Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method. Issue 4 (2nd April 2020)

• Record Type: Journal Article
• Authors: Li, Qitong; Nguyen, Loc Hoang
• Is Part Of: Inverse problems in science and engineering. Volume 28:Issue 4(2020)
• Page Start: 580
• Page End: 598
• Publication Date: 2020-04-02
• Subjects: Initial condition -- parabolic equation -- truncated Fourier series -- approximation -- convergence -- quasi-reversibility method; 35R30 -- 35K15; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2019.1643850
• Languages: English
• ISSNs: 1741-5977

Abstract

ABSTRACT: We consider the problem of computing the initial condition for a general parabolic equation from the Cauchy lateral data. The stability of this problem is well-known to be logarithmic. In this paper, we introduce an approximate model, as a coupled linear system of elliptic partial differential equations. Solution to this model is the vector of Fourier coefficients of the solutions to the parabolic equation above. This approximate model is solved by the quasi-reversibility method. We will prove the convergence for the quasi-reversibility method as the measurement noise tends to 0. The convergent rate is Lipschitz. We present the implementation of our algorithm in details and verify our method by showing some numerical examples.