Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} u_t (x,y,t) = u_{xx} (x,y,t) + u_{yy} (x,y,t), x \in \mathbb{R}, y \in \mathbb{R}, 0 \leqslant t < 1, \hfill \\ u(x,y,1) = \phi (x,y), x \in \mathbb{R},y \in \mathbb{R}. \hfill \\ \end{gathered} \right.$$\end{document} Motivated by Regińska’s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.