Xll

FOREWORD

the operator F satisfies the following conditions:

(0.2) a solution x to Equation (0.1) is unique, i.e., if F(xx) = F(x2), x. e

X, then xx — x2 (uniqueness of solution);

(0.3) a solution x to Equation (0.1) exists for any y e Y , i.e., for any y e Y

there is x e X satisfying Equation (0.1) (existence of solution);

(0.4) a solution to the equation under consideration is stable, i.e., if y — • y ,

then related solutions x — • x (stability of solution).

The problem is said to be well-posed in the sense of Hadamard if, in the

definition given, both X and Y are spaces C or H

,p

or their subspaces

of finite codimensions. Hadamard especially stressed meaning of the stabil-

ity condition. In practice it is important because of inevitable errors when

calculating or measuring something. Unfortunately, many important prob-

lems of mathematical physics, including basic inverse problems, are not well-

posed according to Hadamard. Two examples are the Cauchy problem for

the Laplace equation and the inverse problem of potential theory mentioned

above.

A problem described by Equation (0.1) is said to be conditionally correct

(correct according to Tikhonov) in a correctness class M if the operator F

satisfies the following conditions:

(0.2M) a solution x to Equation (0.1) is unique in M, i.e., if F(xx) =

F(x2), Xj; e M, then x{ = x2 (uniqueness of a solution in M);

(0.4 M) a solution is stable on M, i.e., x — • x i n X i f x , . x e M and

F(x) — • F(x) in Y (conditional stability).

So the requirements (0.2) and (0.4) are replaced by the less restrictive ones

(0.2 M) and (0.4 M). There is no existence requirement at all. We remark

that the convergence x — x in I (y — • y in Y) means that \\x - x\\x — • 0

(||j — JV|| - —* - 0). A theory of conditionally correct problems was created in

the 1950s-1960s by Ivanov, John, Lavrent'ev, Pucci, and Tikhonov. It is

described very briefly in Section 2.3. One can find detailed expositions in

the books and papers of Ivanov, Vasin, Tanana [67], John [71], Lavrent'ev

[87, 88], Nashed [103], and Tikhonov and Arsenin [163]. According to this

theory, any conditionally correct problem can be solved numerically by means

of regularization and the success of this solution process depends on the

correctness class M. Note that if M is compact in X then the condition

(0.4 M) is a consequence of the condition (0.2 M ). Uniqueness questions are

central in the theory of conditionally correct problems; nevertheless, existence

theorems are of importance as well, since they guarantee that we do not use

extra data.

The inverse problem of potential theory has been studied extensively, al-

though many cardinal questions are waiting for answers. In 1943, analyzing

stability of this problem (which is not well-posed in the sense of Hadamard),

Tikhonov introduced certain important concepts of the theory of condition-

ally correct problems. In fact, we have to find the right-hand side (source)

of the Laplace equation. This problem serves as a good pattern for other