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

Identification of the forcing term in hyperbolic equations

We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite-difference methods. Numerical results are presented and discussed for one and two-dimensional problems