Full Text
UDC 517.948:518
MATHEMATICS
G. D. MAISTROVSKII
LOCAL RELAXATION THEORY FOR NONLINEAR EQUATIONS
(Presented by Academician L. V. Kantorovich on 9 I 1967)
Consider, in a real Hilbert space, the equation
\[ Fx=0 \tag{1} \]
with potential operator \(F\): \(Fx=\operatorname{grad}\varphi(x)\), where \(\varphi\) is some functional. To solve equation (1), relaxation methods are usually applied, leading to minimization of the functional \(\varphi\).
In the work of Yu. I. Lyubich \((^{1})\), a general theory of relaxation processes was developed for the case of a quadratic functional. The analogous theory in the general case must evidently be local. Its construction is the aim of the present note.
Suppose that, in a neighborhood of some solution of equation (1), the functional \(\varphi(x)\) has linear Gateaux differentials of the 1st and 2nd orders*. Then Taylor’s formula is valid:
\[
\varphi(x)=\varphi(a)+(Fa,x-a)+\frac12\bigl(F'(a+\xi(x-a))(x-a),x-a\bigr)
\]
\[
(0<\xi<1),
\tag{2}
\]
where \(F'(z)\) is the Gateaux derivative of the operator \(F\) at the point \(z\). Suppose further that the second differential \(D^2\varphi(x,v,v)=(F'(x)v,v)\) is a positive-definite bounded quadratic functional
\[ m\|v\|^2\le (F'(x)v,v)\le M\|v\|^2 . \]
A sequence \(\{x_k\}_0^\infty\subset H\) is called a relaxation sequence (r.s.) if \(\varphi(x_{k+1})\le \varphi(x_k)\) \((k=0,1,2,\ldots)\). We shall call an r.s. well localized (w.l.r.s.) if all its terms belong to that neighborhood \(S(x^0,r)=\{x:\|x-x^0\|\le r\}\), in which the above conditions are fulfilled. In particular, for \(r=\infty\), every r.s. is well localized.
Let \(\{x_k\}_0^\infty\) be a w.l.r.s. Define the vectors \(y_k\) by setting
\[ x_{k+1}=x_k-a_k y_k, \tag{3} \]
where \(a_k=\|x_{k+1}-x_k\|\). Next introduce the relaxation multipliers \(q_k\) and the relaxation angles \(\theta_k\) by the equalities** (cf. with \((^{1})\))
\[ q_k=2+2\Delta\varphi(x_k)/a_k(Fx_k,y_k),\qquad (Fx_k,y_k)=\|F(x_k)\|\cos\theta_k, \]
where \(\theta\varphi(x_k)=\varphi(x_{k+1})-\varphi(x_k)\). The relaxationality of the sequence \(\{x_k\}_0^\infty\) is equivalent to the inequality \(0\le q_k\le 2\).
It can be shown that the representation
\[ \Delta\varphi(x_k)=-q_k(2-q_k)\cos^2\theta_k\Delta_k h_k, \tag{4} \]
* Regarding differential calculus for functionals and operators, see, for example, \((^{2})\).
** We take \(q_k=1\) when \((Fx_k,y_k)=0\), and \(q_k=0\) when \(a_k=0,\ (Fx_k,y_k)\ne0\); the angle \(\theta_k\) is chosen in \([0,\pi/2]\), with \(\theta_k=0\) if \(F(x_k)=0\).
where \(\Delta_k=\varphi(x_k)-\varphi(x^0)\),
\[ h_k=\frac12\|F(x_k)\|^2\left(\Delta_k\bigl(F'(x_k-\alpha_k \xi_k y_k)y_k,y_k\bigr)\right)^{-1} \]
\((0<\zeta_k<1)\), and for the quantities \(h_k\) the estimates * hold
\[ h^{-1}\leqslant h_k\leqslant h, \tag{5} \]
where \(h=Mm^{-1}\) is the measure of conditioning. From formula (4) and inequality (5) there follow, exactly as in \((^1)\), the following results.
Theorem 1. For convergence of the l.r.p. to the solution \(x^0\) it is necessary and sufficient that the series
\[ \sum_{k=0}^{\infty} q_k(2-q_k)\cos^2\theta_k \]
diverge.
Theorem 2. Put
\[ \varkappa=\varlimsup_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\frac{\Delta_{k+1}}{\Delta_k},\qquad \sigma=\varliminf_{n\to\infty}\frac1n\sum_{k=0}^{n-1}q_k(2-q_k)\cos^2\theta_k . \]
For any l.r.p. the inequality holds
\[ 1-h\sigma\leqslant \varkappa\leqslant 1-h^{-1}\sigma . \tag{6} \]
An l.r.p. is called normally convergent if \(\varkappa<1\). From normal convergence there obviously follows convergence to the solution \(x^0\) with the rate of a geometric progression.
Theorem 3. An l.r.p. converges normally to the solution \(x^0\) if and only if \(\sigma>0\).
An l.r.p. is called quasigradient if \(\inf|\cos\theta_k|>0\), and is called strictly relaxation if \(\inf[q_k(2-q_k)]>0\). For quasigradient and strictly relaxation l.r.p., the formulations of Theorems 1–3 are naturally simplified (see \((^1)\)). In particular, the following holds.
Theorem 4. If an l.r.p. is simultaneously quasigradient and strictly relaxation, then it converges normally to the solution \(x^0\).
For illustration let us consider a class of processes of the form
\[ x_{n+1}=x_n-\beta_n A_nF(x_n), \tag{7} \]
where \(A_n=A_n^*>0\) are bounded operators in \(H\), \(\beta_n>0\). The convergence of these processes was studied in the recent paper of M. N. Yakovlev \((^4)\) and in a number of earlier works (see the bibliography in \((^4)\)). It is not difficult to show that, under the conditions imposed in \((^4)\) on the operators \(A_n\) and the numbers \(\beta_n\), and with a good localization, the processes (7) are quasigradient and strictly relaxation. Consequently, they converge normally to the solution with a rate guaranteed by estimate (6).
Let us now consider, in the \(n\)-dimensional Euclidean space \(H_n\), coordinate-relaxation processes defined by formula (3), where \(y_k=e_{i_k}\) (\(\{e_s\}_1^n\) is an orthonormal basis in \(H_n\)). S. Shekhter showed \((^5)\) that if, in the controlling sequence \(\{i_k\}_0^\infty\) of a well-localized strictly relaxation process, each index from \(\Omega^n=\{1,2,\ldots,n\}\) occurs arbitrarily far along (condition \(S\)), then \(\{x_k\}_0^\infty\) converges to \(x^0\). Yu. I. Lyubich, in \((^6)\), under condition \(S\) obtained an estimate of the rate of convergence for the case of a linear equation. We transfer this result below to the nonlinear equation (1).
Notation:
\[ \lambda(N)=\min\{l:\Omega^n\subset\{i_k\}_N^{N+l-1}\}; \]
\[ N_{p+1}=N_p+\lambda(N_p)\quad (p=0,1,2,\ldots;\quad N_0=0); \]
\[ \nu(k)=\min\{p:N_{p+1}>k\};\qquad \varepsilon_p=\min q_k(2-q_k)\quad (N_p\leqslant k<N_{p+1}). \]
* In deriving these estimates, inequalities between successive derivatives are used, analogous to the well-known Landau–Hadamard inequality (see, for example, \((^3)\)).
Theorem 5. For coordinate l.r.p., under condition \(S\), the inequality holds
\[ \Delta_k \leqslant \prod_{p=0}^{\nu(k)-1} (1-\omega\varepsilon_p) \qquad (k=0,1,2,\ldots), \]
where the coefficient \(\omega\) \((0<\omega\leqslant 1)\) depends only on the dimension \(n\) and on the condition measure \(h\), and moreover \(\omega(n,h)\) is a decreasing function of \(h\).
From Theorem 5 there follow, in a purely formal way, consequences analogous to those formulated in \((^6)\), in particular the above-mentioned result of S. Schechter.
The scheme considered does not formally cover relaxation processes for the Rayleigh functional \(\varphi(x)=(Ax,x)/(x,x)\). However, it can be shown that here too theorems analogous to our Theorems 1–5 are valid.
Assumptions and notation: \(A=A^*>0\) is a bounded operator in \(H\); \(\lambda_1=\inf_H \varphi(x)\) is an isolated point of the spectrum, \(U\) is the corresponding eigenspace;
\(G=H\ominus U\); \(\lambda_2=\inf_G \varphi(g)\); \(x^0\in U\), \(\|x^0\|=1\); \(\{x_k\}_0^\infty\) is an r.p., \(\|x_k\|=1\), \(\varphi(x^0)<(\lambda_1+\lambda_2)/2\) (the last condition ensures good localization).
Using the representation \(x=\beta u+\gamma g\) \((x\in H,\ u\in U,\ g\in G,\ \|u\|=\|g\|=1)\), introduce the operator \(P\): \(Px=|\beta|x^0+\gamma g\). Further set
\(Px_{k+1}-Px_k=-\alpha_k y_k\), where \(\alpha_k=\|Px_{k+1}-Px_k\|\). Let us now note that
\[ \operatorname{grad}\varphi(x)=\frac{2}{(x,x)}R(x)x \quad (R(x)=A-\varphi(x)E), \]
and put \(R_k=R(x_k)\). It can be shown that the equality
\[ \Delta\varphi(x_k)=-q_k(2-q_k)\cos^2\theta_k\Delta_k h_k, \tag{8} \]
holds, where \(q_k=\alpha_k(R_ky_k,y_k)/(R_kPx_k,y_k)\); \(\cos\theta_k=(\operatorname{grad}\varphi(x_k),y_k)/\|\operatorname{grad}\varphi(x_k)\|\); \(\Delta_k=\varphi(x_k)-\lambda_1\); \(h_k=\|R_kPx_k\|^2[(R_ky_k,y_k)\Delta_k]^{-1}\). The inequalities
\[ h^{-1}(x_k,A)\leq h_k\leq h(x_k,A) \qquad (k=1,2,\ldots), \tag{9} \]
hold, where \(h(x,A)\) is a certain condition measure, and as \(\varphi(x)\searrow \lambda_1\), \(h(x,A)\searrow(\|A\|-\lambda_1)(\lambda_2-\lambda_1)^{-1}\), the condition measure of the operator \(R(x^0)\big|_G\). Relations (8)–(9) make it possible to obtain, for the Rayleigh functional, Theorems 1–5 in the corresponding formulation.
In conclusion, the author takes this opportunity to express gratitude to Yu. I. Lyubich for posing the problem and for his constant attention.
Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Received
10 XII 1966
CITED LITERATURE
- Yu. I. Lyubich, DAN, 161, No. 6, 1274 (1965).
- M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Moscow, 1956.
- G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, IL, 1948.
- M. N. Yakovlev, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 84, 8 (1965).
- S. Schechter, Trans. Am. Math. Soc., 104, No. 1, 179 (1962).
- Yu. I. Lyubich, DAN, 173, No. 1, 37 (1967).