Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 3

MATHEMATICS

Corresponding Member of the USSR Academy of Sciences A. N. TIKHONOV

ON THE SOLUTION OF ILL-POSED PROBLEMS AND THE REGULARIZATION METHOD

1. Inverse problems of mathematical physics often lead to ill-posed problems. A typical example is the Fredholm equation of the first kind

[
A[x,z(s)] = \int_a^b K(x,s)z(s)\,ds = u(x), \qquad c \le x \le d.
\tag{1}
]

This equation has a solution not for every function (u(x)). It is obvious that if (K(x,s)) has a certain degree of smoothness with respect to (x), then there is no function (z(s)\in L_2) satisfying equation (1) if (u(x)) has a lower degree of smoothness. We shall assume uniqueness of the solution of equation (1), i.e., we shall suppose that if for some function (\bar u(x)) equation (1) has a solution (\bar z(s)), then it has only one.

Let the function (\bar u(x)) be such that equation (1) has a solution (\bar z(s)). The purpose of the present article is to set forth an algorithm for constructing a uniform approximation to the function (\bar z(s)). In what follows we shall assume that (\bar z\in \bar C_1), where (\bar C_1) is the class of continuous piecewise-smooth functions.

Denote by (U) the class of functions (u(x)=A[x,z(s)]), (z(s)\in \bar C_1). As the norm of deviation in (\bar C_1) we shall take

[
|z|=\max |z(s)| \qquad (a\le s\le b)
]

and as the norm of deviation in (U),

[
|u(x)|=\left[\int_c^d u^2(x)\,dx\right]^{1/2}.
]

If the kernel (K(x,s)) is continuous, then the mapping (\bar C_1\to U) is continuous. It should be borne in mind that the inverse problem—the problem of finding (z(s)) from a given function (u(x))—is ill-posed. Indeed, to the functions (z_1(s)) and (z_2(s)=z_1(s)+p\cos\omega s), (z_1(s), z_2(s)\in \bar C_1), where (p) is any fixed number (however large), there will correspond functions (u_1(x)) and (u_2(x)), whose norm of deviation (|u_1(x)-u_2(x)|) is arbitrarily small if (\omega) is sufficiently large. However, if the class of admissible solutions is a compact class (\bar Z), then the inverse mapping (\bar U\to \bar Z) will be stable ((^1)). In other words, for any (\varepsilon>0) there exists a (\delta(\varepsilon,\bar Z)) such that from (|u_1-u_2|<\delta(\varepsilon,\bar Z)) it follows that (|z_1-z_2|<\varepsilon), if (u_1,u_2\in \bar U={u(x)=A[x,z(s)],\, z\in \bar Z}), where (\bar Z) is a compact class of functions.

The construction of an algorithm for obtaining an approximate solution that uniformly approximates (\bar z(s)) is based on the following regularization principle: a family of functions (z^\alpha(s)), depending on a parameter (\alpha), will be called a regularized family of approximate solutions if: 1) (u_\alpha(x)=A[x,z^\alpha(s)]\to \bar u(x)) as (\alpha\to 0); 2) the functions (z^\alpha(s)), for any (\alpha), belong to a compact class of functions (\bar Z) containing (\bar z(s)). The regularized family of approximate solutions converges uniformly to (\bar z(s)) as (\alpha\to 0).

1. Let a function (\bar u(x)) be given. Consider the functional

[
M^\alpha[z(s), \bar u(x)] = N[z(s), \bar u(x)] + \alpha \Omega[z(s)],
\tag{2}
]

where the functional (N) represents the quadratic deviation of (\bar u(x)) from (A[x,z(s)])

[
N[z(s), \bar u(x)] = \int_c^d [A[x,z(s)]-\bar u(x)]^2\,dx,
]

[
\Omega[z(s)] = \int_a^b [k(s)z'(s)^2 + p(s)z^2(s)]\,ds
\quad (k(s)>0,\; p(s)>0).
]

We shall call (\Omega[z]) a regularizing functional and (M^\alpha) a smoothing functional.

Theorem 1. For any function (\bar u(x)\in L_2) there exists a unique continuous, differentiable function (z^\alpha(s)) realizing the minimum of the smoothing functional (M^\alpha[z(s),\bar u(x)]).

The function (z^\alpha(s)) is determined by the Euler equation for the functional (M^\alpha[z,\bar u]):

[
L^\alpha[z] = \alpha\left{\frac{d}{ds}\left[k\frac{dz}{ds}\right]-pz\right}
-\left{\int_a^b \bar K(s,\xi)z(\xi)\,d\xi-\bar b(s)\right}=0,\quad
z'(a)=z'(b)=0,
\tag{3}
]

where

[
\bar K(s,\xi)=\int_c^d K(\xi,s)\,K(\xi,\xi)\,d\xi,\qquad
\bar b(s)=\int_c^d K(\xi,s)\bar u(\xi)\,d\xi.
]

With the aid of the Green’s function for the boundary-value problem

[
L^\omega[z]=\frac{d}{ds}\left[k(s)\frac{dz}{ds}\right]-p(s)z(s)=f(s),\quad
z'(a)=z'(b)=0,
\tag{4}
]

defined by the Euler operator for the regularizing functional, equation (3) can be transformed into a Fredholm equation of the second kind, for which, when (\alpha>0), the homogeneous equation has only the trivial solution; hence the existence of (z^\alpha(s)) follows.

Theorem 2. If (\bar z(s)\in \bar C_1), (u(x)=A[x,\bar z(s)]), then for any (\varepsilon>0) there exists such an (\alpha(\varepsilon,\bar z)) that

[
|z^\alpha(s)-\bar z(s)|<\varepsilon
]

for all (\alpha<\alpha_0(\varepsilon,\bar z)).

Indeed,

[
M^\alpha[z^\alpha(s);\bar u(x)] \leq N[\bar z,\bar u] + \alpha\Omega[\bar z] = \alpha C^2
\quad (C^2=\Omega[\bar z]),
]

whence it follows that (z^\alpha(s)) satisfies the inequality

[
1)\quad \Omega[z(s)]\leq C^2,
]

which determines a compact class of functions (\bar Z), and also

[
2)\quad |u^\alpha(x)-\bar u(x)|\leq \alpha C \to 0
\quad \text{as } \alpha\to 0.
]

Hence theorem 2 follows.

Theorem 3. If (\bar z\in \bar C_1), then for any (\varepsilon>0) and any auxiliary numbers (0<\gamma_1<\gamma_2) there exists such a (\delta_0(\varepsilon,\gamma_1,\gamma_2,\bar z)) that if: 1) the norm of the deviation of the function (u_\delta(x)) from the function (\bar u(x)) is less than (\delta)

[
|u_\delta(x)-\bar u(x)|<\delta;
]

2) (\bar{\alpha}=\bar{\alpha}(\delta)) satisfies the conditions

[
\gamma_1 \leqslant \delta^2/\alpha \leqslant \gamma_2
\quad
(\text{or } \delta^2/\gamma_2 \leqslant \alpha \leqslant \delta^2/\gamma_1),
]

then (\widetilde z_{\delta}^{\,\bar{\alpha}}(s)), realizing the minimum of the smoothing functional (M^{\bar{\alpha}}[z,\widetilde u_\delta(x)]), belongs to the (\varepsilon)-neighborhood of the function (\bar z(s)),

[
\left|\widetilde z_{\delta}^{\,\bar{\alpha}}(s)-\bar z(s)\right|<\varepsilon
]

for (\delta \leqslant \delta_0(\varepsilon,\gamma_1,\gamma_2,\bar z)).

It is not difficult to verify by examples that the function (\widetilde z_{\delta}^{\,\alpha}(s)) corresponding to a fixed function (\widetilde u_\delta(s)), for small (\delta), as (\alpha\to 0), may leave the (\varepsilon)-neighborhood of (\bar z(s)).

1. Let us pass to approximate methods for solving equation (1). Consider the method of finite differences. Take a mesh on ((a,b)): (s_j=jh-0.5h) ((j=1,\ldots,n)), and on ((c,d)): (x_i=ih_1-0.5h_1) ((i=1,\ldots,m)), where (h=\frac{1}{n}(a-b)) and (h_1=\frac{1}{m}(c-d)). Denote (z_j=z(s_j)), and let

[
\sum_{j=1}^{n} K_{ij}z_jh
=
\int_a^b K(x_i,s)z(s)\,ds+O(h^\gamma)
]

be some quadrature formula of order (\gamma).

Consider the difference smoothing functional

[
\widehat M_h^{\alpha}[\widehat z,\widehat u]
=
\sum_{i=1}^{m}
\left{
\sum_{j=1}^{n} K_{ij}\widehat z_jh-\widehat u_i
\right}^{2}h_1
+
\alpha
\sum_{j=1}^{n}
\left{
k_j(\widehat z_{j+1}-\widehat z_j)^2\frac{1}{h}
+
p_j\widehat z_j^{\,2}h
\right},
]

where (\widehat u={\widehat u_i}) is a given mesh function on ({x_i}), (\widehat z={\widehat z_j}) is a mesh function on ({s_j}), and (k_j>0,\ p_j>0).

Analogously to the preceding, the following holds.

Theorem (1'). For any mesh function (\widehat u) and (\alpha>0) there exists a mesh function (\widehat z^{\,\alpha}) realizing the minimum of the smoothing functional (\widehat M_h^{\alpha}[\widehat z,\widehat u]).

The mesh function (\widehat z^{\,\alpha}) is determined from the system of equations

[
\widehat L^{\alpha}[\widehat z]
=
\alpha
\left{
\frac{1}{h^2}
\bigl[
k_j(\widehat z_{j+1}-\widehat z_j)
-
k_{j-1}(\widehat z_j-\widehat z_{j-1})
\bigr]
-
p_j\widehat z_j
\right}
-
\left{
\sum_{l=1}^{n}\overline K_{jl}\widehat z_lh-\widehat b_j
\right}
=0,
\tag{3′}
]

[
\widehat z_0=\widehat z_1,\qquad
\widehat z_{n+1}=\widehat z_n,
]

where

[
\overline K_{jl}=\sum_{i=1}^{m}K_{ij}K_{il}h_1,
\qquad
\widehat b_j=\sum_{i=1}^{m}K_{ij}\widehat u_i h_1,
]

and (k_j) and (p_j) are determined through (k(x)), (p(x)) by means of some homogeneous difference scheme converging to problem (4) (see ((^2))). In particular, for example, (k_j=k(s_j+0.5h)), (p_j=p(s_j)).

Theorem (2'). If (z(s)\in \overline C_1), then for any (\varepsilon>0) and any auxiliary numbers (0<\gamma_1\leqslant\gamma_2) there exist such (\delta_0(\varepsilon,\gamma_1,\gamma_2,\bar z)) and (h_0(\varepsilon,\gamma_1,\gamma_2,\bar z)) that if: 1) the norm of the deviation of the function (\widetilde u_\delta(x)) from (\bar u(x)) is less than (\delta):

[
|\widetilde u_\delta-\bar u|<\delta;
]

2) (\bar\alpha=\bar\alpha(\delta)) satisfies the conditions

[
\gamma_1 \leqslant \delta^2/\alpha \leqslant \gamma_2
\quad
(\text{or } \delta^2/\gamma_2 \leqslant \alpha \leqslant \delta^2/\gamma_1),
]

then (\bar z_{\delta}^{\alpha}(s)), which realizes the minimum of the difference smoothing functional (\hat M_h^\alpha[\hat z,\hat u_\delta]), belongs to the (\varepsilon)-neighborhood of the function (\bar z(s)) for (\delta \leqslant \delta_0(\varepsilon,\gamma_1,\gamma_2,\bar z)), (h < h_0(\varepsilon,\gamma_1,\gamma_2,\bar z)).

Equation (3′) is an algorithm for solving equation (1), giving very effective results with the aid of electronic digital computers.

The construction of the functions (z^\alpha(s)) may also be carried out using expansions in series with respect to orthogonal systems.

The method set forth above is applicable to equations of the type

[
A[x,z(s)] = u(x),
\tag{1′}
]

where (A[x,z(s)]) is a bounded operator. If we denote

[
\alpha(x,s)=A[x,\eta_s(\xi)], \qquad
\eta_s(\xi)=
\begin{cases}
1, & \xi \leqslant s,\
0, & \xi > s,
\end{cases}
]

then equation (3) can be represented in the form

[
\alpha\left{k(s)z'(s)+\int_0^s p(\xi)z(\xi)\,d\xi\right}
-
\left{\int_c^d A[x,z(\xi)]\alpha(x,s)\,dx
-
\int_c^d \alpha(x,s)\bar u(x)\,dx\right}=0,
]

[
z'(a)=z'(b)=0.
]

The regularizing functional (\Omega[z]) may be chosen as a quadratic functional (not necessarily differential) so that the condition (\Omega(z)\leqslant C) determines a compact set and so that the Euler operator for (\Omega(z)) has a completely continuous inverse operator. This also applies to the case where the domain of definition of (z(s)) is a domain (D) of (n) dimensions (see the Sobolev–Kondrashev theorem({}^{3})).

Smoothing functionals provide a convenient apparatus for solving equations of the second kind at an isolated point of the spectrum, as well as for solving nonlinear problems.

Received
17 IV 1963

CITED LITERATURE

({}^{1}) A. N. Tikhonov, Dokl. Akad. Nauk SSSR, 39, No. 5, 195 (1943); M. M. Lavrent’ev, Dokl. Akad. Nauk SSSR, 102, No. 2, 205 (1955); 106, No. 2, 389 (1956); 112, No. 2, 195 (1957). ({}^{2}) A. N. Tikhonov, A. A. Samarskii, Zhurn. Vychisl. Matem. i Matem. Fiz., 1, issue 1, 5 (1961). ({}^{3}) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.