4 Stability

Well-posed and ill-posed problems

A problem is called properly-posed, or well-posed, (in the sense of Hadamard) if its solution satisfies the following three requirements:

(1) existence,

(2) uniqueness,

(3) stability.

This means that the solution exists, that there is only one solution, and that solution depends stably on the data. If one or more of these requirements are violated, then we have an improperly-posed, or ill-posed, problem.

For a long time it was thought that only well-posed problems were physically meaningful. Nowadays we think differently, and ill-posed problems are no longer discriminated. On the contrary, they have become synonyms for mathematically very difficult but particularly interesting problems.

Stability and instability

The meaning of existence and uniquiness is pretty clear. What does stability mean? Its meaning is: small changes in the cause imply small changes in the result. Instability: small changes in the cause may imply large changes in the result. In this sense we speak of unstable equilibrium, of unstable weather, even of the unstable character of a dictator.

STABILITY: small causes small effects

Example of stability

INSTABILITY: small causes large deviations

Example of instability

The solution of "interesting" inverse problems is frequently unstable; therefore we must investigate them with particular care.

In fact, even if the direct problem is "well-posed", the corresponding inverse problem is usually "ill-posed", so that " inverse problems" and " ill-posed problems" are frequently used as synonyms.

Weather forecastis a paradigm in which both the direct and the indirect problem are unstable. Weather as such is unstable, and meteorological measurements, in spite of meteorological satellites, are still insufficient.

Causality and determinism

In the paradigm of classical physics based on Newtonian mechanics, the evolution is stable and continuous. Natura non facit saltus, nature does not make jumps. However, quantum theory and molecular biology have shown us that nature does make jumps. And "modern" (going back to Poincare around 1890!) nonlinear dynamics shows that even Newtonian celestial mechanics may lead to "chaotic" phenomena incompatible with classical ideas about causality; one speaks of "deterministic chaos"! Even classical mechanical systems are not always "well-posed". In a certain sense, "most" classical dynamic systems are ill-posed. We have believed in classical causality or determinism only because we recognized only such simple differential equations which we could solve explicitly without computers, and this has lead us to believe that ALL differential equations were well behaved. So even direct problems of classical mechanics may well be ill-posed, not to speak of inverse problems.

The following pictures show clearly that, by changing the parameters of nonlinear Newtonian systems, an originally "smooth" classical system becomes increasingly "chaotic". (This is still a "direct system", so you can imagine how a corresponding "inverse system" may look like!)

Poincare example (nonlinear Newtonian system)

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