A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data. (4th July 2019)

• Record Type: Journal Article
• Authors: Hinze, Michael; Hofmann, Bernd; Quyen, Tran Nhan Tam
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 9(2019)
• Page Start: 1080
• Page End: 1112
• Publication Date: 2019-07-04
• Subjects: Conjugate gradient method -- convergence rates -- Dirichlet problem -- finite element method -- ill-posedness -- inverse source problem -- Neumann problem -- source condition -- Tikhonov and Lavrentiev regularization; 35R25 -- 47A52 -- 35R30 -- 65J20 -- 65J22; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1596953
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this article, we investigate the problem of identifying the source term f in the elliptic system − ∇ · ( Q ∇ Φ ) = f in Ω ⊂ R d, d ∈ { 2, 3 }, Q ∇ Φ · n → = j on ∂ Ω and Φ = g on ∂ Ω from a single noisy measurement couple ( j δ, g δ ) of the Neumann and Dirichlet data ( j, g ) with noise level δ > 0 . In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn–Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to a sought source function. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution, we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.