Correctly posed problem

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A mathematical problem is correctly set (also well-placed , well-off or properly placed ) if the following conditions are met:

  1. The problem has a solution (existence).
  2. This solution is clearly determined (uniqueness).
  3. This solution always depends on the input data (stability).

If one of these conditions is not met, the problem is called incorrectly posed (or poorly posed ). This definition goes back to Jacques Hadamard , which is why the above conditions are also called "Hadamard's requirements".

motivation

In order to be able to deal with problems from physics, technology or other natural sciences with the help of the methods of mathematics or numerics , the problem must first be formulated as a mathematical model . Do we know for the real process to be described (e.g. from our experience and from the feeling that “nature doesn’t make leaps”) that a solution exists, that it is clearly defined and that it does not change very much if the input data is only used a little changes, we would like such a behavior also for the solution of the corresponding mathematical model. In the mathematical model, all of these properties are by no means clear. They cannot be derived from the properties of the corresponding physical system either, since certain aspects of reality (e.g. friction ) are always masked out in mathematical modeling . It is therefore necessary to use mathematical methods to prove that conditions 1 to 3 are met.

meaning

The third condition (constant dependence of the solution on the input data) states that if there is a small change in the input data, the solution to the problem changes only slightly. This is important in many applications, since the input data is often only available as incorrect measurement data . However, if this third condition is not fulfilled, it even has the consequence that in any “vicinity” of a solvable problem there are an infinite number of problems without a solution.

As a rule, a stable numerical solution algorithm is known for correctly posed problems , poorly posed problems usually have to be reformulated first, for example using regularization techniques .

Examples

The initial value problem for the heat conduction equation , for example, leads to correctly posed problems. In contrast, the corresponding inverse problem (given a solution, determine the initial data) is poorly posed.

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

It has been shown that many interesting mathematical problems (e.g. in computer tomography , deposit exploration) violate these correctness conditions. Measurement errors can contribute to condition 1 being violated. The structure of the problem can cause Condition 3 to be violated.

See also

Individual evidence

  1. ^ Jacques Hadamard : Sur lesproblemèmes aux dérivées partielles et leur signification physique. In: Princeton University Bulletin. Vol. 13, No. 4, 1902, ZDB -ID 1282693-5 , pp. 49-52 .