Convexification Numerical Method for a Coefficient Inverse Problem for the Riemannian Radiative Transfer Equation

Convexification for the viscocity solution for a coefficient inverse problem for the radiative transfer equation

“…(a) Since smallness assumptions are not imposed on the number R > 0 and since v 0 ∈ B is an arbitrary point, then theorem 5.3 guarantees the global convergence of the gradient descent method (5.60) and (5.61). (b) Even though the requirement of our theory is that the parameter λ of the Carleman Weight Function e 2λz 2 should be sufficiently large, we have observed in computational experiments of section 6 that λ = 5 is sufficient, which is the same as in two previous publications of this group [21,22]. Similar observations about reasonable values of λ ∈ [1,3] were made in other publications about the convexification method [20,23].…”

“…The following existence and uniqueness theorem for the forward problem was proven in [21], and a similar theorem was proven in [22] for the Riemannian analog of RTE:…”

“…[20] for a summary of results as of 2021. In particular, in two most recent publications they have solved by convexification CIPs for two versions of the RTE [21,22]. In both these works one obtains first a boundary value problem (BVP) for a nonlinear PDE of the second order, which does not contain the unknown coefficient.…”

“…Publications [21,22] represent the first numerical solutions of CIPs for both RTE [21] and its Riemannian version [22]. Previous numerical results were obtained only for the inverse source problems for RTE in [8][9][10]26].…”