Tran Nhan Tam Quyen

Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I

We investigate the convergence rates for total variation regularization of the problem of identifying (i) the coefficient q in the Neumann problem for the elliptic equation , and (ii) the coefficient a in the Neumann problem for the elliptic equation , when u is imprecisely given by z? in . We regularize these problems by correspondingly minimizing the convex functionals and over the admissible sets, where U(q) (U(a)) is the solution of the first (second) Neumann boundary value problem; ? > 0 is the regularization parameter. Taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems, we obtain the convergence rates of them to a total variation-minimizing solution in the sense of the Bregman distance under relatively simple source conditions without the smallness requirement on the source functions.

Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations

We investigate the convergence rates for Tikhonov regularization of the problem of identifying (1) the coefficient q L fty(?) in the Dirichlet problem ?div(q?u) = f in ?, u = 0 on ??, and (2) the coefficient a L fty(?) in the Dirichlet problem ??u + au = f in ?, u = 0 on ??, when u is imprecisely given by z? H10(?), , We regularize these problems by correspondingly minimizing the strictly convex functionals and where U(q) (U(a)) is the solution of the first (second) Dirichlet problem, ? > 0 is the regularization parameter and q* (or a*) is an a priori estimate of q (or a). We prove that these functionals attain a unique global minimizer on the admissible sets. Further, we give very simple source conditions without the smallness requirement on the source functions which provide the convergence rate for the regularized solutions.

Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem

We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation

Matrix coefficient identification in an elliptic equation with the convex energy functional method

{{-{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}

Convergence rates for total variation regularization of coefficient identification problems in elliptic equations II

on the boundary

Variational method for multiple parameter identification in elliptic PDEs

{{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}