Inverse source problem for the hyperbolic equation with a time-dependent principal part

Abstract

In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

“…More precisely, in [45] the author considered the recovery of source terms of the form f (x)G(x, t), where G is known, while in [44] the analysis of the author is restricted to source terms of the form σ(t)f (x), with σ known. More recently, the approach of [45] has been extended by [28] to hyperbolic equations with time-dependent second order coefficients and to less regular coefficients by [46]. We mention also the work of [13,32] using similar approach for inverse source problems stated for parabolic equations and the result of [43] proved by a combination of geometrical arguments and Carleman estimates.…”

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

“…Now we are well prepared to recast Problem 1.1 into a minimization problem with the Tikhonov regularization where α > 0 denotes the regularization parameter. Unlike the formulation in [15,20], here we penalize the L 2 -norm of ∇p because one can expect certain smoothness of p as the second order coefficient. Meanwhile, there is no need to penalize the H 1 -norm of p due to the boundary condition p = h 0 on ∂Ω.…”

“…With some suitably given boundary condition and the compatibility condition, it is well known that the initial-boundary value problem governed by (1.3) admits a unique solution u(p) which depends continuously on the involved coefficients (see e.g. [15,19]). In order to prove the theoretical stability for Problem 1.1, we have to assume u(p) ∈ 2 k=0 H 4−k (−T, T ; H k (Ω)), (2.2) which satisfies the a priori estimate…”

An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

“…In particular, in [29] the ISP, which is similar with the one of the current paper, is considered, a numerical method is proposed and implemented. The numerical method of [29] is based on the optimization approach. The convergence of regularized solutions to the exact one is not proved in [29].…”

“…The numerical method of [29] is based on the optimization approach. The convergence of regularized solutions to the exact one is not proved in [29]. To contribute to the field, we propose in this paper a numerical method, which is not difficult to implement, without using the straight forward optimal control approach.…”