Correct and Incorrect Problems

classes of mathematical problems that differ by the degree of determinacy of their solutions. Many mathematical problems consist in seeking a solution z satisfying given initial data u. It is assumed that u and z are linked by a functional relation z = R(u). A problem is said to be correct (or correctly posed) if the following conditions hold: (1) the problem has a solution for any admissible initial data (existence of a solution), (2) to each set of initial data u there corresponds only one solution (uniqueness of solution), and (3) the solution is stable.

The first condition signifies that the initial data are free of contradictions (the existence of contradictions would rule out the possibility of solving the problem).

The second condition ensures that there are enough initial data to determine a unique solution of the problem. Conditions (1) and (2) are usually called the conditions of determinacy of the problem.

The third condition states that by choosing sufficiently close initial data u₁ and u₂ we obtain arbitrarily close solutions z₁ = R(u₁) and z₂ = R(u₂). Here it is assumed that in the manifolds U = {u} and Z = {z} of admissible initial data and possible solutions, respectively, there are defined concepts of proximity (distance) ρ(u₁, u₂) and ρ*(z₁, z₂). The third condition is usually viewed as the condition of physical determinacy of the problem. This is explained by the fact that the initial data of a physical problem are given, as a rule, with a certain error; if the third condition were violated, even the smallest perturbations in the initial data could cause large deviations in the solution.

Problems that fail to meet one or more of the conditions (1), (2), and (3) are called incorrect problems (or incorrectly posed problems).

Attention was first drawn to the correctness of problems by the French mathematician J. Hadamard in connection with the solution of boundary value problems for partial differential equations. The concept of correctness of a problem was one of the reasons for the classification of boundary value problems involving such equations.

It had been thought that incorrect problems could not be encountered in the solution of physics and technical problems and that it was impossible to construct approximate solutions for incorrect problems lacking stability. The development of methods of automation in the gathering of experimental data has led to a great increase in the volume of such data; the necessity of basing information about natural objects on such data has led to the consideration of incorrect problems. The development of computer technology and the application of this technology to the solution of mathematical problems have changed the view-point regarding the possibility of constructing approximate solutions of incorrectly posed problems.

The meaning of the concept of an approximate solution is essentially different for correct and incorrect problems. Consider a correct problem. For any neighborhood ∈ of the approximate solution of such a problem, there exists, owing to the stability of condition (3), a neighborhood δ(∈) of the initial data such that if ρ(u, ũ) ≤ δ(∈), then ρ*(z, z̃) ≤ ∈. It follows that as an ap-proximate solution z = R(u) of a correct problem, we may take its exact solution z̃ corresponding to approximate initial data ũ. For incorrect problems it is impossible to consider the exact solution with approximate initial data as an approximate solution. However, the assignment of approximate initial data in the natural sciences may be characterized not only by the original element ũ but also by a measure of its exactness δ. Hence the concept of an approximate solution of a problem z = R(u) is introduced with the help of an operator R_δ(u), which depends on the parameter δ and is called a regularizing (or correcting) operator. If the operator R_δ(u) is defined for all δ > 0 and for all u belonging to the class of admissible inital data and if z = R(u), then for any given ∈ there exists (in principle) a δ(
) such that for any element ũ^δ with ρ(u,ũ^δ) ≤ δ the solution z = R_δ(ũ^δ) differs from z by less than the given ∈, that is, ρ*(z, z̃^δ) < ∈.

Thus, an approximate solution of an incorrect problem may be reduced to finding a regularizing operator R_δ(ũ), which determines a stable approximation to z.

An example of an incorrect classical mathematical problem is the problem of approximate differentiation with a definite (important in practice) degree of accuracy in the assignment of z and u. More specifically, the problem of finding a uniform approximation z̃ to z, given a uniform approximation ũ to u, is an incorrect problem, since neither the first nor third condition is fulfilled: the first condition fails because not every function ũ such that ǀũ(x) — u(x)ǀ ≤ δ has a derivative u′(x); and the third condition fails because even if the derivative u′ exists, the inequality ǀũ(x) — u(x)ǀ ≤ δǀũ(x) — u(x)ǀ ≤ δǀũ(x) — u(x)ǀ ≤ δǀũ(x) — u(x)ǀ ≤ δ does not imply the proximity of the derivatives ũ′(x) and u′(x). However, we can take as a regularizing operator R_δ(ũ(x)) =[u(x + h) − ũ(x)]/h for h ≫ δ. This operator is defined for all ũ(x) regardless of their differentiability, and in a bounded interval it gives a uniform approximation for every continuously differentiable function u(x).

There exist many other examples of classical mathematical problems that are incorrect for a perfectly natural choice of the concepts of measure of accuracy both for the initial data of the problem and for the possible solutions. Here we mention the solution of a system of linear algebraic equations with zero determinant, the problem of optimal planning; the solution of integral equations of the first kind, the problem of analytic continuation, the summation of Fourier series, and a large number of boundary value problems for partial differential equations.

A broad class of incorrectly posed problems in natural science comprises problems of processing observational data without additional (quantitative) information about the properties of the solutions. If an object is being studied whose quantitative characteristics z are inaccessible to direct investigation, then it is usual to study certain manifestations u of this object that are function-ally dependent on z. The problem of processing the observations consists in the solution of the “inverse problem,” that is, in determining the characteristics z of the object from the results of observations of u; moreover, u is given approximately.

There are many works (particularly by Soviet mathematicians) devoted to methods of approximate solution of incorrectly posed problems and their applications to the solution of inverse problems. These works are very important for the automation of the processing of observations, for the solution of control problems, and so forth.