Uniqueness from pointwise observations in a multi-parameter inverse   problem
  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 $(0,\epsilon).$ 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. 
Michel Cristofol (LATP), Jimmy Garnier (LATP, BIOSP), Francois Hamel   (LATP), Lionel Roques (BIOSP)
math.AP