Regularization of 2D Backward Heat Equation Using Heatlets

Heatlets were introduced by Shen and Strang [1999. Journal of Differential Equations, 161, pp 403–421] to solve the heat equation using wavelet expansion of the initial data. In this paper, we regularize the 2D backward heat equation and, in general, ill-posed Cauchy problems using heatlets. We use the quasi-reversibility method to find approximate solutions to ill-posed Cauchy problems and then apply Hölder-continuous dependence inequalities obtained by Ames and Hughes [2005, Semigroup Forum, 70, pp. 127–145] to relate the solutions of original and approximate problem.