2021/01/17 by Barbara Kaltenbacher, Kaltenbacher, Barbara, William Rundell +1 · 1 citation
Computer Science · Mathematics · #35K15 #35K58 #35R30 #80A23 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2101.06696
openalex publication_date 2021/01/17 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
The problem of recovering coefficients in a diffusion equation is one of the\nbasic inverse problems. Perhaps the most important term is the one that couples\nthe length and time scales and is often referred to as it the / diffusion\ncoefficient a in ut - \∇(a\∇ u) = f. In this paper we seek the\nunknown a assuming that a=a(u) depends only on the value of the solution at\na given point. Such diffusion models are the basic of a wide range of physical\nphenomena such as nonlinear heat conduction, chemical mixing and population\ndynamics. We shall look at two types of overposed data in order to effect\nrecovery of a(u): the value of a time trace u(x0,t) for some fixed point\nx0 on the boundary of the region \Ω; or the value of u on an\ninterior curve \Σ lying within \Ω. As examples, these might\nrepresent a temperature measurement on the boundary or a census of the\npopulation in some subset of \Ω taken at a fixed time T>0. In the\nlatter case we shall show a uniqueness result that leads to a constructive\nmethod for recovery of a. Indeed, for both types of measured data we shall\nshow reconstructions based on the iterative algorithms developed in the paper.\n