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

Published electronically:
August 26, 2004

MathSciNet review:
2137003

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and analyze a stable procedure for the approximation of where is the least residual norm solution of the minimal norm of the ill-posed equation , with compact operator between Hilbert spaces, and has some smoothness assumption. Our method is based on a finite number of singular values of 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 .

**1.**M. E. Davison,*A singular value decomposition for the Radon transform in 𝑛-dimensional Euclidean space*, Numer. Funct. Anal. Optim.**3**(1981), no. 3, 321–340. MR**629949**, 10.1080/01630568108816093**2.**A. Caponnetto and M. Bertero,*Tomography with a finite set of projections: singular value decomposition and resolution*, Inverse Problems**13**(1997), no. 5, 1191–1205. MR**1474363**, 10.1088/0266-5611/13/5/006**3.**Heinz W. Engl, Martin Hanke, and Andreas Neubauer,*Regularization of inverse problems*, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR**1408680****4.**C. W. Groetsch,*The theory of Tikhonov regularization for Fredholm equations of the first kind*, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**742928****5.**Bernd Hofmann,*Regularization for applied inverse and ill-posed problems*, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 85, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1986. A numerical approach; With German, French and Russian summaries. MR**906063****6.**M. M. Lavrentiev,*Some Improperly Posed Problems of Mathematical Physics*, Springer, New York, 1967.**7.**Alfred Karl Louis,*Inverse und schlecht gestellte Probleme*, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). MR**1002946****8.**A. K. Louis and P. Maass,*A mollifier method for linear operator equations of the first kind*, Inverse Problems**6**(1990), no. 3, 427–440. MR**1057035****9.**A. K. Louis,*Corrigendum: “Approximate inverse for linear and some nonlinear problems” [Inverse Problems 11 (1995), no. 6, 1211–1223; MR1361769 (96f:65068)]*, Inverse Problems**12**(1996), no. 2, 175–190. MR**1382237**, 10.1088/0266-5611/12/2/005**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 (Oberwolfach, 1996) Springer, Vienna, 1997, pp. 120–133. MR**1603903****11.**A. K. Louis,*A unified approach to regularization methods for linear ill-posed problems*, Inverse Problems**15**(1999), no. 2, 489–498. MR**1684469**, 10.1088/0266-5611/15/2/009**12.**P. Jonas and A. K. Louis,*Approximate inverse for a one-dimensional inverse heat conduction problem*, Inverse Problems**16**(2000), no. 1, 175–185. MR**1741235**, 10.1088/0266-5611/16/1/314**13.**M. Thamban Nair,*An iterated version of Lavrent′iev’s method for ill-posed equations with approximately specified data*, J. Inverse Ill-Posed Probl.**8**(2000), no. 2, 193–204. MR**1769213****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*, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR**856916****17.**Andreas Rieder and Thomas Schuster,*The approximate inverse in action with an application to computerized tomography*, SIAM J. Numer. Anal.**37**(2000), no. 6, 1909–1929 (electronic). MR**1766853**, 10.1137/S0036142998347619**18.**Andreas Rieder and Thomas Schuster,*The approximate inverse in action. II. Convergence and stability*, Math. Comp.**72**(2003), no. 243, 1399–1415 (electronic). MR**1972743**, 10.1090/S0025-5718-03-01526-6**19.**U. Tautenhahn,*On the method of Lavrentiev regularization for nonlinear ill-posed problems*, Inverse Problems**18**(2002), no. 1, 191–207. MR**1893590**, 10.1088/0266-5611/18/1/313**20.**Andrey N. Tikhonov and Vasiliy Y. Arsenin,*Solutions of ill-posed problems*, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR**0455365**

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:
http://dx.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