Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data

Abstract

This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Numerical examples involving smooth and singular data are presented.

Combined energy method and regularization to solve the Cauchy problem for the heat equation

“…Notice that this constant vanishes in case of noise-free compatible Cauchy data. In [11,14,15], the authors determine this threshold in order to propose a stopping criterion depending on the noise level, which avoids the instability by stopping the iterative procedure first time the criterion is satisfied.…”

“…[1][2][3][4] We propose a specific regularization procedure for the Cauchy data and the identified data provided by the energy-like method. This latter was introduced in [5][6][7][8][9][10][11][12] for different applications of elliptic partial differential equations and in [13][14][15] for the parabolic heat equation and damped elastodynamic one. In this approach, two distinct fields were introduced, they are solutions of two wellposed problems, each of them meeting only one of the overspecified data and has an unknown boundary condition (Dirichlet or Neumann).…”

Regularization of the noisy Cauchy problem solution approximated by an energy‐like method

“…From the trace theorem, we know the functional spaces containing the Cauchy data and the associated norms [24]. In the sequel, these norms will be be approximated by the norm in the space of squared integrable functions on…”

“…A numerical validation of these estimates was performed in [24]. Then when the functional attains its threshold, its variations fall by an order of magnitude below the functional itself.…”

An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

“…There are lots of literature on both the theoretical and methodological developments in this field. For theoretical aspects, the readers can refer to [6,[17][18][19][20]. For computational aspects, the readers can refer to [5,[21][22][23][24][25][26][27][28].…”