Abstract
Full Text
MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR A. N. TIKHONOV
ON THE REGULARIZATION OF ILL-POSED PROBLEMS
Numerous practically important problems lead to ill-posed problems, such as, for example, Fredholm integral equations of the first kind, the Cauchy problem for an elliptic equation, the problem of analytic continuation, etc. Ill-posed problems have recently attracted much attention (see, for example, (¹), where the literature on this question is given).
The problem \(R\) of determining a function \(z(s) \in Z\) from a given function \(u(x) \in U\):
\(z(s) = R[s,u(x)]\) (where \(Z\) and \(U\) are the corresponding functional spaces) is called well-posed if:
1°. To every function \(u(x) \in U\) there corresponds a solution \(z(s)\) of the problem.
2°. The solution \(z(s)\) is uniquely determined by the data \(u(x)\).
3°. The solution \(z(s)\) of the problem depends continuously on \(u(x)\) in the metrics of \(Z\) and \(U\).
Let an ill-posed problem \(z = R[s,u]\) be given. As an illustrative example we shall take the Fredholm equation of the first kind
\[ A[x,z(s)] = \int_a^b K(x,s) z(s)\,ds = u(x), \qquad c \leq x \leq d. \tag{1} \]
Not every function \(\bar u(x)\) corresponds to a solution of the problem \(\bar z(s)\).
Let us consider the following question concerning the solution of ill-posed problems.
It is known that to some function \(\bar u(x)\) there corresponds a solution of the problem
\(\bar z(s) = R[s,\bar u(x)]\); suppose that a function \(\tilde u(x)\) is given, an approximation to \(\bar u(x)\) with known accuracy \(\delta\):
\(\|\bar u(x)-\tilde u(x)\| < \delta\). Determine \(\tilde z(s)\)—an approximate value of \(\bar z(s)\) with prescribed accuracy
\(\|\tilde z(s)-\bar z(s)\|_z \leq \varepsilon\), provided that \(\delta\)—the accuracy with which \(\tilde u(x)\) is specified—is sufficiently small. In this case \(\tilde z(s)\) is by no means required to be equal to \(R[s,\tilde u(x)]\), which may in general not exist.
We shall call an operator (or algorithm) \(R_\delta[s,u(x)]\) regularizing if:
1°. \(R_\delta[s,\tilde u(x)]\) is defined for all \(\tilde u \in U\) and \(\delta > 0\).
2°. If for \(\bar u(x)\) there exists
\(\bar z(s)=R[s,\bar u(x)]\), then for every \(\varepsilon\) there exists a \(\delta(\varepsilon,\bar z)\) such that, if
\(\|\bar u(x)-\tilde u(x)\|_U < \delta\), then
\(\|\tilde z_\delta(s)-\bar z(s)\|_Z \leq \varepsilon\), where
\(\tilde z_\delta(s)=R_\delta[s,\tilde u]\).
We shall call the problem \(z(s)=R[s,u(x)]\) regularizable if it admits at least one regularizing algorithm.
It is obvious that if the problem \(z=R[s,u]\) is well-posed, then it is regularizable, since, putting \(R_\delta[s,u]=R[s,u]\) for any \(\delta\), we obtain a regularizing algorithm.
Depending on the norm of \(z\), we may distinguish weak regularization, uniform regularization, and regularization of the \(n\)-th order of smoothness.
Regularizing algorithms provide a practical method for solving ill-posed problems.
In (²) a uniformly regularizing algorithm is given for equations of the first kind. In the present article, for the same class of problems, regularizing algorithms of the \(n\)-th order of smoothness are presented.
- Consider the integral equation of the first kind (1), and for simplicity suppose that the kernel is continuous and that for \(\bar u(x)=0\) there is only the unique solution \(\bar z(\delta)\equiv 0\). Consider the smoothing functional
\[ M_n^\alpha [z(s);\bar u(x)] = N[z(s);\bar u(x)] + \alpha \Omega^{(n)}[z], \tag{2} \]
where
\[ N[z(s);\bar u(x)] = \int_c^d |A[x,z(s)]-\bar u(x)|^2\,dx \]
and the regularizing functional
\[ \Omega^{(n)}[z]= \int_a^b \left\{ \sum_{i=0}^{n+1} K_i(s)\bigl(z^{(i)}(s)\bigr)^2 \right\}\,ds, \]
where \(K_i(s)\) are continuous functions, \(K_i(s)\ge 0\).
Theorem 1. For any function \(\bar u(x)\in L_2\) and any \(\alpha>0\), there exists a unique \(2(n+1)\)-times continuously differentiable function \(z_n^\alpha(s)\) realizing the minimum of the smoothing functional \(M_n^\alpha[z,\bar u(x)]\).
The function \(z_n^\alpha(s)\) is determined by the Euler equation
\[ L_n^\alpha[z] = \alpha \left\{ \sum_{i=0}^{n+1} (-1)^{i+1}\frac{d^i}{ds^i} \left(K_i(s)\frac{d^i z}{ds^i}\right) \right\} - \left\{ \int_a^b \bar K(s,\zeta)z(\zeta)\,d\zeta-\bar b(s) \right\} =0 \tag{3} \]
with boundary conditions
\[ \pi^l(s)= \left\{ \sum_{i=l+1}^{n+1} (-1)^{i-l-1} \bigl[K_i(s)z^i(s)\bigr]^{(i-l-1)} \right\}\Big|_{a,b} =0 \quad (l=1,\ldots,n+1), \tag{4} \]
where
\[ \bar K(s,\zeta)= \int_a^b K(\xi,s)K(\xi,\zeta)\,d\xi, \qquad \bar b(s)= \int_c^d K(\xi,s)\bar u(\xi)\,d\xi. \]
Using the Green function for the boundary-value problem
\[ \tilde L_n[z]= \sum_{i=0}^{n+1} (-1)^{i+1}\frac{d^i}{ds^i}\bigl[K_i z^{(i)}\bigr] =f, \qquad \pi^l(a)=\pi^l(b)=0, \quad l=1,\ldots,n+1, \]
equation (3) is transformed into a Fredholm equation of the second kind, which for \(\alpha>0\) has only the trivial solution, and this proves the existence of the function \(z_n^\alpha(s)\).
Theorem \(1'\). For any function \(\bar u(x)\in L_2\) and any \(\alpha>0\), there exists a function \(z_{(-1)}^\alpha(s)\in L_2\) realizing the minimum of the functional \(M_{(-1)}^\alpha[z,\bar u]\).
This function is determined as the solution of the equation
\[ \alpha K_0(z)z(s) = \int_a^b \bar K(s,\xi)z(\xi)\,d\xi-\bar b(s), \tag{3*} \]
for which the corresponding homogeneous equation has only the trivial solution.
Theorem 2. If \(\bar u(x)=A[x,\bar z(s)]\), \(\bar z\in \bar C^{(n+1)}\), then for any \(\varepsilon>0\) and auxiliary numbers \(0<\gamma_1<\gamma_2\) there exists such a \(\delta(\varepsilon,\gamma_1,\gamma_2,\bar z)\) that if: 1) \(\|u_\delta(x)-\bar u(x)\|_{L_2}\le \delta\), where \(u_\delta(x)\in L_2\); 2) \(\alpha=\alpha(\delta)\) has pos-
has order \(\delta^2\): \(\gamma_1 \leqslant \delta^2/\bar{\alpha}(\delta) \leqslant \gamma_2\), then \(\bar{z}^{\bar{\alpha}}_{\delta,n}(s)\), realizing the minimum of the functional \(M_n^{\bar{\alpha}}\bigl[z,\widetilde{u}_{\delta}(x)\bigr]\), are such that
\[ \bigl|\bar{z}^{\bar{\alpha}}_{\delta,n}(s)^{(i)}-\bar{z}(s)^{(i)}\bigr|\leqslant \varepsilon,\qquad a\leqslant s\leqslant b,\qquad i=0,\ldots,n, \]
for \(\delta<\delta_0(\varepsilon,\gamma_1,\gamma_2,\bar{z})\).
Theorem \(2'\). If \(\bar{u}(x)=A[x,\bar{z}(s)]\), \(\bar{z}\in L_2\), then for any \(\varepsilon\) and auxiliary numbers \(0<\gamma_1\leqslant\gamma_2\) there exists such a \(\delta_0(\varepsilon,\gamma_1,\gamma_2,\bar{z})\) that, if: 1) \(\|\widetilde{u}_{\delta}(x)-\bar{u}(x)\|_{L_2}\leqslant\delta\), where \(\widetilde{u}_{\delta}(x)\in L_2\); 2) \(\bar{\alpha}=\bar{\alpha}(\delta)\) has order \(\delta^2\): \(\gamma_1\leqslant\delta^2/\bar{\alpha}(\delta)\leqslant\gamma_2\), then \(\bar{z}^{\bar{\alpha}}_{\delta,n}(s)\), realizing the minimum of the functional \(M_n^{\bar{\alpha}(\delta)}[z,\widetilde{u}_{\delta}(x)]\), converges weakly to \(\bar{z}(s)\).
From Theorems 1, 2 and \(1'\), \(2'\) it follows that the solution of the boundary-value problem (3), (4) constitutes a regularizing algorithm of the \(n\)-th order of smoothness if \(\bar{z}\in C^{n+1}\), and the solution of equation \((3^*)\) constitutes an algorithm of weak regularization for \(\bar{z}\in L_2\).
Analogously \((^2)\), all these results carry over to multidimensional problems and to operator equations of the first kind
\[ A[x,z(s)]=\bar{u}(x), \]
where \(A[x,z]\) is an operator acting from \(C\) to \(L_2\), bounded in the sense that, for almost every \(x\),
\[ |A[x,z(s)]|\leqslant A(x)\|z\|_C,\qquad \int_c^d A^2(x)\,dx=A_0<+\infty . \]
A finite-dimensional approximation of the problem under consideration is treated analogously \((^2)\).
- Let us consider a very important special case of the problem under consideration, when
\[ \int_a^b K(x,s)\,z(s)\,ds=\bar{u}(x),\qquad a\leqslant x\leqslant b \]
(i.e., when \(a=c,\ b=d\)) and when for \(K(x,s)\) there exists a kernel of half-order, i.e.
\[ K(x,s)=\int_{c'}^{d'} \widehat{K}(\xi,x)\widehat{K}(\xi,s)\,d\xi . \]
In this case the algorithm for obtaining \(z^\alpha(s)\) is simplified. The equation for determining \(z^\alpha(s)\) can be written in the form
\[ L_n^{(\alpha)}[z]=\alpha \widetilde{L}_n[z]-\left\{\int_a^b K(s,\xi)\,z(\xi)\,d\xi-\bar{u}(s)\right\}=0 \tag{3'} \]
with the conditions
\[ \pi^l(a)=\pi^l(b)=0\qquad (l=1,\ldots,n+1). \tag{4'} \]
Indeed, consider the function
\[ v(\xi)=\int_a^b K(\xi,s)\,z(s)\,ds,\qquad c'\leqslant \xi\leqslant d'. \]
Let us pose the problem: regarding \(v(\xi)\) as a given function, find \(z(s)\). This problem is equivalent to the original one and determines the same function \(z(s)\). Applying to this
problem, by the basic algorithm set forth above, we obtain for determining \(z^x(s)\) the problem \((3^*),(4^*)\), determined directly by the specification of \(K(x,s)\), \(\bar u(x)\).
- The special case under consideration (Sec. 2) includes the problem of continuation of a potential toward the perturbing masses, determined by the Poisson equation
\[ \frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{h}{(x-s)^2+h^2}\,z(s)\,ds=\bar u(x),\qquad c\leq x\leq d. \]
If \(z(s)\) is different from zero only for \(a\leq s\leq b\) and the domain \((c,d)\) in which \(\bar u(x)\) is specified contains \((a,b)\), then one may use the special algorithm \((3'),(4')\). If, however, \((c,d)\) does not contain \((a,b)\), then one must use the general algorithm \((3),(4)\).
Analogous considerations apply both to the inverse problem of heat conduction of the usual type
\[ \frac{1}{2\sqrt{\pi}}\int_{-\infty}^{+\infty}\frac{1}{\sqrt{a^2t}}e^{-(x-s)^2/4a^2t}z(s)\,ds=u(x), \]
and to the inverse problem of the second type, corresponding to the problem of determining the historical climate \({}^{(3)}\).
The problem of analytic continuation from the arc \(L_1\) to the contour \(L_2\), determined by the equation
\[ w(z)=\frac{1}{2\pi i}\int_{L_2}\frac{w(\zeta)}{\zeta-z}\,d\zeta \qquad (z\subset L_1,\ \zeta\subset L_2) \]
is solved with the aid of the basic algorithm.
Problems of optimal regulation lead to ill-posed variational problems, and the regularization method also finds application in solving these problems.
The methods set forth have been tested on electronic computers and have yielded very effective results.
Received
15 VIII 1963
REFERENCES
\({}^{1}\) M. M. Lavrent’ev, On the solution of certain ill-posed problems, Novosibirsk, 1963. \({}^{2}\) A. N. Tikhonov, DAN, 151, No. 3, 501 (1963). \({}^{3}\) A. N. Tikhonov, Matem. sborn., 42, 199 (1935).