Reports of the Academy of Sciences of the USSR
1962. Volume 145, No. 2

MATHEMATICS

V. K. IVANOV

ON LINEAR ILL-POSED PROBLEMS

(Presented by Academician S. L. Sobolev on 26 I 1962)

1. Let \(X\) be a linear metric space; \(Y\) a Banach space; \(A\) a linear continuous operator from \(X\) into \(Y\) such that \(A^{-1}\) exists but is unbounded. Many linear ill-posed problems of mathematical physics and function theory can be reduced to solving the equation

\[ Ax = y, \tag{1} \]

where \(y \in Y\) is the given element and \(x \in X\) the sought element. Because of the unboundedness of the operator \(A^{-1}\), a solution does not exist for all \(y\) and is unstable: arbitrarily small variations of \(y\) may correspond to arbitrarily large changes in \(x\).

As was shown by A. N. Tikhonov \((^{1})\), stability can be achieved if the solution is sought in a prescribed compact set \(M \subset X\). Put \(AM = N\). Then, according to a well-known topological theorem, the mapping \(M = A^{-1}N\) is continuous on \(N\). Therefore, if \(y \in N\) and, as \(y\) varies, we do not leave \(N\), then \(x\) will depend continuously on \(y\). The estimate of stability is determined by the modulus of continuity of the mapping \(A^{-1}\) on \(N\):

\[ \omega(\delta)=\sup \rho(x,x') \quad \text{for } x,x' \in N,\ \|Ax-Ax'\|\leq \delta. \tag{2} \]

If \(M\) is symmetric with respect to the zero of the space \(X\), then for sufficiently small \(\delta\) in (2) one may take \(x'=\theta\).

M. M. Lavrent’ev \((^{2})\) constructed an approximate method for solving equation (1), in which it is assumed that instead of the exact value \(y\) its approximation \(y_\delta\) is known with accuracy \(\delta\), and the function \(\omega(\delta)\), or its majorant, is known. A similar device is used in the work of F. John \((^{3})\).

Usually there are no effective criteria that make it possible to establish whether \(y\) belongs to \(N\); this has to be assumed known a priori (see \((^{1,2})\)). At the same time, in approximate solution, instead of \(y\) one operates with its approximate value \(y_\delta\), which may not lie in \(N\), so that \(A^{-1}y_\delta\) may not belong to \(M\) or may have no meaning.

In connection with this it is natural to change the formulation of the problem and, instead of the exact solution of equation (1), to seek a quasisolution (see below). For the quasisolution the classical well-posedness conditions are preserved (Theorem 1), and to find it one can indicate, by modifying known methods \((^{2,5})\), convergent processes. If, for the given \(y\), there exists a true solution in \(M\), then the quasisolution coincides with it; in other cases it gives the best approximation to the solution.

2. Definition. We shall call a quasisolution of equation (1) on a given compact set \(M\) of the space \(X\), and for a given \(y_0 \in N\), a point \(x_0 \in M\) for which \(\|Ax-y_0\|\) attains its minimum on \(M\).

In application to the Cauchy problem for the Laplace equation, the idea of best approximation is contained in the work of S. N. Mergelyan \((^{4})\).

Theorem 1. A quasi-solution of equation (1) exists for any nonempty compact set \(M \subset X\) and any \(y \in Y\). If \(M\) is convex and the sphere in the space \(Y\) is strictly convex, then the quasi-solution is unique and depends continuously on \(y\).

Proof. Existence follows from the compactness of \(M\). If \(M\) is convex, then \(AM\) is also convex; \(q = Ax\), where \(x\) is a quasi-solution, is the point \(N\) closest to \(y\) (the projection of \(y\) onto \(N\)).

Under the assumptions of the theorem, the projection of a point onto a convex set is uniquely determined; hence the uniqueness of the quasi-solution. Continuous dependence follows from uniqueness and compactness.

If, under the assumptions of Theorem 1, for a given \(y_0\) equation (1) has in \(M\) a true solution \(x_0\), and a sequence \(\{y_n\}\) of points of \(Y\) converges to \(y_0\), then the sequence of the corresponding quasi-solutions \(\{x_n\}\) converges to \(x_0\). Therefore, if \(y_\delta\) is an approximate value of \(y_0\), then for \(y_\delta \to y_0\), for the quasi-solution \(x_\delta\) we shall have \(x_\delta \to x_0\), independently of whether \(y_\delta\) belongs to \(N\).

If \(Y\) is Hilbert and \(M\) is convex, then from geometric considerations it follows that if \(y\) and \(y'\) are elements of \(Y\), and \(q\) and \(q'\) are their projections onto \(N\), then \(\|q-q'\| \leqslant \|y-y'\|\). Therefore, if in Hilbert \(Y\), \(\|y-y'\| \leqslant \delta\), and \(x\) and \(x'\) are the corresponding quasi-solutions, then \(\rho(x,x') \leqslant \omega(\delta)\).

In \((^5)\) an approximate method for solving equation (1) is constructed under the assumption that \(y \in N\). It can be shown that, in the case of linearity of \(A\), for arbitrary \(y\), the approximations constructed there converge to the quasi-solution. Thus \((^5)\) gives an approximate method for finding quasi-solutions.

1. More definite results can be obtained by specializing the spaces \(X\) and \(Y\). We shall assume that \(X\) and \(Y\) are Hilbert spaces, \(M=\Omega_R\) is the ball \(\|x\| \leqslant R\), and \(A\) is a completely continuous operator from \(X\) into \(Y\). Since the ball is weakly compact in a Hilbert space and this space is metrizable with respect to the weak topology, the results of Section 2 for the weak topology in \(X\) are applicable to the case under consideration.

Let \(A^*\) be the operator adjoint to \(A\). Then \(A^*A\) is a self-adjoint, positive, completely continuous operator from \(Y\) into \(Y\). Denote by \(\lambda_1 \geqslant \lambda_2 \geqslant \cdots \geqslant \lambda_n \geqslant \cdots\) the complete system of its eigenvalues, and by \(u_1, u_2, \ldots, u_n, \ldots\) the complete orthonormal system of its eigenvectors. Let

\[ A^*y=\sum_n \beta_n u_n . \tag{3} \]

Theorem 2. The quasi-solution of equation (1) on \(\Omega_R\) is expressed by the formula

\[ x=\sum_n \frac{\beta_n}{\lambda_n+\lambda}\,u_n, \tag{4} \]

where \(\lambda=0\), if

\[ \sum_n \frac{\beta_n^2}{\lambda_n^2}\leqslant R^2; \tag{5} \]

\(\lambda\) is the positive root of the equation

\[ \sum_n \frac{\beta_n^2}{(\lambda_n+\lambda)^2}=R^2, \]

if

\[ \sum_n \frac{\beta_n^2}{\lambda_n^2}>R^2. \tag{6} \]

Proof. The finding of quasi-solutions on \(\Omega_R\) reduces to finding in \(\Omega_R\) a vector \(x\) minimizing on \(\Omega_R\) the quadratic functional \((Ax-y, Ax-y)\). If (5) holds, the minimum is unconditional, and its finding reduces to solving the equation

\[ A^*Ax=A^*y. \tag{7} \]

The solution has the form (4) with \(\lambda=0\); it is the quasi-solution (the exact solution).

If (5) is not satisfied, then one must seek the minimum under the condition \((x,x)=R^2\).

Applying the method of Lagrange multipliers, we arrive at an equation of the second kind

\[ A^*Ax+\lambda x=A^*y, \tag{8} \]

whose solution gives (4) with \(\lambda>0\).

When \(y\) is given approximately, owing to errors, as a rule, (6) will hold. The presence of positive \(\lambda\) in the denominator ensures strong convergence of the series (4).

The method of Theorem 2 is a development of the device of M. M. Lavrent'ev (see (²)); however, unlike the latter, for the application of this method there is no need to know the function \(\omega(\delta)\) defined by (2). This makes it possible to restrict oneself to weak compactness of the set \(M\).

Ural State University
named after A. M. Gorky

Received
22 I 1962

References

¹ A. N. Tikhonov, DAN, 39, No. 35, 195 (1943).
² M. M. Lavrent'ev, DAN, 127, No. 1, 31 (1959).
³ F. John, Ann. di Mat. pura ed appl., 40, 129 (1955).
⁴ S. N. Mergelyan, UMN, 11, No. 5, 3 (1956).
⁵ V. K. Ivanov, DAN, 142, No. 5 (1962).