UDC 518:517.948

MATHEMATICS

V. A. MOROZOV

ON A STABLE METHOD FOR COMPUTING THE VALUES OF UNBOUNDED OPERATORS

(Presented by Academician A. N. Tikhonov, 10 VII 1968)

1. The stable computation of the values of unbounded operators is one of the most important problems of computational mathematics. Let \(A\) be an operator with domain of definition \(D_A \subset F\) and range \(Q_A \subset U\), where \(F\) and \(U\) are certain linear normed spaces and \(\|A\|=+\infty\). Then there certainly exists a sequence of elements \(f_n \in F\), \(\|f_n\|_F=1\), such that \(\|Af_n\|_U \to +\infty\). Let \(\bar f \in D_A\) and \(\bar u=A\bar f\). Put \(f_{n,\delta}=\bar f+\delta f_n\), where \(\delta>0\) is any arbitrarily small number. Then, obviously, \(\|u_{n,\delta}-\bar u\|_U \to +\infty\) as \(n \to +\infty\), where \(u_{n,\delta}=A(\bar f+\delta f_n)\), whereas \(\|f_{n,\delta}-\bar f\|_F=\delta\).

Thus, the problem of computing the values of the operator \(A\) in the case under consideration is ill-posed. If one has arbitrary \(\delta\)-approximations to the element \(\bar f\), i.e., elements \(f_\delta \in F\), \(\|f_\delta-\bar f\|_F \le \delta\), then it may also happen that the values of the operator \(A\) are not even defined on the elements \(f_\delta\), i.e., \(f_\delta \notin D_A\).

The problem considered below consists in the effective construction of elements \(\hat f_\delta \in F\) from the given \(\delta\)-approximations to the element \(\bar f\), satisfying the following two basic requirements:

1) \(\hat f_\delta \in D_A\);

2) \(\displaystyle \lim_{\delta\to 0}\|\hat u_\delta-\bar u\|_U=0\), where \(\hat u_\delta=A\hat f_\delta\).

2. In what follows we shall assume throughout that \(U=H\) and \(F\) are Hilbert spaces, and that the operator \(A\) is linear and closed \((^2)\), i.e., from the simultaneous fulfillment of the relations

\[ f_\nu \in D_A,\qquad \lim_{\nu\to\infty} f_\nu=f,\qquad \lim_{\nu\to\infty} Af_\nu=u \]

it follows that

\[ f\in D_A,\qquad u=Af. \]

Consider the following auxiliary parametric variational problem \((\alpha>0\) is a parameter) \((^1)\):

\[ \Phi_\alpha[f]=\|f-g\|_F^2+\alpha\|Af\|_H^2,\qquad f\in D_A \ — \min . \tag{1} \]

Theorem 1. For any given element \(g\in F\), problem (1) has a unique solution \(f_\alpha \in D_A\).

Proof. For any \(f_1\) and \(f_2 \in D_A\) we have

\[ \left\|\frac{f_1-f_2}{2}\right\|_F^2 +\alpha\left\|\frac{A(f_1-f_2)}{2}\right\|_H^2 = {}^1/_2\,\Phi_\alpha[f_1]+{}^1/_2\,\Phi_\alpha[f_2] -\Phi_\alpha\left[\frac{f_1-f_2}{2}\right]. \tag{2} \]

Let

\[ m_\alpha=\inf_{f\in D_A}\Phi_\alpha[f]. \]

and \(f_\nu \in D_A\) is a minimizing sequence for the functional (1):

\[ \lim_{\nu\to\infty}\Phi_\alpha[f_\nu]=m_\alpha . \tag{3} \]

Putting \(f_1=f_\nu,\ f_2=f_{\nu+p}\), where \(p\) is an arbitrary natural number, we obtain, by virtue of (3),

\[ \left\|\frac{f_\nu-f_{\nu+p}}{2}\right\|_F^2 +\alpha\left\|\frac{Af_\nu-Af_{\nu+p}}{2}\right\|_H^2 = \frac12\Phi_\alpha[f_\nu]+\frac12\Phi_\alpha[f_{\nu+p}] \]

\[ -\Phi_\alpha\left[\frac{f_\nu+f_{\nu+p}}{2}\right] \le \frac12\Phi_\alpha[f_\nu]+\frac12\Phi_\alpha[f_{\nu+p}]-m_\alpha \to 0, \]

as \(\nu\to\infty\), independently of \(p\). But then

\[ \lim_{\nu\to\infty}\|f_\nu-f_{\nu+p}\|_F=0, \qquad \lim_{\nu\to\infty}\|Af_\nu-Af_{\nu+p}\|_H=0, \]

i.e., the sequences \(f_\nu\) and \(Af_\nu\) are fundamental. Let

\[ \lim_{\nu\to\infty} f_\nu=f_\alpha,\qquad \lim_{\nu\to\infty} Af_\nu=u_\alpha . \tag{4} \]

Then, by the closedness of the operator \(A\), we have

\[ f_\alpha\in D_A,\qquad Af_\alpha=u_\alpha . \tag{5} \]

From (3), (4), and (5) it follows that

\[ m_\alpha=\Phi_\alpha[f_\alpha], \]

i.e., \(f_\alpha\) is a solution of problem (1). If \(\hat f_\alpha\) is some other solution, then, again using (2), we obtain

\[ \left\|\frac{\hat f_\alpha-f_\alpha}{2}\right\|_F^2 +\alpha\left\|\frac{A\hat f_\alpha-Af_\alpha}{2}\right\|_H^2 = m_\alpha-\Phi_\alpha\left[\frac{\hat f_\alpha+f_\alpha}{2}\right]\le 0, \]

whence it follows that \(\hat f_\alpha=f_\alpha\). The theorem is proved.

Corollary. From the theorem just proved it follows that an operator \(R_\alpha\) with values in \(D_A\) is defined on \(F\):

\[ f_\alpha=R_\alpha g,\qquad g\in F. \]

Let us prove that the operator \(R_\alpha\) is bounded: \(\|R_\alpha\|\le 2\). Indeed,

\[ \|f_\alpha-g\|_F^2+\alpha\|Af_\alpha\|_H^2 \le \|f-g\|_F^2+\alpha\|Af\|_H^2,\qquad f\in D_A . \]

Putting here \(f=0\), we have

\[ \|R_\alpha g\|_F=\|f_\alpha\| \le \|f_\alpha-g\|_F+\|g\|_F \le 2\|g\|_F, \]

which proves our assertion. It can also be shown that the operator \(R_\alpha\) is linear.

Next put \(f=f_\delta\) in (1) and \(f_\alpha^\delta=R_\alpha f_\delta\). The following auxiliary assertion is valid:

Lemma. Let \(\delta<\|\operatorname{pr}_{N^\perp}f_\delta\|_F\), where \(N^\perp\) is the orthogonal complement of the subspace \(N\) of solutions of the homogeneous equation \(Af=0\). Then there exists a unique value of the parameter \(\alpha>0\) such that \(\rho(\alpha)=\delta^2\), where \(\rho(\alpha)=\|f_\alpha^\delta-f_\delta\|_F^2\).

It can be shown, as in (3), that the values of the function \(\rho(\alpha)\) exhaust the interval \([\delta^2,\|\operatorname{pr}_{N^\perp}f_\delta\|_F^2]\), \(0<\alpha\le+\infty\), and \(\rho(\alpha)\) increases strictly monotonically. Hence the assertion follows.

Theorem 2. Let \(\hat u_\delta=A\hat f_\delta\), where \(\hat f_\delta=f_\alpha^\delta\). Then

\[ \lim_{\delta\to 0}\|\hat u_\delta-\bar u\|_H=0 . \tag{6} \]

i.e., the values of the operator \(A\) on the elements \(\hat f_\delta\) constructed above approximate the sought value of the operator \(A\bar f=\bar u\).

Proof. We have

\[ \|f_\alpha^\delta-f_\delta\|_F^2+\alpha\|Af_\alpha^\delta\|_H^2 \leq \|f-f_\delta\|_F^2+\alpha\|Af\|_H^2,\qquad f\in D_A. \]

Putting here \(\alpha=\hat\alpha\), \(f=\bar f\), we obtain

\[ \|\hat f_\delta-f_\delta\|_F^2+\hat\alpha\|\hat u_\delta\|_H^2 =\delta^2+\hat\alpha\|\hat u_\delta\|_H^2 \leq \|\bar f-f_\delta\|_F^2+\hat\alpha\|\bar u\|_H^2 < \delta^2+\hat\alpha\|\bar u\|_H^2, \]

whence it follows that

\[ \|\hat u_\delta\|_H\leq \|\bar u\|_H,\qquad \|\hat f_\delta-\bar f\|_F\leq 2\delta, \tag{7} \]

and, consequently, the family \(\hat u_\delta=A\hat f_\delta\) is weakly compact. Let \(u_0\) be some weak limit point of the family \(\{\hat u_\delta\}\) and let the subfamily \(\{u_{\delta'}\}\subseteq\{\hat u_\delta\}\) be such that, for any \(u\in H\),

\[ \lim_{\delta'\to 0}(u_{\delta'},u)_H=(u_0,u). \]

Then we have

\[ \hat f_{\delta'}\to \bar f,\qquad \hat u_{\delta'}=A\hat f_{\delta'}\xrightarrow{\mathrm{w}}u_0. \]

Using the fact of weak closedness of the closed operator \(A\), we obtain

\[ A\bar f=u_0. \]

On the other hand, \(A\bar f=\bar u\), i.e. the family \(\{\hat u_\delta\}\) has a unique weak limit point, which is the element \(\bar u\). Consequently, the element \(\bar u\) is weakly limiting for the whole family \(\{\hat u_\delta\}\). Using the known properties of weakly convergent sequences and the first of inequalities (7), we obtain

\[ \|\bar u\|_H\leq \varliminf_{\delta\to 0}\|\hat u_\delta\|_H \leq \varlimsup_{\delta\to 0}\|\hat u_\delta\|_H \leq \|\bar u\|_H, \]

whence it follows that

\[ \lim_{\delta\to 0}\|\hat u_\delta\|_H=\|\bar u\|_H. \]

Together with the weak convergence of the family \(\{\hat u_\delta\}\) to \(\bar u\), the last relation proves the validity of (6). The theorem is proved.

Theorem 2 completely solves the problem of effective computation of the values of the operator \(A\); here the required algorithm can be written in the form

\[ \hat f_\delta=R_{\hat\alpha}f_\delta,\qquad \hat u_\delta=A\hat f_\delta. \]

1. As an example of the use of the proposed method, consider the problem of numerical differentiation.

Let \(H=F=L_2[a,b]\), and let the operator \(Af=d^n f(x)/dx^n\) be the operator of generalized differentiation in the sense of Sobolev. If it is known that \(\bar f(x)\in D_A\) and \(\delta\)-approximations to \(\bar f(x)\) in \(L_2[a,b]\) are given:

\[ \|\bar f-f_\delta\|_{L_2} = \left\{\int_a^b[\bar f(x)-f_\delta(x)]^2\,dx\right\}^{1/2} <\delta, \]

then problem (1) reduces to the necessity of solving the Euler equation:

\[ \alpha\,d^{2n}f/dx^{2n}+f=f_\delta,\qquad d^{n+i}f(x)/dx^{n+i}\big|_{x=a,x=b}=0,\qquad i=0,1,\ldots,n-1. \tag{8} \]

If the values of the function or of its derivatives at the points \(a\) or \(b\) are known a priori, then it is necessary to solve problem (8) with the corresponding boundary conditions. The value of the parameter \(\alpha\) is found from the equation

\[ \int_a^b [f_\alpha^\delta(x)-f_\delta(x)]^2 dx=\delta^2, \]

after which, as an approximation to \(\bar u=d^n\bar f/dx^n\), one may take the function

\[ \hat u_\delta=d^n f_\alpha^\delta(x)/dx^n. \]

Numerical calculations carried out in application to the problem of differentiation have shown the sufficient effectiveness and simplicity of the proposed method.

Received
10 VII 1968

REFERENCES

\(^{1}\) A. N. Tikhonov, DAN, 153, No. 1 (1963). \(^{2}\) N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, “Nauka,” 1966. \(^{3}\) V. A. Morozov, Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki, 8, No. 2 (1968).