Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS

1943, Volume XXXIX, N 5

 

 

MATHEMATICS PHYSICS

ON THE STABILITY OF INVERSE PROBLEMS

By A.N.TIKHONOV, Corrsponding Member of the Academy

Many objects of Nature have properties that are not well fitted to, or even altogether defy, direct investigation.In such cases some of their characteristics are studies whose manifestation can be measured. Our judgment on the structure of the earth crust, for instance, may in certain case be based on the investigation of its characteristics, such as density, or electric conductivity, which determine the respective physical field accessible to measurement at the surface of the earth -the gravitational field, the field of electric current.

Given a certain characteristics of the medium (distribution of density or electic conductivity), we are usially able to compute (precisely or approximately) the physical field determine by this structure in the region where the measurements are made. Yet, the problem to be solved here is the reverse to this. Namely, the physical field is known, while the structure of the mediumwhich determines it is sought.

The customary way to solve the inverse problems is by selection. Within as arbitrary chosen (sufficiently wide) class f possible structures of the medium the corresponding physical fields are computed and the solution of the problem arrived at by selecting some admissible medium, for which the calculated phisical fields shows but a small derivation from observation.

In order to put the method of selection upon a solid foundation, it is necessary to establish ( or admit ) the existence of certain regularities: 1) One has to establish the uniqueness theorem for the direct correspondence, i.e. to prove that no two different types of medium have a single corresponding field. Then we have also the right to speak of a reverse correspondence. Within this the method of selection has no sense at all. 2) The coincidence between the calculated and observed field is not an absolute one ( if only because the selectin s made in an approximate way). We therefore have, moreover, to prove the stability of thr inverse problem or the continuity of the inverse mapping), that is , to make sure that with slight deviation of the auxiliary field from observation the respective structure of the medium cannot possibly deviate strongly from the actual.

In studing the stability of the inverse problem, a number of qualitative and quantitative character may be raised.

In the present paper set theoretical conceptions are applied to one of these problems, which consists in proving that under certain conditions the stability of the inverse problem is a direct consequence of the uniqueness theorem. We shall also apply it to the inverse problem to the potential and to the study of the continuous dependence of the solutions of ordinary differential equations upon the parameter.

1. In the theory of continuous mappings there takes place the following theorem (1,2).

Let a set of elements forming a metric space be mapped continuously upon another set of elements forming a metric space . If the mapping is one-to-one and continuous and if the space is compact, then the inverse mapping is also continuous.

Definition . Let a set of elements be mapped by function on to another set of elements : .This mapping is said to be one-to-one at the point if for any element distinct from

It is easy to prove the following theorem.

Let a metric space be mapped continuously on another

metric space

If this mapping is one-to-one at the point and the space is compact, then the inverse mapping is likewise continuous at the point .

The continuity of the inverse mapping is understood to mean that for any there exists a such that if

then

where is any prototype of the point .

Though we formulate these theorems for metric spaces, they also hold in a more general case.

2. Among the direct problems of the potential theory there is one that consists in computing for the surface the potential of a hounder body Tfilled up with homogeneous mass of density and lying beneath that surface . We shall demonstrate that the inverse problem has a stable solution.

Suppose approximately that the position of the disturbing body is known to be inside a given surface . Let us examine the totality of bodies satisfying the following conditions:

10. Each of the bodies belongs to a given bounded surface lying in the region .

20. Each of the bodies is stellate with respect to its centre of gravity, so that the equation of surface å , bounded the body , may in the spherica system of coordinate with their centre at the point be represented in the form

30. Function has its derivative numbers bounded by a number , common to all bodies of the class .

Let us determine the degree of proximity of two different bodies and from the class R by means of the number

where and are functions defining the surface equations of the bodies and with respect to their centres of gravity band .

We shall prove the following theorem.

Whatever may be the degree of accuracy of and the class of the bodies, such number may be indicated that if the values of the potentials ( or of their derivatives) and for any two bodies and of the class differ from each other at by less than

then these bodies are separate by a distance less then

.

In fact, with the notion of distance determined as , the totality of bodies of the considered class form a metric space. If the distance between the potentials (or their derivatives) and be evaluated by the number

(the muximum exists indeed), then the set of functions {V} presents a metric space .

The correspondence between the bodies and the values of their potentials (or derivatives) determines the mapping

of the space on the space .

This mapping is continuous and one-to-one, for, in vertue of well known theorem by P.S. Novikov, no two different bodies and , stellate with respect to their centre of gravity, may have equal potentials corresponding to them. And the centre of gravity has its position determined by the value of the potential at the surface .

The class under consideration, , forms a compact family. Direct application of the theorem mentioned in 1 Just proves the theorem.

Modifying the notion of distance between the bodies or that of proximity of potentials (or of potential derivatives), one may easily establish other theorems of stability.

It is well to note that without the conditions of type or type the stability of the inverse problem does not hold any longer.

3. Carathedory has established the following theorem on the continuous dependence of the solutions of s system of ordinary difference equations upon the parameter .

Let the functions be given, satisfying the conditions:

a) For every value of t inside a certain neighbourhood of the point every function is measurable with respect to when are fixed , and is continuous with respect to when is fixed, varying within region ; and, moreover, there exists such measurable ( in the region ) function, independent of , that

b) For and arbitrary values of the function are continuous with respect to .

c) For and thre exists a sole system of functions satisfying the equations

If, under these conditions

then for any system of solution (1) at any point of the region under consideration

Let us examine the totality of all possible systems of solutions of equation (1) when Defining the distance between the elements as

we obtain a metric space R which is compact by virtue of obvious fact that the functions are uniformly bounded and uniformly continuous in their totality [ this is a consequence of the existence of M(x) and the condition (2) if only is small enough].

To every elements of the space R we take correspond a point of an n-dimensional Euclidean space putting

The distance between the images of two elements being less than that between the elements, it will be obvious that this is a continuous mapping.

In virtue of the conditions of the theorem this mapping is also one-to-one at the point .

Applying the theorem of 1, one sees that the inverse correspondence is continuous in the sense that for any there exists such that if

then, whatever the solution corresponding to may be, we shall have

 

and this proves the theorem.

Many examples can easily be pput on the stability of inverse problems.

Recieved

12.IV.1943

REFERENCES

1 Hausdorff F., Grundzuge der Mengenlehre. Leipzig, Verlag von Veit & Co. 1914. S.476. Hausdorff F., Mengenlehre. Zweite, neubearb. Auflage. Berlin und Leipzig, Walter de Grugter & Co. 1927, S.282. 2Alexandroff-Hopf, Topologie, I, p.95 (1935). 3P.S. Novikov, C.R.Acad.Sci. RSS, XVIII, 3(1938); G.A. Gamburzev, Bull.Acad.Sci. URSS, ser.geophys., 4(1938); A.A. Zamorev, ibid., 3(1939). 4Caratheodory, Vorlesungen uber relle unktionen, S. 678 (1918).