We treat the stability issue for an inverse problem arising from nondestructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem.We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential illposedness, the stability estimate obtained is optimal.

Stability properties of an inverse parabolic problem with unknown boundaries / M. DI CRISTO, L. Rondi, S. Vessella. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 185:2(2006), pp. 223-255.

Stability properties of an inverse parabolic problem with unknown boundaries

M. DI CRISTO;L. Rondi;
2006

Abstract

We treat the stability issue for an inverse problem arising from nondestructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem.We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential illposedness, the stability estimate obtained is optimal.
inverse problems; stability; parabolic equations; Bessel functions; unique continuation
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/612294
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