Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A finite dimensional realization of the mollifier method for compact operator equations


Authors: M. T. Nair and Shine Lal
Journal: Math. Comp. 74 (2005), 1281-1290
MSC (2000): Primary 65J10; Secondary 65R10
DOI: https://doi.org/10.1090/S0025-5718-04-01707-7
Published electronically: August 26, 2004
MathSciNet review: 2137003
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and analyze a stable procedure for the approximation of $\langle f^\dagger, \varphi\rangle $ where $f^\dagger$ is the least residual norm solution of the minimal norm of the ill-posed equation $Af=g$, with compact operator $A:X\to Y$ between Hilbert spaces, and $\varphi\in X$ has some smoothness assumption. Our method is based on a finite number of singular values of $A$ and some finite rank operators. Our results are in a more general setting than the one considered by Rieder and Schuster (2000) and Nair and Lal (2003) with special reference to the mollifier method, and it is also applicable under fewer smoothness assumptions on $\varphi$.


References [Enhancements On Off] (What's this?)

  • 1. M.E. Davison, A singular value decomposition for the Radon transform in $n$-dimensional Euclidian space, Numer. Funct. Anal. Optim., 3 (1981), 321-340. MR 83c:65274
  • 2. A. Caponnetto and M. Bertero, Tomography with a finite set of projections, Inverse problems, 13 (1997), 1191-1205. MR 98j:44001
  • 3. H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrect, 1996. MR 97k:65145
  • 4. C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman Publishing, Boston, London, Melbourne, 1984. MR 85k:45020
  • 5. B. Hofmann, Regularization of Applied Inverse and Ill-Posed Problems, Leipzig: Teubner, 1986. MR 88i:65001
  • 6. M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer, New York, 1967.
  • 7. A.K. Louis, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989. MR 90g:65075
  • 8. A.K. Louis and P. Maass, A mollified method for linear operator equation of first kind, Inverse problems, 6 (1990), 427-440.MR 91g:65130
  • 9. A.K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse problems, 12 (1996), 175-190. MR 96m:65063
  • 10. A.K. Louis, Application of the approximate inverse to 3D X-ray CT and Ultrasound Tomography, Inverse problems in medical imaging and nondestructive testing; Springer- Verlag Wien, New York. Eds: H.W. Engl, A.K. Louis and W. Rendell (1997), 120-133. MR 99c:65113
  • 11. A.K. Louis, A unified approach to regularisation methods for linear ill-posed problems, Inverse problems, 15 (1999), 489-498. MR 2000b:65111
  • 12. A.K. Louis, P.Jonas, Approximate inverse for one dimensional inverse heat conduction problem, Inverse problems, 16 (2000), 175-185.MR 2000k:35290
  • 13. M.T. Nair, An iterated version of Lavrentiev's method for ill-posed equations with approximately specified data, J. Inverse and Ill-Posed Problems, 8(2) (2000), 193-204. MR 2001f:65072
  • 14. M.T. Nair, Functional Analysis: A First Course, Prentice-Hall of India, New Delhi, 2002.
  • 15. M.T. Nair and Shine Lal, Finite Dimensional Realization of Mollifier Method: A New Stable Approach, J. Inverse and Ill-Posed Problems 12(5) (2004), 1-7.
  • 16. F. Natterer, The Mathematics of Computerized Tomography, Wiley, Chichester, UK, 1986. MR 88m:44008
  • 17. A. Rieder and T. Schuster, The Approximate Inverse in Action with an Application to Computerized Tomography, SIAM.J. Numer. Anal., 37 (2000), 1909-1929. MR 2001a:65072
  • 18. A. Rieder and T. Schuster, The Approximate Inverse in Action II: Convergence and Stability, Math. Comput., 72 (2003), 1399-1415. MR 2004c:65059
  • 19. U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), 191-207. MR 2002m:47079
  • 20. A. N. Tikhonov and V.Y. Arsenin, Solution of Ill-Posed Problems, Wiley, New York, 1977. MR 56:13604

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65J10, 65R10

Retrieve articles in all journals with MSC (2000): 65J10, 65R10


Additional Information

M. T. Nair
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
Email: mtnair@iitm.ac.in

Shine Lal
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
Email: lalshine@hotmail.com

DOI: https://doi.org/10.1090/S0025-5718-04-01707-7
Received by editor(s): December 10, 2003
Published electronically: August 26, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society