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A few geometrical features of inverse and ill-posed problems. (English) Zbl 0637.47003

Inverse and ill-posed problems, Alpine-U.S. Semin. St. Wolfgang/Austria 1986, Notes Rep. Math. Sci. Eng. 4, 1-18 (1987).
[For the entire collection see Zbl 0623.00010.]
A few simple geometrical remarks help to understand the strategies currently used for “solving” inverse or ill posed problems. Geometry obviously explains what is a quasisolution (the parameter which is the reciprocal image of the closest point to the measured result inside the image of the set of parameters) and an approximate solution (any point which gives a value sufficiently small for a given convex combination of an a priori measure of the reliability of the parameter and of the value of the fit discrepancy). They explain also why all “methods” to obtain approximate solutions are roughly equivalent when the set of approximate solutions is narrow enough and are unreliable when it is too large, or disconnected (what we call a strongly ill posed problem). Then they suggest to replace the research of an approximate solution by the research of a set of “good questions” (which give sections of the set of solutions in a stable way). In mathematically well understood ill- posed problems, techniques using sets of geometrical transformations enable one to span the set of parameters and the set of results and to make a joint exploration of both, so as to discover ambiguities, to understand the nature of ill-posedness, or to create new inverse methods for solving nonlinear partial differential equations. Examples illustrating these remarks and methods are shown from recent works of the author’s laboratory.
Reviewer: P.C.Sabatier

MSC:

47A40 Scattering theory of linear operators
47F05 General theory of partial differential operators
47A50 Equations and inequalities involving linear operators, with vector unknowns

Citations:

Zbl 0623.00010

Software:

FLIPS