UDC 517.948 : 513.88 + 518 : 517.948

MATHEMATICS

Yu. E. Boyarintsev, V. G. Vasil’ev

ON THE THEORY OF CONVERGENCE OF APPROXIMATE SOLUTIONS OF AN OPERATOR EQUATION

(Presented by Academician M. A. Lavrent’ev on 9 VIII 1967)

Recently many works have appeared on the study of the convergence of approximate solutions of ill-posed problems \((^{1-5})\). In the present note essentially the same questions are considered, but under different initial assumptions.

Consider in a Hilbert space \(H\) the linear equation

\[ A\varphi = f \in H,\qquad A = A^* . \tag{1} \]

Let \(e_1, e_2, \ldots, e_n, \ldots\) be an orthonormal system of eigenvectors of the operator \(A\), with \(Ae_i = 0\), \((f,e_i)=0\) for \(i=1,2,\ldots,n\). We associate with equation (1) the equation

\[ A_h\varphi_h = f_h \in H,\qquad h>0, \tag{2} \]

where the linear operator \(A_h\) acts in the same Hilbert space \(H\),

\[ \|f-f_h\| = O(h^\alpha),\qquad \alpha>0, \tag{3} \]

and on the solutions of equation (1) corresponding to \(f \in H_1 \subset H\), the equalities

\[ \|(A-A_h)\varphi\| = O(h^\beta),\qquad \beta>0; \tag{4} \]

\[ (A_h\varphi,e_i)=O(h^\varepsilon),\qquad i=1,2,\ldots,n,\qquad \varepsilon>0. \tag{5} \]

hold. Suppose, moreover, that

\[ \|A_h^{-1}\| = O(h^{-\gamma}),\qquad \gamma\geq 0; \tag{6} \]

\[ (f_h,e_i)=O(h^{\varepsilon_1}),\qquad i=1,2,\ldots,n,\qquad \varepsilon_1>0. \tag{7} \]

We represent the solution \(\varphi_h\) of equation (2) in the form

\[ \varphi_h=\varphi_0+\varphi_1+\varphi_2. \tag{8} \]

Here \(\varphi_0\) is a solution of equation (1), and \(\varphi_1\) is a solution of the equation

\[ A\varphi_1=-f_1, \tag{9} \]

where \(f_1\) satisfies the equality

\[ A_h\varphi_0-f_h=f_1+\tilde f_1, \tag{10} \]

\[ \|\tilde f_1\|=O(h^\mu),\qquad \mu=\min(\varepsilon,\varepsilon_1). \]

From conditions (5), (7) it follows that the solution of equation (9) exists for \(h^{-\nu}f_1\in H_1\), and, by virtue of relations (3), (4),

\[ \|f_1\|=O(h^\nu),\qquad \nu=\min(\alpha,\beta) \tag{11} \]

and, consequently, for \(h^{-\nu}f_1\in H_1\)

\[ \|\varphi_1\|=O(h^\nu). \tag{12} \]

Substituting (8) into equation (2), by virtue of the equalities (4), (6), (10), (11), (12), we obtain

\[ \|A_h\varphi_2\|=\|f_h-A_h\varphi_0-A_h\varphi_1\| =\|-f_1-\tilde f_1-A_h\varphi_1+A\varphi_1-A\varphi_1\|\le \]
\[ \le \|\tilde f_1\|+\|(A-A_h)\varphi_1\|+\|(A\varphi_1+f_1)\| =O\!\left(h^{\min(\mu,\,2\nu)}\right), \tag{13} \]

\[ \|\varphi_2\|=O\!\left(h^{\min(\mu,\,2\nu)-\gamma}\right). \tag{14} \]

From (8), (12), (14) it follows that

\[ \|\varphi_h-\varphi_0\| =O\!\left(h^{\min[\nu,\,\min(\mu,\,2\nu)-\gamma]}\right). \tag{15} \]

Thus, we have proved the following

Theorem (convergence criterion). If \(f, h^{-\nu}f_1\in H_1\),

\[ \lim_{h\to 0}\|\varphi_0-\varphi_h\|=0, \]

if

\[ \mu=\min(\varepsilon,\varepsilon_1)>\min(\alpha,\beta)=\nu,\qquad \min(\mu,2\nu)>\gamma. \]

Remark. It follows from relation (15) that the larger the instability exponent \(\gamma\), the larger the approximation exponents \(\mu\) and \(\nu\) must be.

Example. As an example, let us consider the system of two linear algebraic equations

\[ A\varphi=f=\binom{1}{1},\qquad A=\begin{pmatrix}1&1\\[2pt]1&1\end{pmatrix}. \tag{16} \]

It is obvious that the eigenvectors of the matrix of the system of equations (16) are

\[ y_1=\frac1{\sqrt2}\binom{1}{1},\qquad y_2=\frac1{\sqrt2}\binom{1}{-1}. \tag{17} \]

We approximate equation (16) by the equation

\[ A_h\varphi_h=f_h=\binom{1}{1},\qquad A_h=\begin{pmatrix}1+h+h^2&1\\[2pt]1&1+h\end{pmatrix}. \tag{18} \]

As a solution of the system (16) we take

\[ \varphi_0=\binom{1/2}{1/2}. \tag{19} \]

If \(H_1\) is the set of vectors of the form

\[ x=\binom{a}{a},\qquad |a|<a_0=\mathrm{const}<\infty, \tag{20} \]

then \(f,h^{-1}f_1\in H_1\) and \(\mu=2\), \(\nu=1\), \(\gamma=1\), \(\min(\mu,2\nu)=2\). Consequently, according to the theorem, the solution of the system of equations (18) converges as \(h\to 0\) to the solution (19) of equation (16).

The scheme of reasoning presented above is readily applied to the proof of convergence theorems also for evolution equations with unbounded operators.

The authors express their gratitude to M. M. Lavrent’ev for his attention to this work.

Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
24 VII 1967

REFERENCES

1. A. N. Tikhonov, DAN, 151, No. 3 (1963).
2. A. N. Tikhonov, DAN, 153, No. 1 (1963).
3. M. M. Lavrent’ev, On the solution of certain ill-posed problems of mathematical physics, Novosibirsk, 1962.
4. V. K. Ivanov, Matem. sborn., 61 (103), 2 (1963).
5. L. A. Chudov, DAN, 143, No. 4 (1962).