A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations

A wavelet multiscale iterative regularization method is proposed for the parameter estimation problems of partial differential equations. The wavelet analysis is introduced and a wavelet multiscale method is constructed based on the idea of hierarchical approximation. The inverse problem is decomposed into a sequence of inverse problems which rely on the scale variables and are solved successively according to the size of scale from the longest to the shortest. By combining multiscale approximations, the problem of local minimization is overcomed, and the computational cost is reduced. At each scale, based on the wavelet approximation, the problem of inverting the parameter is transformed into the problem of estimating the finite wavelet coefficients in the scale space. A novel iterative regularization method is constructed. The efficiency of the method is illustrated by solving the coefficient inverse problems of one- and two-dimensional elliptical partial differential equations.

[1]  P. Maass,et al.  Wavelet-Galerkin methods for ill-posed problems , 1996 .

[2]  Jochen Fröhlich,et al.  An Adaptive Wavelet-Vaguelette Algorithm for the Solution of PDEs , 1997 .

[3]  R. Ghanem,et al.  A wavelet-based approach for model and parameter identification of non-linear systems , 2001 .

[4]  Ailin Qian,et al.  A new wavelet method for solving an ill-posed problem , 2008, Appl. Math. Comput..

[5]  P. Maass,et al.  AN OUTLINE OF ADAPTIVE WAVELET GALERKIN METHODS FOR TIKHONOV REGULARIZATION OF INVERSE PARABOLIC PROBLEMS , 2003 .

[6]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[7]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[8]  Ju-Hong Lee,et al.  Comparison of generalization ability on solving differential equations using backpropagation and reformulated radial basis function networks , 2009, Neurocomputing.

[9]  Albert Cohen,et al.  Adaptive Wavelet Galerkin Methods for Linear Inverse Problems , 2004, SIAM J. Numer. Anal..

[10]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[11]  S. B. Childs,et al.  INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS. , 1968 .

[12]  Yi Sun,et al.  Adaptive wavelet-Galerkin methods for limited angle tomography , 2010, Image Vis. Comput..

[13]  G. Beylkin,et al.  On the representation of operators in bases of compactly supported wavelets , 1992 .

[14]  L. Eldén,et al.  Stability and convergence of the wavelet-Galerkin method for the sideways heat equation , 2000 .

[15]  Ling-Yun Chiao,et al.  Multiresolution parameterization for geophysical inverse problems , 2003 .

[16]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[17]  H.S. Fu,et al.  A wavelet multiscale method for the inverse problems of a two-dimensional wave equation , 2004 .

[18]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .

[19]  David Luengo,et al.  Novel fast random search clustering algorithm for mixing matrix identification in MIMO linear blind inverse problems with sparse inputs , 2012, Neurocomputing.

[20]  Fionn Murtagh,et al.  Wavelet-based nonlinear multiscale decomposition model for electricity load forecasting , 2006, Neurocomputing.

[21]  Jun Liu.  A Multiresolution Method for Distributed Parameter Estimation , 1993, SIAM J. Sci. Comput..

[22]  Giovanni Naldi,et al.  A wavelet-based method for numerical solution of nonlinear evolution equations , 2000 .

[23]  W. Dahmen.  Wavelet methods for PDEs — some recent developments , 2001 .

[24]  Y. Meyer.  Wavelets and Operators , 1993 .

[25]  Qiuqi Ruan,et al.  Shift and gray scale invariant features for palmprint identification using complex directional wavelet and local binary pattern , 2011, Neurocomputing.