An inverse potential problem for subdiffusion: stability and reconstruction*The work of B Jin is supported by UK EPSRC Grant EP/T000864/1, and the research of Z Zhou is supported by Hong Kong RGC Grant (No. 15304420). (3rd December 2020)

• Record Type: Journal Article
• Authors: Jin, Bangti; Zhou, Zhi
• Is Part Of: Inverse problems. Volume 37:Number 1(2021)
• Publication Date: 2020-12-03
• Subjects: inverse potential problem -- subdiffusion -- stability -- numerical reconstruction; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abb61e
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this work, we study the inverse problem of recovering a potential coefficient in the subdiffusion model, which involves a Djrbashian–Caputo derivative of order α ∈ (0, 1) in time, from the terminal data. We prove that the inverse problem is locally Lipschitz for small terminal time, under certain conditions on the initial data. This result extends the result in [6 ] for the standard parabolic case to the fractional case. The analysis relies on refined properties of two-parameter Mittag–Leffler functions, e.g., complete monotonicity and asymptotics. Further, we develop an efficient and easy-to-implement algorithm for numerically recovering the coefficient based on (preconditioned) fixed point iteration and Anderson acceleration. The efficiency and accuracy of the algorithm is illustrated with several numerical examples.