Regularized solution for nonlinear elliptic equations with random discrete data

Abstract

The aim of this paper is to study the Cauchy problem of determining a solution of nonlinear elliptic equations with random discrete data. A study showing that this problem is severely ill posed in the sense of Hadamard, ie, the solution does not depend continuously on the initial data. It is therefore necessary to regularize the in‐stable solution of the problem. First, we use the trigonometric of nonparametric regression associated with the truncation method in order to offer the regularized solution. Then, under some presumption on the true solution, we give errors estimates and convergence rate in L2‐norm. A numerical example is also constructed to illustrate the main results.