Uniqueness from pointwise observations in a multi-parameter inverse problem

TL;DR

This paper proves a uniqueness result for identifying polynomial reaction terms in 1D reaction-diffusion equations using minimal pointwise measurements, and demonstrates its practical numerical application.

Contribution

It establishes a new uniqueness criterion based on single-point measurements for inverse problems involving polynomial reaction terms in reaction-diffusion equations.

Findings

Uniqueness is guaranteed under specific pointwise measurement conditions.

Counter-examples show the optimality of the assumptions.

Numerical methods effectively determine polynomial reaction terms for degrees 2 and 3.

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 $(0,\epsilon).$ 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.