Heuristic regularization methods for numerical differentiation

AbstractIn this paper, we use smoothing splines to deal with numerical differentiation. Some heuristic methods for choosing regularization parameters are proposed, including the L-curve method and the de Boor method. Numerical experiments are performed to illustrate the efficiency of these methods in comparison with other procedures

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate

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

AbstractWe 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 −div(q∇u)=f in Ω, q∂u/∂n=g on ∂Ω, (ii) the coefficient a in the Neumann problem for the elliptic equation −Δu+au=f in Ω, ∂u/∂n=g on ∂Ω, Ω⊂Rd, d⩾1, when u is imprecisely given by zδ∈H1(Ω), ‖u−zδ‖H1(Ω)⩽δ, δ>0. We regularize these problems by correspondingly minimizing the strictly convex functionals12∫Ωq|∇(U(q)−zδ)|2dx+ρ(12‖q‖L2(Ω)2+∫Ω|∇q|), and12∫Ω|∇(U(a)−zδ)|2dx+12∫Ωa(U(a)−zδ)2dx+ρ(12‖a‖L2(Ω)2+∫Ω|∇a|) over admissible sets, where U(q) (U(a)) is the solution of the first (second) Neumann boundary value problem, ρ>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 the solution of the inverse problem in the sense of the Bregman distance and in the L2-norm under relatively simple source conditions without the smallness requirement on the source functions

Towards Tikhonov regularization of non-linear ill-posed problems: a dc programming approach

International audienc

On the ill-posedness of the trust region subproblem

International audienc

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

International audienc

D.C. (Difference of Convex) Functions Programming for Solving an Inverse Problem for an Elliptic Equation

International audienc