Uniqueness from pointwise observations in a multi-parameter inverse problem
Michel Cristofol - Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France (email)
Abstract: In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.
Keywords: Reaction-diffusion, inverse problem, multi-parameter, heterogeneous media, uniqueness.
Received: March 2010; Revised: November 2010; Published: September 2011.
2011 Impact Factor.692