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UDC 517.948.32
MATHEMATICS
V. N. STRAKHOV
ON THE NUMERICAL SOLUTION OF ILL-POSED PROBLEMS REPRESENTED BY INTEGRAL EQUATIONS OF CONVOLUTION TYPE
(Presented by Academician A. N. Tikhonov on 13 III 1967)
Many problems of exploration geophysics \(\left({}^{1,2}\right.\) and others) lead to linear integral equations of convolution type
\[ f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\varphi(\xi)K(x-\xi)\,d\xi,\qquad -\infty \leq x \leq +\infty, \tag{1} \]
whose kernels satisfy the conditions:
\[ \begin{aligned} &1)\quad |k(t)|\leq 1,\qquad -\infty \leq t \leq +\infty;\\ &2)\quad k(t)\ne 0\ \text{for almost all } t,\ -\infty<t<+\infty. \end{aligned} \tag{2} \]
Here
\[ k(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}K(x)e^{-itx}\,dx \tag{3} \]
is the spectrum (Fourier transform) of the kernel \(K(x)\).
We write equation (1) in operator form:
\[ f=A\varphi;\qquad \|A\|=\sup_t |k(t)|=1, \tag{4} \]
where \(A\) is a linear integral operator acting in the space \(L^2\), from the set \(P\), singled out by condition 2); \(f\) and \(\varphi\) are functions from the space \(L^2\). Obviously, there exists an inverse operator \(A^{-1}\) to \(A\), but
\[ \|A^{-1}\|=\sup_t |1/k(t)|=+\infty, \tag{5} \]
so that the solutions of equation (1) possess instability: small (in the norm in \(L^2\)) variations \(\delta f\) may correspond to large variations \(\delta\varphi\).
A. N. Tikhonov \(\left({}^{3,4}\right)\) proposed a method for the approximate solution of problems of the type (4)—(5)—the regularization method. It consists in introducing, instead of equation (4), a linear operator \(T_\alpha\) in the space \(L^2\), depending on a parameter \(\alpha\), \(0<\alpha<\alpha_0\), and satisfying the conditions:
\[ \begin{aligned} &1)\quad \|T_\alpha\|<+\infty,\qquad 0<\alpha<\alpha_0;\\ &2)\quad \|\varphi-T_\alpha A\varphi\|\xrightarrow[\alpha\to 0]{}0\quad (\text{for any } \varphi\in L^2). \end{aligned} \tag{6} \]
\[ \tag{7} \]
The function \(T_\alpha f\), for some \(\alpha=\alpha^*\), is taken as an approximate solution of equation (4). V. K. Ivanov \(\left({}^{5}\right)\) introduced the notion of uniform regularization. Namely, the operator \(T_\alpha\) uniformly regularizes equation (4) on some set \(m\subseteq L^2\), if, as \(\alpha\to 0\), \(\|\varphi-T_\alpha A\varphi\|\to 0\) uniformly for all \(\varphi\in m\).
If an equation \(f_\delta=A\varphi\) is given, where \(f_\delta=f+\delta f\), \(\|\delta f\|\leq \delta\), and moreover \(f\in M\), \(\delta f\in M\), then \(f_\delta\in M\), and a solution of the equation does not exist for any \(\delta>0\). However, an approximate solution can always be constructed, pro-
than
\[ \|\varphi-T_{\alpha}f_{\delta}\|\leq \|\varphi-T_{\alpha}A\varphi\|+\|T_{\alpha}\|\delta, \tag{8} \]
and by an appropriate choice of \(\alpha\) one can achieve sufficiently good results.
Definition 1. The value of the parameter \(\alpha\) determined from the condition
\[ \sup_{\varphi\in m}\|\varphi-T_{\alpha}A\varphi\|+\|T_{\alpha}\|\delta=\min \tag{9} \]
is called optimal on the set \(m\) of uniform regularization for a given \(\delta>0\).
Accordingly, \(T_{\alpha}f_{\delta}\), where \(\alpha\) is chosen according to (9), will be an optimal (on \(m\) for the given \(\delta\)) solution.
Definition 2. A linear operator \(L\), \(\|L\|<+\infty\), acting in the space \(L^2\), will be called computationally finite-dimensional if, in order to find the function \(Lf\), \(f(x)\in L^2\), at one value of the argument \(x\), a finite number of values \(f(x)\) is used, on which a finite number of arithmetic operations is performed. Operators \(L\) of the opposite kind will be called computationally infinite-dimensional.
If \(T_{\alpha}\) is a regularizing operator, but computationally infinite-dimensional, then constructing the function \(T_{\alpha}f\) even for a single \(x\) is practically impossible. Therefore one proceeds as follows: the operator \(T_{\alpha}\) is approximately approximated by some computationally finite-dimensional operator \(\widetilde T_{\alpha}\), and the approximate solution of the problem is taken to be not \(T_{\alpha}f\), but \(\widetilde T_{\alpha}f\). However, if the method for constructing the operator \(\widetilde T_{\alpha}\) from the operator \(T_{\alpha}\) is not regular, in a certain way connected with the value of \(\alpha\) used, then convergence may be lost: although \(T_{\alpha}A\varphi\to\varphi\) as \(\alpha\to0\), it may be that \(\widetilde T_{\alpha}A\varphi\to\widetilde\varphi\), \(\|\varphi-\widetilde\varphi\|>0\). Hence it follows that the procedure of passing from the computationally infinite-dimensional operator \(T_{\alpha}\) to the computationally finite-dimensional \(\widetilde T_{\alpha}\) must also be regularized. Such an operation is most simply carried out by introducing a second parameter \(\beta=n\) (\(n\) natural) and putting \(\widetilde T_{\alpha}=T_{\alpha,n}\), where, as \(n\to\infty\), \(T_{\alpha,n}f\to T_{\alpha}f\) for any \(f\in L^2\).
Definition 3. A linear operator \(T_{\alpha,n}\), \(0\leq \alpha\leq \alpha_0\), \(n_0\leq n\leq +\infty\), is called regularizing equation (4) on the space \(L^2\) if: 1) for all \(n\) and \(\alpha\), except the case when simultaneously \(n=+\infty\) and \(\alpha=0\),
\[ \|T_{\alpha,n}\|<+\infty; \tag{10} \]
2) if at least for one sequence of values \(\alpha\) and \(n\)
\[ \|\varphi-T_{\alpha,n}A\varphi\|\xrightarrow[\alpha\to0,\ n\to\infty]{}0 \quad \text{for any } \varphi\in L^2. \tag{11} \]
Definition 4. The operator \(T_{\alpha,n}\) is called uniformly regularizing equation (4) on the set \(m\in L^2\) if, for all \(\varphi\in m\) and at least for one sequence \((\alpha,n)\), \(\|\varphi-T_{\alpha,n}A\varphi\|\to0\) uniformly.
Definition 5. The values of the parameters \(\alpha\) and \(n\), determined from the condition
\[ \sup_{\varphi\in m}\|\varphi-T_{\alpha,n}A\varphi\|+\|T_{\alpha,n}\|\delta=\min, \tag{12} \]
are called optimal on the set \(m\) of uniform regularization for a given \(\delta>0\).
Theorem 1. In order that the two-parameter operator \(T_{\alpha,n}\) be regularizing, it is sufficient that \(T_{\alpha,\infty}=T_{\alpha}\) be a one-parameter regularizing operator and
\[ \|T_{\alpha}-T_{\alpha,n}\|\xrightarrow[n\to\infty]{}0 \tag{13} \]
uniformly on any interval \(0<\alpha_0^*\leq \alpha\leq \alpha_0\).
Proof follows from the inequality
\[ \|\varphi-T_{\alpha,n}A\varphi\|\leq \|\varphi-T_\alpha A\varphi\|+\|T_\alpha-T_{\alpha,n}\|\,\|A\varphi\|. \tag{14} \]
Let us proceed to the construction of methods for the approximate solution of integral equations of convolution type (1)—(4) by means of computationally finitely supported two-parameter regularizing operators. We shall seek the latter in the class of operators
\[ S_{\alpha,n}=\sum_{-n}^{+n} c_k E_\alpha^k, \tag{15} \]
where \(E_\alpha\) is the shift operator, \(E_\alpha^k\{f(x)\}=f(x+k\alpha)\), and the \(c_k\) are numerical coefficients depending on \(\alpha\) and \(n\). The approximate solution \(\varphi_{\alpha,n}(x)\) of equation (1) on the basis of the operator \(S_{\alpha,n}\) is determined by the expression
\[ \varphi_{\alpha,n}(x)=\sum_{-n}^{+n} c_k f(x+k\alpha). \tag{16} \]
Clearly, under the conditions for the existence of a solution \(\varphi(x)\in L^2\) of equation (1),
\[ \|\varphi-\varphi_{\alpha,n}\|_{L_2} = \left( \int_{-\infty}^{+\infty} |F(t)|^2 \left| \frac{1}{k(t)}-\sum_{-n}^{+n} c_k e^{-ik\alpha t} \right|^2 \,dt \right)^{1/2}, \tag{17} \]
where \(F(t)\) is the Fourier transform of the function \(f(x)\).
A natural method is the choice of coefficients from the conditions of expanding the function \(1/k(t)\) in a Fourier series on the interval \(|t|\leq \pi/\alpha\):
\[ c_k=c_k(\alpha)=\frac{\alpha}{2\pi} \int_{-\pi/\alpha}^{+\pi/\alpha} \frac{e^{ik\alpha t}}{k(t)}\,dt, \qquad k=0,\pm 1,\pm 2,\ldots \tag{18} \]
Consider the subclass of integral equations (1)—(4) satisfying the additional conditions, fulfilled for all \(\alpha>0\):
\[ \mu(\alpha)=\sup_{|t|\leq \pi/\alpha}\left|\frac{1}{k(t)}\right|<+\infty; \tag{19} \]
\[ |1-k(t)/k_{\mathrm{per}}^{(\alpha)}(t)|\leq C,\qquad |t|>\pi/\alpha, \tag{20} \]
where \(C\) is an absolute constant, and \(k_{\mathrm{per}}^{(\alpha)}(t)\) is the \(2\pi/\alpha\)-periodic repetition of the values of the function \(k(t)\) on the interval \(|t|\leq \pi/\alpha\).
Theorem 2. In order that, under conditions (19)—(20), the operator \(S_{\alpha,n}\) (15) with coefficients (18) be regularizing, it is sufficient that the expansion
\[ \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} = \sum_{-\infty}^{+\infty} c_k(\alpha)e^{-ik\alpha t}, \qquad -\infty\leq t\leq +\infty, \tag{21} \]
converge uniformly for all \(\alpha>0\).
Proof. The operator \(S_{\alpha,n}\) is bounded for arbitrary \(\alpha>0\), \(n<+\infty\):
\[ \|S_{\alpha,n}\|=\sup_t \left| \sum_{-n}^{+n} c_k e^{-ik\alpha t} \right| <+\infty. \tag{22} \]
The operator \(S_\alpha=S_{\alpha,\infty}\) is regularizing. Indeed,
\[ \|S_\alpha\| = \sup_{\|f\|\leq 1} \left( \int_{-\infty}^{+\infty} |F(t)|^2 \left| \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} \right|^2 \,dt \right)^{1/2} = \sup_{|t|\leq \pi/2} \left| \frac{1}{k(t)} \right| = \mu(\alpha)<+\infty \tag{23} \]
and \((\Phi(t)\) is the Fourier transform of the function \(\varphi(x))\):
\[ \|\varphi-S_\alpha A\varphi\| = \left( \int_{-\infty}^{+\infty} |F(t)|^2 \left| \frac{1}{k(t)}-\frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} \right|^2 \,dt \right)^{1/2} \leq \]
\[ \leq C\left(\int_{-\infty}^{-\pi/\alpha} |\Phi(t)|^2\,dt+\int_{\pi/\alpha}^{+\infty} |\Phi(t)|^2\,dt\right)^{1/2}, \tag{24} \]
and since
\[ \int_{-\infty}^{+\infty} |\Phi(t)|^2\,dt<+\infty, \]
as \(\alpha\to 0\) the right-hand side in (24) tends to zero. Since, in addition,
\[ \begin{aligned} \|S_\alpha-S_{\alpha,n}\| &=\sup_{\|f\|\leq 1} \left( \int_{-\infty}^{+\infty} |F(t)|^2 \left| \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} -\sum_{-n}^{+n} c_k e^{-ikat} \right|^2 dt \right)^{1/2} \\ &= \sup_{|t|\leq \pi/\alpha} \left| \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} -\sum_{-n}^{+n} c_k e^{-ikat} \right|, \end{aligned} \tag{25} \]
and, by virtue of the uniform convergence (21), tends to zero as \(n\to\infty\), the validity of Theorem 2 follows from Theorem 1.
Let \(\varphi_\sigma(x)\in L^2\) be a collection of functions whose spectra \(\Phi_\sigma(t)\) satisfy the boundedness condition:
\[ \Phi_\sigma(t)\equiv 0 \quad \text{for } |t|>\sigma. \tag{26} \]
The set \(m\) of functions \(\varphi\) on which the two-parameter regularizing operator \(S_{\alpha,n}\) admits uniform regularization can be specified as the collection of functions with bounded spectrum (26), with \(\|\varphi_\sigma\|\leq N\).
In this case, for \(\varphi=\varphi_\sigma\in m\), we have the estimate
\[ \|\varphi_\sigma-S_{\alpha,n}A\varphi_\sigma\| \leq N\sup_{|t|\leq \sigma} \left| \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} -\sum_{-n}^{+n} c_k e^{-ikat} \right| + \delta\sup_{|t|\leq \sigma} \left| \sum_{-n}^{+n} c_k e^{-ikat} \right|. \tag{27} \]
The operator \(S_{\alpha,n}\) optimal on the set \(\varphi_\sigma\in m\), for a given \(\delta\), is determined by the condition
\[ \frac{\delta}{N}\sup_{|t|\leq \sigma} \left| \sum_{-n}^{+n} c_k e^{-ikat} \right| + \sup_{|t|\leq \sigma} \left| \frac{1}{k_{\mathrm{per}}^{(\alpha)}(t)} -\sum_{-n}^{+n} c_k e^{-ikat} \right| = \min. \tag{28} \]
Up to now we have considered regularization on the whole space \(L^2\). However, regularization may also be considered on certain sets \(E\) of functions from \(L^2\) admitting a stronger metrization. In these cases, the operator regularizing on the set \(E\) is defined analogously, with the norm from \(L^2\) in (7) and (11) replaced by the stronger norm of this set.
Theorem 3. Let \(L^2_{(k+1)}\) be the set of functions differentiable \(k+1\) times, with derivatives belonging to \(L^2\). Under the condition \(\varphi(x)\in L^2_{(k+1)}\) and the fulfillment of the restrictions (19)—(21), the operator \(S_{\alpha,n}\) is regularizing in the sense of uniform convergence
\[ \max_x \left| d^r\varphi(x)/dx^r - d^r\varphi_{\alpha,n}(x)/dx^r \right| \longrightarrow 0,\quad r\leq k, \tag{29} \]
as \(\alpha\to 0,\ n\to\infty\), and convergence
\[ \left\| d^{k+1}\varphi(x)/dx^{k+1} - d^{k+1}\varphi_{\alpha,n}(x)/dx^{k+1} \right\|\to 0 \]
in the space \(L^2\).
Schmidt Institute of Physics of the Earth,
Academy of Sciences of the USSR
Received
16 II 1967
CITED LITERATURE
- M. M. Lavrent’ev, On Some Ill-Posed Problems of Mathematical Physics, Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, 1962.
- M. M. Lavrent’ev, V. G. Vasil’ev, Sibirsk. Mat. Zhurn., 7, No. 3, 559 (1966).
- A. N. Tikhonov, DAN, 151, No. 3, 501 (1963).
- A. N. Tikhonov, DAN, 153, No. 1, 49 (1963).
- V. K. Ivanov, Sibirsk. Mat. Zhurn., 7, No. 3, 546 (1955).
- V. N. Strakhov, DAN, 153, No. 3, 533 (1963).