Jin, Bangti;
Zhou, Zhi;
(2023)
Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis.
IMA Journal of Numerical Analysis
, 43
(3)
pp. 2496-2531.
10.1093/imanum/drac051.
Preview |
Text
2109.06789.pdf - Accepted Version Download (1MB) | Preview |
Abstract
In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(\varOmega ))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell ^2(L^2(\varOmega ))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.
| Type: | Article |
|---|---|
| Title: | Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imanum/drac051 |
| Publisher version: | https://doi.org/10.1093/imanum/drac051 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10170812 |
Archive Staff Only
 |
View Item |
