Application of Wavelet Shrinkage to Solving the Sideways Heat Equation

Abstract

The sideways heat equation is considered in terms of the ill-posed operator equation Au = g, g ∈ R(A) ⊂ L ²(ℝ), with a given noisy right hand side. For a reconstruction of the solution from indirect data the dual least square method generated by the family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found under which convergence results including error estimation are obtained. Next, a certain simple nonlinear modification of the method based on local refinements of the wavelet expansion of the noisy data is investigated. Moreover, it is shown that an accuracy of the solution reconstruction can be improved by adding some sufficiently large coefficients on the level higher than that indicated by the convergence theorem.