A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log -type error estimate under an additional source condition. Numerical results show that our method is feasible.

[1]  Xiang-Tuan Xiong,et al.  Semidiscrete central difference method in time for determining surface temperatures , 2005, Int. J. Math. Math. Sci..

[2]  H. Engl,et al.  A posteriori parameter choice for general regularization methods for solving linear ill-posed problems , 1988 .

[3]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[4]  A. Neubauer.  An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error , 1988 .

[5]  Jin Qi-nian,et al.  On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems , 1999 .

[6]  R. Gorenflo,et al.  Fractional diffusion: probability distributions and random walk models , 2002 .

[7]  Qinian Jin,et al.  On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems , 1999, Numerische Mathematik.

[8]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[9]  Ulrich Tautenhahn.  Optimal Stable Solution of Cauchy Problems for Elliptic Equations , 1996 .

[10]  M. Z. Nashed,et al.  On weakly bounded noise in ill-posed problems , 2009 .

[11]  Thomas Bonesky.  Morozov's discrepancy principle and Tikhonov-type functionals , 2008 .

[12]  G. Blanchard,et al.  Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration , 2012 .

[13]  Xiang-Tuan Xiong,et al.  A spectral regularization method for solving surface heat flux on a general sideways parabolic , 2008, Appl. Math. Comput..

[14]  Shaher Momani,et al.  Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation , 2007 .

[15]  S. George,et al.  An a posteriori parameter choice for simplified regularization of ill-posed problems , 1993 .

[16]  Peter Mathé,et al.  Regularization under general noise assumptions , 2011 .

[17]  H. Engl.  Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates , 1987 .

[18]  R. Gorenflo,et al.  Some recent advances in theory and simulation of fractional diffusion processes , 2007, 0801.0146.

[19]  Alexandra Smirnova,et al.  ON APPLICATION OF GENERALIZED DISCREPANCY PRINCIPLE TO ITERATIVE METHODS FOR NONLINEAR ILL-POSED PROBLEMS , 2005 .

[20]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[21]  Ting Wei,et al.  Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem , 2010 .

[22]  B. Hofmann,et al.  Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation , 2013 .

[23]  D. Colton,et al.  A simple method using Morozov's discrepancy principle for solving inverse scattering problems , 1997 .

[24]  R. Ramlau,et al.  Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators , 2010 .

[25]  A. Leonov.  Approximate calculation of a pseudoinverse matrix using a generalized discrepancy principle , 1986 .

[26]  Ulrich Tautenhahn,et al.  Morozov's Discrepancy Principle under General Source Conditions , 2003 .

[27]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[28]  Otmar Scherzer,et al.  The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems , 1993, Computing.

[29]  Sergei V. Pereverzev,et al.  Morozov's discrepancy principle for tikhonov , 2000 .

[30]  B. Hofmann,et al.  Optimization aspects of the generalized discrepancy principle in regularization , 1986 .