A Haar wavelets-based direct reconstruction method for the Cauchy problem of the Poisson equation

Abstract

In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part. The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely ill-posed. The new method is mathematically simple to implement and can be easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics. To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed. Comparing the numerical results produced by a meshless method based on the polynomial expansion with those produced by the proposed technique illustrates the superiority of the latter. Other numerical results show its effectiveness.