A preconditioning method of ill conditioned matrices using wavelet bases

After discretizations with respect to two different wavelet bases of the partial differential equations (PDEs), we obtain a big sparse ill-conditioned linear system of equations. For discretizing of PDEs with wavelet method , this paper present a preconditioning techinque for linear systems involving the operator such that the system becomes a sparse systems in the wavelet basis. In fact  the condition number of the matrix involved in the solution of PDEs, after a diagonal preeonditioning appears to be bounded. The orthogonal property of the wavelets is used to cnstruct efficient  iterative methods for the solution of the resultant linear algebraic systems.