UDC 517.948.35

MATHEMATICS

A. V. GONCHARSKII, A. G. YAGOLA

ON THE UNIFORM APPROXIMATION OF A MONOTONE SOLUTION OF ILL-POSED PROBLEMS

(Presented by Academician A. N. Tikhonov, 6 VI 1968)

In the present paper we consider the question of solving the equation

\[ A[x,z(s)] = u(x), \qquad a \leq s \leq b;\ c \leq x \leq d;\ u(x) \in U;\ z(s) \in Z, \tag{1} \]

where \(u(x)\) is given, and \(z(s)\) is an unknown function, belonging respectively to the functional normed spaces \(U\) and \(Z\); \(A\) is a continuous operator.

Suppose that for every \(\bar u(x) \in U\) there exists, and moreover uniquely, a \(z(s) \in Z\) such that \(A[x,\bar z(s)] = \bar u(x)\). Suppose that the operator \(A^{-1}\) is not a bounded operator on \(U\), i.e., problem (1) is ill-posed \((^1)\). The availability of some additional information about the solution makes it possible in a number of cases to make the problem well-posed in the sense of Tikhonov \((^2)\). Such natural information for certain physical problems is the boundedness and monotonicity of the sought function \(z(s)\) \((^3)\).

1. Suppose an ill-posed problem (1) is given and it is known that the exact solution \(\bar z(s)\) of the problem, corresponding to \(\bar u(x)\), is a monotone (for definiteness, nonincreasing) function bounded by some constant \(B\):

Introduce the set of functions \(Z\downarrow\) such that \(z(s) \in Z\downarrow\) if: a) \(z(s)\) is a monotone nonincreasing function; b) \(|z(s)| \leq B;\ a \leq s \leq b\).

Theorem 1. The set \(Z\downarrow\) is compact in the metric of the space \(L_\alpha\).

Take an arbitrary sequence of functions \(z_1(s), z_2(s), \ldots, z_n(s), \ldots\). According to Helly’s selection theorem \((^4)\), there exist a sequence of indices \(n_0, n_1, \ldots, n_k, \ldots\) and a function \(\bar z(s) \in Z\downarrow\) such that

\[ \lim_{n_k\to\infty} z_{n_k}(s)=\bar z(s) \]

everywhere except for at most a countable number of discontinuity points of \(\bar z(s)\). From convergence almost everywhere and the uniform boundedness of the norms in \(L_\alpha\) there follows convergence in \(L_\alpha\) \((^5)\). The closedness of \(Z\downarrow\) in \(L_\alpha\) is evident.

2. Usually, when solving equation (1), not the exact value of the function \(\bar u(x)\) is known, but only some approximation \(\tilde u(x)\) to it (obtained, say, from an experiment) and an error \(\delta\) such that

\[ \|\tilde u(x)-\bar u(x)\|_U \leq \delta. \]

Consider the set \(U_\delta\) such that \(\tilde u_\delta(x) \in U_\delta\), if

\[ \|\tilde u_\delta(x)-\bar u(x)\|_U \leq \delta, \]

and introduce the set of functions \(\tilde z_\delta(s) \in Z_\delta\downarrow \in Z\downarrow\) such that

\[ \|A[x,\tilde z_\delta(s)]-\tilde u_\delta(x)\|_U \leq \delta^2, \qquad \tilde u_\delta(x) \in U_\delta. \tag{2} \]

Theorem 2. For any \(\varepsilon>0\) there exists a \(\delta_0(\varepsilon)\) such that, for any \(\tilde u_\delta(x)\in U_\delta\),

\[ \|\tilde z_\delta(s)-\bar z(s)\|_{L_2}\leq \varepsilon \]

for all \(z_\delta(s)\in Z_\delta\downarrow\), if \(\delta\leq \delta_0(\varepsilon)\).

Lemma 1. Let \(\tilde u_\delta(x)\in U_\delta,\ \tilde z_\delta(s)\in Z_\delta\downarrow\). Then

\[ \bigl\|A[x,\tilde z_\delta(s)]-A[x,\bar z(s)]\bigr\|_U^2 \leq 2\delta^2. \]

The proof is obvious:

\[ \left\| A[x,\tilde z_\delta(s)]-A[x,\bar z(s)] \right\|_U^2 \le \left\| A[x,\tilde z_\delta(s)]-\tilde u_\delta(x) \right\|_U^2 + \left\| \tilde u_\delta(x)-\bar u(x) \right\|_U^2 \le 2\delta^2 . \tag{3} \]

From Lemma 1 and the compactness of \(Z\downarrow\), Theorem 2 follows.

1. Let \(\bar z(s)\) be a piecewise smooth function. Take a sequence \(\tilde u_{\delta_k}(x)\), where \(\delta_k\to 0\). For each \(\delta_k\), choose some function \(\tilde z_{\delta_k}(s)\in Z_{\delta_k}\). In this case the sequence \(\tilde z_{\delta_k}(s)\) converges to \(\bar z(s)\) uniformly on each segment \([\gamma,\sigma]\subset [a,b]\) that contains no discontinuity points of \(\bar z(s)\) and no boundary points \(a,b\).

2. For the practical determination of a function \(\tilde z_\delta(s)\) such that

\[ \left\| A[x,\tilde z_\delta(s)]-\tilde u_\delta(x) \right\|_U \le \delta, \qquad \tilde u_\delta(x)\in U_\delta, \tag{4} \]

it is natural to pass to grid functions. Then the problem of finding a function \(\tilde z_\delta(s)\) satisfying relation (4) is the usual problem of quadratic programming \(\left({}^{6}\right)\). The only difference is that it is not necessary to seek the minimum of \(\|A[x,z_\delta(s)]-\tilde u_\delta(x)\|\), but it is sufficient to construct a minimizing sequence until a function satisfying relation (4) is found.

This method was used by the authors in solving certain problems of astronomy \(\left({}^{3}\right)\).

Solving problem (4), one can find a function \(\tilde z_\delta(s)\)—an approximation to the true solution \(\bar z(s)\)—such that \(\|A[x,\tilde z_\delta(s)]-\tilde u_\delta(x)\|_U\le \delta\). To estimate the error of this approximation, introduce into the set \(Z_\delta\downarrow\) the metric in which we are interested in estimating the approximation error. The approximation error will be obtained if we find

\[ \sup_{z\in Z_\delta\downarrow} \rho\bigl(z(s),\tilde z_\delta(s)\bigr). \tag{5} \]

It is easy to see that
\[ \rho(\tilde z_\delta(s);\bar z(s))\le \sup_{z\in Z_\delta\downarrow}\rho(z,\tilde z_\delta) \]
(\(\tilde z_\delta(s)\) is a certain fixed function belonging to \(Z_\delta\downarrow\); \(\rho(\tilde z_\delta,z)\) is the distance between \(z\) and \(\tilde z_\delta\) in the metric of interest to us).

For the practical determination of the error it is necessary to introduce grids in \(s\) and \(x\) and a finite-dimensional approximation of (5). Then this problem is a quadratic programming problem \(\left({}^{5}\right)\).

The error estimate by this method was carried out by the authors in solving certain astronomical problems \(\left({}^{3}\right)\).

1. Of great interest is the question of the rate of convergence of the approximations to the true solution.

Let problem (1) be a Fredholm integral equation of the first kind

\[ A[x,z(s)] = \int_a^b K(x,s)z(s)\,ds = u(x); \qquad a\le s\le b;\quad c\le x\le d, \tag{6} \]

with kernel

\[ K(x,s)= C\sum_{n=1}^{\infty}\frac{\sin ns\cdot \sin nx}{n^p} + D\sum_{n=1}^{\infty}\frac{\cos ns\cdot \cos nx}{n^p}. \]

Theorem. For any \(\tilde z_\delta(s)\in Z_\delta\downarrow\) that is an approximate solution of problem (6), the following estimate holds:

\[ \left\|\tilde z_\delta(s)-\bar z(s)\right\|_{L_2} = O\left(\delta^{1/2(p+1)}\right), \]

where \(O\) depends on \(C,D,p,B\); \(B\) is a constant bounding the solution.

In conclusion, the authors consider it their pleasant duty to express their gratitude to Acad. A. N. Tikhonov for posing the problem and supervising the work, to V. B. Glasko for supervising the work, and also to Sh. A. Alimov for numerous discussions.

Moscow State University
named after M. V. Lomonosov

Received
20 V 1968

REFERENCES

1. A. N. Tikhonov, DAN, 151, No. 3, 501 (1963).
2. M. M. Lavrent'ev, On Some Ill-Posed Problems of Mathematical Physics, Novosibirsk, 1962.
3. A. M. Cherepashchuk, A. V. Goncharskii, A. G. Yagola, Astronomical Journal, 45, No. 6 (1968).
4. É. Titmarsh, Expansions in Eigenfunctions Associated with Differential Equations of the Second Order, 2, Moscow, 1961.
5. M. A. Krasnosel'skii, P. P. Zabreiko et al., Integral Operators in Spaces of Summable Functions, Moscow, 1964.
6. S. I. Zukhovitskii, L. I. Avdeeva, Linear and Convex Programming, Moscow, 1964.