Well-posed problem

A mathematical problem is called set correctly (also well placed or appropriately placed ) if the following conditions are met:

If one of these conditions is not met, ie the problem incorrectly placed (or even ill-posed ). This definition goes back to Jacques Hadamard.

Motivation

In order to deal with problems in physics, engineering or other natural sciences using the methods of mathematics or numerical analysis, the problem must first be formulated as a mathematical model. Do we know where to describing real process ( for example, from our experience and from the feeling of "Nature does not make leaps " ) that a solution exists, it is uniquely determined and does not change much when the input data changes only slightly, so we would wish such a behavior for the solution of the mathematical model. In the mathematical model, all these properties are far from clear. They can not be derived from the properties of the corresponding physical system also, as in the mathematical modeling always certain aspects of reality (eg friction) are hidden. One must therefore prove with mathematical methods that the conditions 1 to 3 are met.

Importance

The third condition ( continuous dependence of the solution on the input data ) means straight, that the solution of the problem varies with a small change in the input data very little. This is important in many applications, as they often are present, the input data only when erroneous data. However, this even has the consequence that lie in any " close " a detachable problem infinitely many problems without solution.

For correct identified problems usually requires a stable numerical solution algorithm is known ill-posed problems usually have to first be reformulated, for example by means of regularization.

Examples

The initial value problem for the heat equation leads, for example, to correct any problems. In contrast, the corresponding inverse problem ( given a solution, the initial data determine ) ill-posed.

In general, partial differential equations are only set correctly if the basic type appropriate initial and / or boundary conditions are specified. Thus, the wave equation, for example, found correct as an initial value problem as a pure boundary value problem, however, does not necessarily exist a solution. A similar situation exists with the Laplace equation: Here, the boundary value problem is set correctly, but the initial value problem (where one coordinate the function of time takes over ) is not.

It has been shown that many interesting mathematical problems (eg, in computed tomography, the deposit exploration ) violate these correctness conditions. Thus, measurement errors can help to ensure that Condition 1 is violated. The structure of the problem can lead to Condition 3 is violated.