2018/03/13 by Nguyen Huy Tuan, Tuan, Nguyen Huy, Vo Anh Khoa +3
Computer Science · Engineering · Mathematics · #35K92 #65J05 #65J20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1803.04641
openalex publication_date 2018/03/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This paper presents a modified quasi-reversibility method for computing the\nexponentially unstable solution of a nonlocal terminal-boundary value parabolic\nproblem with noisy data. Based on data measurements, we perturb the problem by\nthe so-called filter regularized operator to design an approximate problem.\nDifferent from recently developed approaches that consist in the conventional\nspectral methods, we analyze this new approximation in a variational framework,\nwhere the finite element method can be applied. To see the whole skeleton of\nthis method, our main results lie in the analysis of a semi-linear case and we\ndiscuss some generalizations where this analysis can be adapted. As is\nomnipresent in many physical processes, there is likely a myriad of models\nderived from this simpler case, such as source localization problems for brain\ntumors and heat conduction problems with nonlinear sinks in nuclear science.\nWith respect to each noise level, we benefit from the Faedo-Galerkin method to\nstudy the weak solvability of the approximate problem. Relying on the\nenergy-like analysis, we provide detailed convergence rates in L2-H1 of\nthe proposed method when the true solution is sufficiently smooth. Depending on\nthe dimensions of the domain, we obtain an error estimate in Lr for some\nr>2. Proof of the backward uniqueness for the quasi-linear system is also\ndepicted in this work. To prove the regularity assumptions acceptable, several\nphysical applications are discussed.\n