Abstract Generated abstract
The paper studies convergence of approximate solutions to a linear self-adjoint operator equation in a Hilbert space, with attention to ill-posed cases and perturbations of both the operator and the right-hand side. Under assumptions controlling approximation errors, components along null eigenvectors, and the growth of the inverse approximate operator, the authors decompose the approximate solution and derive an explicit error estimate in powers of the discretization parameter. This yields a convergence criterion relating the approximation exponents to the instability exponent, showing that stronger instability requires higher-order consistency conditions. A two-dimensional algebraic example illustrates the criterion, and the authors note that the same reasoning can be applied to evolution equations with unbounded operators.
Full Text
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
- A. N. Tikhonov, DAN, 151, No. 3 (1963).
- A. N. Tikhonov, DAN, 153, No. 1 (1963).
- M. M. Lavrent’ev, On the solution of certain ill-posed problems of mathematical physics, Novosibirsk, 1962.
- V. K. Ivanov, Matem. sborn., 61 (103), 2 (1963).
- L. A. Chudov, DAN, 143, No. 4 (1962).