, and Gr P and called the domain of data, the domain of unknowns, and the condition of the problem P. The containment (a, b) € Gr P is written as P(a, b) and is treated as the condition expressing the fact that the unknown b corresponds to the data a.
P
Given data a € Dom P, find unknowns b € Im P which meet the condition P(a, b).
The image P[X] and preimage P-1[Y] of subsets X C Dom P and Y C Im P with respect to P
P[X] = {b € ImP : (3x € X) P(x,b)}, P-1[Y] = {a € DomP : (3y € Y) P(a,y)}.
1.2. A solution to a problem P for a data instance a € Dom P is an arbitrary unknown b € Im P which meets the condition P(a, b). The set of solutions to P for a is denoted by P[a]. Therefore,
P[a] = P[{a}] = {b € Im P : P(a, b)}, a € Dom P.
A problem P is solvable for a € Dom P whenever P[a] = 0, i. e., given a, the problem P has
P
dom P := {a € Dom P : P[a] = 0}
is called the domain of solvability of the problem P. If dom P = Dom P, the problem P is called solvable or, more precisely, everywhere solvable.
1.3. A problem P is said to be uniquely solvable for a € Dom P if, given a, the problem P has a unique solution, i. e., P[a] = {b} for some b € Im P. The corresponding solution b is denoted by Ps (a) Therefore, if P is uniquely solvable for a then
P [a] = {Ps (a)}.
The set
dom Ps := {a € Dom P : P is uniquely solvable for a}
P
Ps: dom Ps ^ Im P, a ^ Ps(a)
P dom P C dom P C Dom P
problem P is uniquely solvable on a set D C Dom P if D C dom Ps. The problem P is called uniquely solvable or, more precisely, everywhere uniquely solvable if it is uniquely solvable on Dom P dom P = Dom P P
P
1.4. Given a problem P = (Dom P, ImP, GrP), the inverse problem is the inverse correspondence
P-1 := (ImP, Dom P, (Gr P)-1), where (Gr P)-1 = {(b,a) : (a, b) € Gr P}.
Remark. If a problem P models a real physical process, consideration of the inverse problem P-1 is motivated by the search of a relatively simple formal law which describes the process with adequate accuracy. The data of the inverse problem are experimentally measurable characteristics of the process, while the unknowns are, for instance, the coefficients of a differential equation describing the process under observation.
P
inverse problem P-1 are functions of the corresponding class, while, in practice, the role of data of the inverse problem is not played by the functions themselves but rather by some of their characteristics which can be measured, i. e., by certain finite sets of numbers.
The inverse problem can be suitably corrected by means of the composition (see 1.5) of the problem P-1 and a simple auxiliary problem which represents the relation between functions and their characteristics being measured. (An example of such correction is presented in 2.3.)
1.5. The composition of problems P and Q is the composition of the correspondences, which is the problem
Q o P := (Dom P, Im Q, Gr Q o Gr P)
with condition
Gr Q o Gr P = {(a, c) € Dom P x Im Q : (3 b € Im P n Dom Q) P (a, b) & Q(b,c)}.
The composition Q o P is usually considered in the case when Im P = Dom Q.
1.6. The restriction of a problem P onto subsets A C Dom P and B C Im P is the problem
P |B := (A, B, Gr P n (A x B)).
The restrictions P |a := P|Am and P|B := P|DomP are particular cases.
The restriction of a problem can be defined by means of composition with the corresponding embedding problems. Given arbitrary sets X and Y, consider the problem IdX := (X, Y,IX), where
= {(z,z) : z € X n Y} = {(x,y) € X x Y : x = y}.
P A C Dom P B C Im P
P |a = P O Id D0m P, P |B = Id fm P O P, P |B = Id fm P O P O Id A0m P.
1.7. An isomorphism between problems P and Q is a pair (f, g) of bijective mappings f: Dom P ^ Dom Qg: Im P ^ Im Q such that
GrQ = {(f(a),g(b)) :(a,b) € GrP}.
Two problems are called isomorphic if there is an isomorphism between them. P Dom P
Im P
(f, g) fg
All the notions introduced here, which are related to topologies or continuity, admit natural analogs for the case of uniformities and uniform continuity. (Metric and, in particular, normed spaces are examples of uniform spaces.) We will not present the corresponding clarified definitions, which are rather obvious.
1.9. A topological problem P is called stable at a point a € dom P if the correspondence P is upper semi-continuous at the point, i. e., for every neighborhood V of the set P[a] in Im P, the preimage P-1[V] is a neighborhood of the point a in dom P. The problem P is stable on a set D C dom P if P is stable at each point a € D. The problem P is called stable
P dom P
a dom P dom P
open set G C Dom P such that a € G n dom P C dom Ps), the stability of the problem P at a
P a D
dom P dom P G C Dom P
D C G n dom P C dom P P D
PD is equivalent to its continuity.
P
a € Dom P a dom P P a
aa
aP
is said to be correct (or, more precisely, conditionally correct) on a set D C Dom P if P is correct at each point a € D. A problem P is called correct if P is correct on Dom P. Therefore, the correctness of a problem means its unique solvability and stability (or, which is the same, continuity).
1.11. By a family (vj)ie/ we traditionally mean a function defined on I, and the term vi denotes the value of the function at a point i € I. Given an arbitrary family (V)ieI, the symbol niei V stands for the corresponding Cartesian product, which is the set of families (vi)ieI such th at vi € Vi for all i € I. If n: X ^ n i£I Vi, i € I, and J C I, the functions
ni: X ^ Vi, nj: X ^ JJ Vj
jeJ
are defined by the formulas
ni(x) := n(x)i € Vi, nj(x) := n(x)| j € Vj, x € X.
jeJ
1.12. A parametrization of a set X is an arbitrary injective mapping n defined on Dom n := dom n = X and acting into the Cartesian product Im n := HieI Vi of some family (Vi)ieI. In this case, I is called the set of parameters and denoted by Par n, the elements i € Par n are called parameters, the set Im ni := Vi is called the range of the parameter i, and ni(x) € Im ni is the value of the parameter i for an object x € X. The product HjeJ Vj is called the range of the set of parameters J C Par n and denoted by Im nj.
Note that the range Im ni of a parameter i need not coincide with the set im ni = ni[X] of the values of the parameter, i. e., the inclusion im ni C Im ni can be strict. In the case of equality im ni = Im ni5 the range of the parameter i is called exact.
A set endowed with a parametrization is called a parametrized set. By default, the parametrization of X is denoted bv n or, more explicitelv, by
1.13. When considering a parametrization n of a topological space X, it is natural to endow the set Im nj where J C Par n, with the image of the topology of X with respect to nj, i. e., to assume open those subsets U C Imnj whose preimage n-1[U] is open in X. In this case, n occurs a continuous mapping from X into Im n and a topological isomorphism between X and im n.
The ranges Im n» of the parameters i € Par n usually have their own natural topologies which make the mappings ^continuous. Otherwise, Im n can be endowed with the image of the topology of X with respect to n or with the topology induced from Im n in which the open subsets of Im n» are the sets of the form (u : u € U}, where U is open in Imn.
The ranges of parameters are often Banach spaces. In this case, parametrized topological spaces are close analogs of Banach bundles (see, for instance, [3]), where the domain I of a bundle V plays the role of the set of parameters, and the stalks V(i) are the ranges of parameters i € L
P
Dom P Im P
X
trivial parametrizations having single parameter: n1(x) = x for all x € X.
As is easily seen, the pair (nA,nB) is an isomorphism between a parametrized problem (A, B,C) and the probtem (A',B', C"), where A' = im B' = im and C' = { (nA(a),nB(b)) : (a, b) € C}. Furthermore, if the problem (A, B,C) is topological then so are the problem (A', B', C') and the isomorphism (nA, ).
1.15. Let n be a parametrization of a set A a € A J C Par n J' := Par n\J. Denote by ResJ(A) the problem (Im nJ, A, Rj), where
Rj = {(v, b) : v € Imnj, b € A, nj(b) = v, nj/(b) = j(a)},
AJ
on assuming fixed the values of the rest parameters. In the case J = {i}, we write ResJ(A) instead of ResJi}(A).
Since n is injective, the problem Resj(A) is uniquely solvable on the set
domRes}(A) = (nj'(b) : b € A, nj'(b) = nj'(a)} and its solution for every v € dom Resj (A) is determined by the formula
{V if i £ J •
W, if i € J-
1.16. Let n ^e a of a topological space A a € A J C Par n. A set of parameters J is locally free at the point a, if the domain of solvability dom Resj (A) of the problem Resj (A) is a neighborhood of th e point nj (a) in the topological space Im nj. Therefore, a locally free set of parameters realizes all sufficiently small changes of values with the values of the rest parameters fixed. A parameter i is locally free at a if so is the set {i}.
1.17. Let P be a parametrized topological problem, a € dom P, and let J C Par n, where n := ndom p. The problem P is stable at the point a with respect to J, if the problem P o Res}(dom P) is stable at the point nJ(a) Stability of a problem at a with respect to J
Ja The problem P is stable on a set D C dom P with respect to J, if P is stable at each point a € D with respect to J. The problem P is stable with respect to J if P is stable on dom P with respect to J. In the case J = {i}, the term stability with respect to the parameter i is used.
If the natural topology on im nJ is considered and a is an interior point of dom Ps relative dom P P a J
a
The latter, in its turn, means that the solution Ps(b) continuously depends on the values nJ(b) of the parameters J as nJ(b) tend to nJ(a) with the equalitv j (b) = j (a) preserved.
1.18. Let P be a parametrized topological problem, i € Parn. The problem P is called a "problem with small parameter i" if Im ni C R, the number 0 is a limit point of Im ni, and
Pi
to 0 for instance, about the stability of P with respect to i at a point a with ni(a) = 0.
As an illustration, we consider a singularly perturbed system of ordinary differential equations which arises in modeling certain processes of chemical kinetics and burning (see, for instance, [4, 5]). Within the study of the corresponding inverse problem, a criterion will be established for linear independence of functions in terms of finite sets of their values (see 2.5).
2.1. Suppose that m,n € N X := Rm, Y is a domain in Rn, T := R, 0 < eo € R. Put E := {e € R : 0 < e < e0}, F := C(X x Y x T x E, Rm), G := C(X x Y x T x E, Rn).
Consider the problem P with domain of data Dom P = F x G x E, domain of unknowns Im P = C 1(T,X) x C 1(T, Y), and condition
where f € F, g € G, e € Ex € C 1(T,X), y € C1 (T, Y).
P
a convenient tool for studying multidimensional singularly perturbed systems of differential equations which makes it possible to lower the dimension of the system under study.
Pe system into "slow" and "fast" subsystems:
P
e
results of A. N. Tikhonov (see, for instance, [9]) on passing to a solution to the degenerate problem as a small parameter tends to zero.
v € [dom R] ^ Ps (R(v)) , where R := Res}(Dom P).
2. The inverse problem of chemical kinetics
for all t € T,
x(t) = f(x(t),y(t),t,e) and ey(t) = g(x(t), y(t), t, e).
2.2. The inverse problem to P consists in finding the unknown functions on the right-hand side of the system, given some data on the solution to the direct problem P. The close connection of the initial problem with the degenerate system motivates the study of the case e = 0. We additionally assume that the "slow surface" defined by the equation
g(x,y,t, 0) = 0
consists of a single sheet (with respect to the dependence of y on x) and that the function g € G meets the condition of the implicit function theorem, which fact allows us to replace the equation
g(x(t),y(t),t, 0) =0
by the equivalent equation of the form
y(t) = h(x(t),t)-
We also assume that the right-hand side f of the main differential equation is a polynomial (which is natural for problems of chemical kinetics).
So, consider the partial case of the problem P in which m = n = 1, E = (0}, and the functions f € F are polynomials in two variables of degree at most p € N:
f (x,y,t,e) = Y1 Yij xV,
(ij)eK(p)
where Yj € R, (i, j) € K(p),
K(p) := ((i,j) : 0 < i,j € Z, i + j < p} .
Introduce the notation
Kip) := fctifcta
for the number of elements of the set K(p) and fix an arbitrary enumeration
K (p) = Unj'l^ (i2 ,j2 ),--.,(iK(p) ,jK(p))} -
Therefore, the expression Yfc xjfcyjfc is the general form of a polynomial in two variables
x, y p
As a result of the above agreements, we arrive at the problem Q with domain of data
Dom Q = RK(p), domain of unknowns Im Q = C 1(R)2, and condition
{K(p)
x(t)= E 7. x(t)ik y(t)jk ^ralli € R, y(t) = h(x(t),t)
where Yi, 72, - - -, 7k(p) € R, x,y € C 1(R) h € C 1(R2).
2.3. The formal inverse problem Q-1, which has pairs of functions (x,y) € C 1(R)2 as data, is very simple and impractical. For representing the domain of data, finite collections of the values of functions or their derivatives are more adequate than everywhere defined functions. The corresponding correction of the inverse problem is realized by composition of
the problem Q 1 and the auxiliary problem R with domain of data Dom R = (Rk(p))3, domain of unknowns Im R = C 1(R)2, and condition
R ((T,a,ft), (x,y)) ^
x(ti) = ai, x(T2) = a2,
x(Tl) = xX(T2) = ft2,
x(tk(p)) aK(p), XX(TK(p)) = ^k(p)!
where t, a, ft € Rk(p), x, y € C 1(
As compared to the formal inverse Q-1, the composition Q-1 o R is more practical and amounts to the following problem: Given t, a, ft € Rk(p), find the coefficients 7 € Rk(p) for which there exist functions x,y € C 1(R) subject to the condition
x(T1) = a1, x(T2) = a2, xX(T1) = xX(T2) = ft2,
k(p)
x(t) = É 7fc x(t)ik y(t)jk for all t € R
, x(tk(p)) ak(p), , x(tk(p)) ftk(p),
fc=1
y (t) = h(x(t),t) for all t € R.
2.4. The following assertion can be proven for arbitrary p € N in the same way as the case p = 1 which is considered in [10, 11].
Theorem. If t, a € RK(p) meet tie condition
A(T,a) :=
a^ h(a1 ,T1 )j1 a^ h(a2 ,T2)j1
a12 h(a1,T1)j2 a22 h(a2,T2)j2
a1 h(a1 ,T1)jK(P) a2K(p) h(a2 ,T2)jK(p)
<(p) h(aK(p), tk(p) ) j1 aK2(p) h(aK(p), tk(p) ) j2 • • • aS h(aK(p), tk(p))
<p)
(p) (p)
)jK(p)
= 0,
then, given arbitrary ft € RK(p), the problem Q 1 o R is uniquely solvable for the data (t, a, ft), and its solution (y1 ,72,..., YK(p)) = (Q-1 oR)s(t, a, ft) can be calculated by Cramer's formulas
7k = A; = 1,2,..., /«(p),
A(т, a)
where Ak(t, a, ft) is the determinant of the matrix formed from the above matrix by replacing the kth column (a^ h(a1 ,T1)jfc, a2k h(a2 ,t2 )jk, ..., a^p h(aK(p) , TK(p))jfc) with tie column
ft = (ft1, ft2, . . . ,ftK(p))-
2.5. The following criterion clarifies the case in which there exist numbers t1, ..., tk(p) satisfying the hypothesis of Theorem 2.4.
Theorem. Let n € N, let T be an arbitrary set, and let : T ^ R, i = 1,...,n. The family of functions ip1,..., ipn is linearly independent in the vector space RT if and only if there are points t1,..., tn € T satisfying the condition
<Mt1) ^2 (t1) ••• ^n(t1)
^1(t2) ^2 (t2) ••• ^n(t2)
n ) ^2(t n)
^n (tn)
=0
(1)
< For convenience, introduce a notation for the matrix in (1) Mra(p1, - - -, ¿1, - - - ,tn) :=
( ^l(tl) ••
V ^l(in) ^(tn) ••
^n(tl) \ ^nfe)
^n (tn) /
The case n = 1 is trivial: if {^1} is linearly independent then < =0 and, hence, for some point t1 € T we have ^1(t1) = 0, i. e., |M1 (<; t1)| = 0.
Let n € N and assume that for every linearly independent family <1,---,<n: T — R there exist points t1, - - - ,tn € T satisfying (1). Now consider a linearly independent family - - -, <n, <n+1: T — R. By the induction hypothesis, there are points t1, - - -, tn € T such that the matrix
M := M„(<£1, - - - ,<n; ¿1, - - - ,i„) is invertible. We are to find a point t € T which ensures invertibilitv of the matrix
M(t) := Mn+1(<1, - - - ,<n,<n+1; ¿1, - - - ,tn,t)-
Assume to the contrary that |M(i)| = 0 for all t € T. Then, for each t € T, there is a tuple 0 = (a1(t), - - -, a«+1(t)) € Rn+1 satisfying the condition
M(t)(al(t), • • •, a«+l(t)) =0
or, which is the same,
<£l (tl) al (t) +-----+ ^n(tl) a«(t) + <£n+l(tl) a«+l (t) = 0,
<£l (t2) al (t) +-----+ ^n(t2) a«(t) + ^n+l(t2) a«+l (t) = 0,
(tn) a (t) +-----+ (tn) an(t) + (tn) an+1(t) = 0,
<1(t) a1(t) +-----h <n(t) an(t) + <n+1(t) an+1(t) =0-
The subsystem (2) is equivalent to the equality
M(a1(t), - - -, an(t)) + an+1(t)(<n+1 (t1), - - -, <n+1(tn)) = 0
which implies
(a1(t), - - -, an(t)) = -an+1(t) M-1 (<n+1(t1), - - -, <n+1 (tn)) -
(2)
(3)
(4)
Due to (4), in the case an+1(t) = 0 we would have a1(t) = ••• = an+1(t) = 0, which contradicts the condition (a1 (t), - - -, an+1(t)) = 0. Consequently, an+1(t) = 0 and
(■5)
, fjn := depend on t. It remains
According to (5), the numbers ¡3\ := .
to observe that (3) implies
(t) + ■ ■ ■ + ^n<n(t) + <n+1 (t) = 0 for all t € T
contrary to the linear independence of the family <1, - - -, <n, >
2.6. Theorems 2.4 and 2.5 directly imply the following condition for unique solvability of the "corrected inverse problem" Q-1 o R.
Theorem. Let x € C 1(R) h € C 1(R2). If the family of functions
t — x(t)ik h(x(t), t)jfc, k = 1, 2, - - - , k(p),
is linearly independent in the vector space Rr then there exist t1, - - - ,TK(p) € R sucA that, for all - - -, ^K(p) € R, the pr obi em Q-1 o R is uniquely solvable for the data T1, - - -, rK(p),
x(T1),---,x(TK(p)) ^1,---,^K(p)-
References
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Received July 3, 2018
Vladikavkaz Mathematical Journal 2018, Volume 20, Issue 3, P. 37-47
БИНАРНЫЕ СООТВЕТСТВИЯ И ОБРАТНАЯ ЗАДАЧА ХИМИЧЕСКОЙ КИНЕТИКИ
Гутман А. Е.1'2, Кононенко Л. И.1'2
1 Институт математики им. С. Л. Соболева СО РАН, Россия, 630090, Новосибирск, пр. Академика Коптюга, 4; 2 Новосибирский государственный университет, Россия, 630090, Новосибирск, ул. Пирогова, 1 E-mail: gutman@math.nsc.ru; larakon2@gmail.ru, larak@math.nsc.ru
Аннотация. Показано, как бинарные соответствия могут быть использованы для простой формализации понятия задачи, определения основных компонентов задач, их свойств и конструкций. В частности, предложена формализация следующих понятий: условие, данные, искомые и решения задачи, разрешимость и однозначная разрешимость, обратная задача, композиция и ограничение задач, изоморфизм между задачами. Рассмотрены топологические задачи и связанные с ними понятия устойчивости и корректности. Указана связь между устойчивостью и непрерывностью однозначно разрешимой топологической задачи. Дано определение параметризации множества. Введены понятия параметризованной задачи, задачи восстановления объекта по значениям параметров, а также понятия локально свободного набора параметров и устойчивости относительно набора параметров.
В качестве иллюстрации рассмотрена сингулярно возмущенная система обыкновенных дифференциальных уравнений, описывающая процесс химической кинетики и горения. Для такой системы сформулированы прямая и обратная задача. Изучаемый класс задач расширен за счет рассмотрения многочленов произвольной степени в качестве правых частей дифференциальных уравнений. Показано, как обратная задача химической кинетики может быть скорректирована и приближена к практике посредством композиции с простой вспомогательной задачей, реализующей связь между функциями и конечными наборами измеряемых числовых характеристик. Приведены формулы решения и указаны условия однозначной разрешимости скорректированной обратной задачи. В рамках исследования разрешимости получен критерий линейной независимости вещественных функций в терминах конечных наборов их значений. С помощью установленного критерия уточнена реализуемость условия однозначной разрешимости обратной задачи химической кинетики.
Ключевые слова: бинарное соответствие, обратная задача, разрешимость, композиция, устойчивость, корректность, дифференциальное уравнение, химическая кинетика, линейная независимость.
Mathematical Subject Classification (2000): 34А55.
