Stability rates for linear ill-posed problems with compact and non-compact operators.

*(English)*Zbl 0941.65055Two types of ill-posedness of linear operator equations \(Ax= y\) in Hilbert spaces are considered: type I, if \(A\) is not compact, but for the range \(R(A)\) we have \(R(A)\neq \text{cl }R(A)\), and type II for compact operators \(A\). Ill-posedness of type I means that range \(R(A)\) contains a closed infinite-dimensional subspace. Conditions are presented that can be ensured for ill-posed equations of type II by appropriate choice of an orthonormal system, but for ill-posed equations of type I are never satisfied. Since noncompact operators \(A\) have no singular values, the authors introduce stability rates in order to have a common measure for both, compact and noncompact ones. Properties of these rates are analyzed for compact convolution operators and for noncompact multiplication operators \(A\).

Reviewer: R.Lepp (Tallinn)

##### MSC:

65J10 | Numerical solutions to equations with linear operators |

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

45B05 | Fredholm integral equations |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

65R20 | Numerical methods for integral equations |

65R30 | Numerical methods for ill-posed problems for integral equations |

##### Keywords:

ill-posed problems; linear operator equations; Hilbert spaces; compact operators; noncompact operators; compact convolution operators; noncompact multiplication operators
PDF
BibTeX
XML
Cite

\textit{B. Hofmann} and \textit{G. Fleischer}, Z. Anal. Anwend. 18, No. 2, 267--286 (1999; Zbl 0941.65055)

**OpenURL**

##### References:

[1] | Baumeister, J.: Stable Solution of Inverse Problems. Braunschweig: Vieweg 1987. (2] Berg, L.: Operatorenrechnung. Part II: Funktionentheoretische Methoden. Berlin: Dt. Verlag Wiss. 1974. |

[2] | Demmel, J. W.: On condition numbers and the distance to the nearest ill-posed problem. Numer. Math. 51(1987), 251 - 289. · Zbl 0597.65036 |

[3] | [5] EngI, H. W., Hanke, M. and A. Neubauer: Regularization of Inverse Problems. Dordrecht: Kluwer 1996. |

[4] | Fleischer, C.: Multiplication operators and its ill-posedness properties. Preprint. Chem- nitz: Techn. Univ., Fac. Math. Preprint 97-26 (1997), 1 - 27. |

[5] | Gorenflo, R. and B. Hofmann: On autoconvolution and regularization. Inverse Problems 10 (1994), 353-373. · Zbl 0804.45003 |

[6] | Hansen, P. C.: Rank-Deficient and Discrete Ill-Posed Problems. Philadelphia: SIAM 1998. |

[7] | Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems. Leipzig: B. G. Teubner 1986. · Zbl 0606.65038 |

[8] | Hofmann, B. and U. Tautenhahn: On ill-posedness measures and space change in Sobolev scales. Z. Anal. Anw. 16 (1997), 979 - 1000. · Zbl 0896.65043 |

[9] | Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer 1996. · Zbl 0865.35004 |

[10] | Kress, R.: Linear Integral Equations. Berlin: Springer 1989. · Zbl 0671.45001 |

[11] | Liu, J., Guerrier, B. and C. Bernard: A sensitivity decomposition for the regularized solution of inverse heat conduction problems by wavelets. Inverse Problems 11 (1995), 1177- 1187. · Zbl 0844.35135 |

[12] | Louis, A.: Inverse und schlecht gestellte Probleme. Stuttgart: B. G. Teubner 1989. · Zbl 0667.65045 |

[13] | Nashed, M. Z.: A new approach to classification and regularization of ill-posed operator equations. In: Inverse and Ill-posed Problems (eds.: H. W. Engl and C. W. Groetsch). Orlando: Acad. Press 1987, pp. 53 - 75. · Zbl 0647.47013 |

[14] | Neubauer, A.: On converse and saturation results for Tikhonov regularization of linear ill-posed problines. SIAM J. Numer. Anal. 34 (1997), 517 - 527. · Zbl 0878.65038 |

[15] | Rudin, W.: Functional Analysis, 2nd ed. New York: McGraw-hill 1991. · Zbl 0867.46001 |

[16] | Schock, E.: What are the proper condition numbers of discretized ill-posed problems ? Lin. AIg. AppI. 81(1986), 129 - 136. · Zbl 0608.65032 |

[17] | Vainikko, C.: On the discretization and regularization of ill-posed problems with non- compact operators. Numer. Funct. Anal. Optimiz. 13 (1992), 381 - 396. · Zbl 0759.65030 |

[18] | Vu Kim Tuan and R. Gorenflo: Asymptotics of singular values of fractional integral oper- ators. Inverse Problems 10 (1994), 949 - 955. · Zbl 0808.45001 |

[19] | Wahba, C.: Ill-posed problems: Numerical and statistical methods for mildly, moderately and severely ill-posed problems with noisy data. Report. Madison: University of Wiscon- sin, Techn. Rep. No. 595 (1980), 1 - 69. |

[20] | Wing, G.M.: Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind. J. mt. Equ. 9 (Suppl.) (1985), 191 - 204. · Zbl 0578.65134 |

[21] | Wolfersdorf, L. von and J. Janno: On Tskhonov regularization for identifying memory kernels in heat conduction and viscoelasticity. Freiberg: Tcchn. Univ. Bergakademie, F’ac. Math. Inf. Preprint 98-01 (1998), 1 - 19. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.