2020/01/29 by Vo Anh Khoa, Khoa, Vo Anh, Pham Truong Hoang Nhan +1
Computer Science · Engineering · Mathematics · #35K92 #65J05 #65J20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2001.10656
openalex publication_date 2020/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the recent developments of regularization theory for inverse and ill-posed problems, a variational quasi-reversibility (QR) method has been designed to solve a class of time-reversed quasi-linear parabolic problems. Known as a PDE-based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward-like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second-order initial value problem for a wave-type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman-like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.