Abstract
Full Text
S. G. Mikhlin
On the Stability of Certain Computational Processes
(Presented by Academician V. I. Smirnov on 15 II 1964)
Let \(X_n\) and \(Y_n\), \(n=1,2,\ldots\), be Banach spaces and let \(A_n\) be linear operators acting from \(X_n\) into \(Y_n\), where for every \(n\) the operator \(A_n^{-1}\) exists and is defined on the entire space \(Y_n\). Consider the computational process consisting in solving the sequence of equations
\[ A_n x^{(n)} = y^{(n)} . \tag{1} \]
Alongside equations (1), consider the sequence of equations
\[ (A_n+\Gamma_n) z^{(n)} = y^{(n)}+\delta^{(n)}, \tag{2} \]
where the operators \(\Gamma_n\) are also linear.
We shall say that the computational process (1) is stable (with respect to changes in the operator \(A_n\) and the free term \(y^{(n)}\)) if there exist constants \(p,q,r\), independent of \(n\), such that for \(\|\Gamma_n\|\le r\) and for arbitrary \(\delta^{(n)}\), equations (2) are solvable and the inequality holds
\(\eta^{(n)}=z^{(n)}-x^{(n)}\)
\[ \|\eta^{(n)}\|\le p\|\Gamma_n\|+q\|\delta^{(n)}\|. \tag{3} \]
Theorem 1. For the stability of the computational process (1), it is necessary and sufficient that, independently of \(n\), the norms of the operators \(A_n^{-1}\) and the elements \(A_n^{-1}B_nA_n^{-1}y^{(n)}=A_n^{-1}B_nx^{(n)}\) be bounded, where \(B_n\) is an arbitrary operator of unit norm acting from \(X_n\) into \(Y_n\).
Necessity. 1) Let \(\Gamma_n=0\). Then \(A_n\eta^{(n)}=\delta^{(n)}\) and
\(\|\eta^{(n)}\|\le q\|\delta^{(n)}\|\). Hence \(\|A_n^{-1}\|\le q\).
2) Let now \(\delta^{(n)}=0\) and \(\Gamma_n=\varepsilon B_n\), where \(\|B_n\|=1\) and \(\varepsilon\) is a constant, \(0<\varepsilon<r\). In this case \(\eta^{(n)}\) satisfies the equation
\[ \eta^{(n)}+\varepsilon A_n^{-1}B_n\eta^{(n)} = -\,\varepsilon A_n^{-1}B_nx^{(n)} . \]
Hence
\[ \|\eta^{(n)}\|\ge \frac{\varepsilon\|A_n^{-1}B_nx^{(n)}\|}{1+\varepsilon q}. \]
Comparing this with inequality (3), which in the present case has the form
\(\|\eta^{(n)}\|\le p\varepsilon\), we find that
\(\|A_n^{-1}B_nx^{(n)}\|\le p(1+\varepsilon q)\). Since \(\varepsilon\) is arbitrarily small, \(\|A_n^{-1}B_nx^{(n)}\|\le p\).
Sufficiency. Let \(\|A_n^{-1}\|\le C_1\), \(\|A_n^{-1}B_nx^{(n)}\|\le C_2\), where the constants \(C_1\) and \(C_2\) do not depend on \(n\). Put \(r=\beta C_1^{-1}\), where \(\beta\) is a constant, \(0<\beta<1\), and require that \(\|\Gamma_n\|\le r\). Form the equations
\[ (A_n+\Gamma_n) z_1^{(n)} = y^{(n)}, \qquad (A_n+\Gamma_n) z_2^{(n)} = \delta^{(n)}; \]
then \(\eta^{(n)}=(z_1^{(n)}-x^{(n)})+z_2^{(n)}\). It is easy to see that
\[ \|z_2^{(n)}\|\le \|(I_n+A_n^{-1}\Gamma_n)^{-1}\|\,\|\delta^{(n)}\|C_1 \le \frac{C_1}{1-\beta}\,\|\delta^{(n)}\|; \]
here \(I_n\) is the identity operator.
Further, putting \(\Gamma_n=\|\Gamma_n\|B_n,\ \|B_n\|=1\), we have
\[ z_1^{(n)}-x^{(n)}=-\|\Gamma_n\|(I_n+A_n^{-1}\Gamma_n)^{-1}A_n^{-1}B_nx^{(n)} \]
and, consequently,
\[ \|z_1^{(n)}-x^{(n)}\|\leq \frac{\|\Gamma_n\|}{1-\beta}\|A_n^{-1}B_nx^{(n)}\| \frac{C_2}{1-\beta}\|\Gamma_n\|; \]
inequality (3) is thus satisfied for the following values of the constants
\[ p=\frac{C_2}{1-\beta},\qquad q=\frac{C_1}{1-\beta},\qquad r=\frac{\beta}{C_1}. \]
If the quantity \(\|x^{(n)}\|=\|A_n^{-1}y^{(n)}\|\) is bounded, then for the stability of the computational process (1) it is necessary and sufficient that the norms \(\|A_n^{-1}\|\) be bounded. The quantity \(\|x^{(n)}\|\) will be bounded if, for example, \(X_n\), \(n=1,2,\ldots\), are subspaces of some Banach space \(X\), and in this space \(\lim x^{(n)}\) exists.
Theorem 2. If the spaces \(X_n\) are Hilbert spaces and the norms \(\|A_n^{-1}\|\) are bounded below by a positive constant independently of \(n\), then for the stability of process (1) it is necessary and sufficient that the quantities \(\|A_n^{-1}\|\) and \(\|x^{(n)}\|=\|A_n^{-1}y^{(n)}\|\) be bounded above independently of \(n\).
Theorem 2 will be proved if we establish that from the conditions
\[ \|A_n^{-1}B_nx^{(n)}\|\leq C_2,\qquad \|A_n^{-1}\|\geq C_3, \]
where \(C_2,C_3\) are positive constants, there follows the boundedness of the norms \(\|x^{(n)}\|\). Suppose the contrary, and let \(\|x^{(n_k)}\|\to\infty\). Choose an element \(t^{(n_k)}\in Y_{n_k}\) so that \(\|t^{(n_k)}\|=1\) and \(\|A_{n_k}^{-1}t^{(n_k)}\|\geq \frac12\|A_{n_k}^{-1}\|\). Construct an operator \(B_{n_k}\), \(\|B_{n_k}\|=1\), which, on elements of the form \(\lambda x^{(n_k)}\), acts according to the formula
\[ B_{n_k}\bigl(\lambda x^{(n_k)}\bigr)=\lambda\|x^{(n_k)}\|t^{(n_k)}, \]
and which annihilates the elements orthogonal to \(x^{(n_k)}\). Then
\[ \|A_{n_k}^{-1}B_{n_k}x^{(n_k)}\|\geq \frac12\|A_{n_k}^{-1}\|\,\|x^{(n_k)}\|\to\infty, \]
contrary to the condition.
Let us give several examples.*
Example 1. Let \(A\) be a positive definite operator acting in a Hilbert space \(H\), and suppose the problem of minimizing the functional
\[ F(u)=\|u\|_A^2-(u,f)-(f,u) \tag{4} \]
is solved by the Ritz process. Let \(\varphi_k,\ k=1,2,\ldots\), be coordinate elements subject to the usual conditions ([1]), and
\[ u_n=\sum_{k=1}^{n}a_k^{(n)}\varphi_k, \tag{5} \]
the Ritz approximate solution of the above variational problem. The vector \(x^{(n)}=(a_1^{(n)},a_2^{(n)},\ldots,a_n^{(n)})'\) is determined from the equation
\[ R_nx^{(n)}=y^{(n)}, \tag{6} \]
where \(R_n\) is the matrix \(\|[\varphi_k,\varphi_j]_A\|_{j,k=1}^{j,k=n}\); \(y^{(n)}=(f_1,f_2,\ldots,f_n)'\); \(f_k=(f,\varphi_k)\).
* On the terminology and notation, see [1].
Equation (6) falls under type (1) if one sets \(X_n = Y_n, E_n\), where \(E_n\) is an \(n\)-dimensional unitary space, and \(A_n = R_n\). It is not difficult to prove that \(\|R_n^{-1}\| \geq |\varphi_1|_A^2\), and then it follows from Theorem 2 that the process of computing the Ritz coefficients from equations (6) is stable if and only if the coordinate system is strongly minimal \({}^{(2)}\) in the space \(H_A\), i.e., if \(\lambda_1^{(n)} \geq \lambda_0 = \mathrm{const} > 0\), where \(\lambda_1^{(n)}\) is the smallest eigenvalue of the matrix \(R_n\); the sufficiency of this condition was proved in \({}^{(3)}\).
Example 2. Let \(Y_n = E_n\), and let \(X_n\) be the subspace of the space \(H_A\) that is the linear span of the elements \(\varphi_1, \varphi_2, \ldots, \varphi_n\). The approximate solution \(u_n\) of the variational problem (4) is determined from the equation
\[ (S_n^*)^{-1}u_n = y^{(n)}, \tag{7} \]
where \(S_n\) is the operator which assigns to the element \(u_n\) (formula (5)) the vector \(x^{(n)}\). Since \(u_n\) tends in the metric of \(H_A\) to the limit \((1)\), the norms \(|u_n|_A\) are bounded in the aggregate, and for the stability of the process (7) it is necessary and sufficient that the quantities \(\|S_n\|\) be bounded in the aggregate. The relation \((S_n^*)^{-1}S_n^{-1} = R_n\) holds; hence \(\|S_n\| = [\lambda_1^{(n)}]^{-1/2}\), and the necessary and sufficient condition for stability of the process (7) reduces to the strong minimality of the coordinate system in the space \(H_A\). By definition, the inequality characterizing the stability of the named process has the form
\[ |\eta_n|_A \leq p\|\Gamma_n\| + q\|\delta^{(n)}\|, \]
where \(\Gamma_n\) and \(\delta^{(n)}\) may be regarded as the errors of the operator \((S_n^*)^{-1}\) and of the vector \(y^{(n)}\), respectively.
Suppose that these errors arise only as a result of inexact computation of the scalar products \((f,\varphi_k)\) and the energy products \([\varphi_k,\varphi_j]_A\). Then the operator \(S_n^{-1}\) is free of error, as is seen from the formula
\[ S_n^{-1}x^{(n)} = u_n = \sum_{k=1}^{n} a_k^{(n)}\varphi_k. \]
Denoting by \(\overline{\Gamma}_n\) the error of the matrix \(R_n\), we have \(\Gamma_n = \overline{\Gamma}_n S_n\). Hence \(\|\Gamma_n\| \leq \|\overline{\Gamma}_n\|[\lambda_1^{(n)}]^{-1/2}\). If the coordinate system is strongly minimal in \(H_A\), then, putting \(p\lambda_0^{-1/2}=p_1,\ q=q_1\), we obtain the inequality established in \({}^{(4)}\)
\[ |\eta_n|_A \leq p_1\|\overline{\Gamma}_n\| + q_1\|\delta^{(n)}\|. \]
- Let the operator \(A\) be defined as above, and let the operator \(K\) be such that the product \(A^{-1}K\) can be extended to an operator that is completely continuous in the space \(H_A\). From Theorem 1 follow the theorems of the article \({}^{(4)}\), by virtue of which the processes of determining the vector of coefficients and the approximate solution in the Bubnov–Galerkin method for the equation
\[ (A+K)u=f \]
are stable if the coordinate system is strongly minimal in the space \(H_A\).
Received
13 II 1964
REFERENCES CITED
\({}^{1}\) S. G. Mikhlin, Variational Methods in Mathematical Physics, Moscow, 1957.
\({}^{2}\) S. G. Mikhlin, DAN, 135, No. 1 (1960).
\({}^{3}\) A. T. Taldykin, Matem. sbornik, 29 (71), 1 (1951).
\({}^{4}\) G. N. Yakovleva, M. N. Yakovlev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66, 182 (1962).