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 {ut(x,y,t)=uxx(x,y,t)+uyy(x,y,t),x.R,y.R,0.t<1,u(x,y,1)=.(x,y) ,x.R,y.R. Motivated by Regi.ska.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.