A fast multiscale Galerkin method for the first kind ill‐posed integral equations via Tikhonov regularization

References

• Beylkin , G. , Coifman , R. and Rokhlin , V. 1991 . Fast wavelet transforms and numerical algorithms I . Comm. Pure Appl. Math. , 44 : 141 – 183 .
   Web of Science ®Google Scholar
• Chen , Z. , Micchelli , C. A. and Xu , Y. 1997 . The Petrov-Galerkin methods for second kind integral equations II: Multiwavelet scheme . Adv. Comput. Math. , 7 : 199 – 233 .
   Web of Science ®Google Scholar
• Chen , Z. , Micchelli , C. A. and Xu , Y. 2002 . Discrete wavelet Petrov-Galerkin methods . Adv. Comput. Math. , 16 : 1 – 28 .
   Web of Science ®Google Scholar
• Chen , Z. , Micchelli , C. A. and Xu , Y. 2002 . Fast collocation methods for second kind integral equations . SIAM J. Numer. Anal. , 40 : 344 – 375 .
   Web of Science ®Google Scholar
• Chen , Z. , Wu , B. and Xu , Y. 2005 . Multilevel augmentation methods for solving operator equations . Numer. Math. J. Chinese Univ. , 14 : 31 – 55 .
   Google Scholar
• Chen , Z. , Xu , Y. and Yang , H. 2006 . Multilevel augmentation methods for solving ill-posed operator equations . Inverse Problems , 22 : 155 – 174 .
   Web of Science ®Google Scholar
• Chen , Z. , Xu , Y. and Yang , H. Fast collocation methods for solving Fredholm integral equations of the first kind . osubmitted for publication ,
   Google Scholar
• Dahmen , W. , Proessdorf , S. and Schneider , R. 1993 . Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solutions . Adv. Comput. Math. , 1 : 259 – 335 .
   Google Scholar
• Dicken , V. and Maass , P. 1996 . Wavelet–Galerkin methods for ill-posed problems . J. Inverse Ill-posed Probl. , 4 : 203 – 221 .
   Google Scholar
• Engl , H. W. and Neubauer , A. 1985 . Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind . Int. Ser. Numeri. Math. , 73 : 120 – 141 .
   Google Scholar
• Engl , H. W. , Hanke , M. and Neubauer , A. 1996 . Regularization of Inverse Problems , Dordrecht : Kluwer .
   Google Scholar
• George , S. and Nair , M. T. 1994 . Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates . J. Optim. Theory Appl. , 83 : 217 – 222 .
   Web of Science ®Google Scholar
• Gfrerer , H. 1987 . An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates . Math. Comput. , 49 : 507 – 522 .
   Web of Science ®Google Scholar
• Groetsch , C. W. 1984 . The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics 105 , Boston : Pitman .
   Google Scholar
• Guacaneme , J. E. 1988 . An optimal parameter choice for regularized ill-posed problems . Integr. Equat. Oper. Th. , 11 : 610 – 613 .
   Web of Science ®Google Scholar
• Hämarik , U. 1992 . “ On the discretization error in regularized projection methods with parameter choice by discrepancy principle ” . In Ill-posed Problems in Natural Sciences , Edited by: Tikhonov , A. N. 24 – 29 . Moscow : VSP, Utrecht/TVP .
   Google Scholar
• Hämarik , U. 1992 . Quasioptimal error estimate for the regularized Ritz-Galerkin method with the a-posteriori choice of the parameter . Acta et Comment. Univ. Tartuensis , 937 : 63 – 76 .
   Google Scholar
• Hämarik , U. 1993 . On the parameter choice in the regularized Ritz-Galerkin method . Proc. Estonian Acad. Sci. Phys.–Math. , 42 : 133 – 143 .
   Google Scholar
• Harbrecht , H. , Pereverzev , S. and Schneider , R. 2003 . Self-regularization by projection for noisy pseudodifferential equations of negative order . Numer. Math. , 95 : 123 – 143 .
   Web of Science ®Google Scholar
• Kaltenbacher , B. 2001 . On the regularizing properties of a full multigrid method for ill-posed problems . Inverse Problems , 17 : 767 – 788 .
   Web of Science ®Google Scholar
• Maass , P. , Pereverzev , S. V. , Ramlau , R. and Solodky , S. G. 2001 . An adaptive discretization for Tikhonov-Phillips regularization with a posteriori parameter selection . Numer. Math. , 87 : 485 – 502 .
   Web of Science ®Google Scholar
• Micchelli , C. A. and Xu , Y. 1998 . Using the matrix refinement equation for the construction of wavelets on invariant sets . Appl. Comput. Harmon. Anal. , 1 : 391 – 401 .
   Google Scholar
• Nair , M. T. 1992 . A generalization of Arcangeli's method for ill-posed problems leading to optimal rates . Integr. Equat. Oper. Th. , 15 : 1042 – 1046 .
   Web of Science ®Google Scholar
• Nair , M. T. and Rajan , M. P. 2001 . Arcangeli's discrepancy principle for a modified projection scheme for ill-posed problems . Numer. Funct. Anal. Optimiz. , 22 : 177 – 198 .
   Web of Science ®Google Scholar
• Nair , M. T. and Schock , E. 1998 . A discrepancy principle for Tikhonov regularization with approximately specified data . Ann. Polon. Math. LXIX , 3 : 197 – 205 .
   Google Scholar
• Plato , R. 1997 . The Galerkin scheme for Lavrentiev's m-times iterated method to solve linear accretive Volterra integral equations of the first kind . BIT , 37 : 404 – 423 .
   Web of Science ®Google Scholar
• Rieder , A. 1997 . A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization . Numer. Math. , 75 : 501 – 522 .
   Web of Science ®Google Scholar