On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

TL;DR

This paper investigates the uniqueness and reconstruction of a nonlinear diffusion coefficient in a parabolic PDE, using boundary or interior measurements, and proposes iterative algorithms for recovering the coefficient.

Contribution

It establishes uniqueness results and develops constructive iterative algorithms for recovering a nonlinear diffusion coefficient depending on the solution.

Findings

Proves uniqueness of the diffusion coefficient from boundary or interior data.

Develops iterative algorithms for reconstructing the nonlinear diffusion term.

Provides constructive methods for practical recovery of the coefficient.

Abstract

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as {\it the\/} diffusion coefficient $a$ in $u_t - \nabla(a\nabla u) = f$. In this paper we seek the unknown $a$ assuming that $a=a(u)$ depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of $a(u)$: the value of a time trace $u(x_0,t)$ for some fixed point $x_0$ on the boundary of the region $\Omega$; or the value of $u$ on an interior curve $\Sigma$ lying within $\Omega$. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of $\Omega$ taken at a fixed time $T>0$. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of $a$. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.