##
**A direct method for nonlinear ill-posed problems.**
*(English)*
Zbl 1453.65125

Author’s abstract: We propose a direct method for solving nonlinear ill-posed problems in Banach-spaces. The method is based on a stable inversion formula we explicitly compute by applying techniques for analytic functions. Furthermore, we investigate the convergence and stability of the method and prove that the derived noniterative algorithm is a regularization. The inversion formula provides a systematic sensitivity analysis. The approach is applicable to a wide range of nonlinear ill-posed problems. We test the algorithm on a nonlinear problem of travel-time inversion in seismic tomography. Numerical results illustrate the robustness and efficiency of the algorithm.

Reviewer: Xiaolong Qin (Chengdu)

### MSC:

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

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

65J15 | Numerical solutions to equations with nonlinear operators |

86A15 | Seismology (including tsunami modeling), earthquakes |

### Software:

KAIRUAIN
PDF
BibTeX
XML
Cite

\textit{A. Lakhal}, Inverse Probl. 34, No. 2, Article ID 025002, 17 p. (2018; Zbl 1453.65125)

Full Text:
DOI

### References:

[1] | Abraham R and Robin J 1967 Transversal Mapping and Flows (New York: Benjamin) |

[2] | Abazid M A, Lakhal A and Louis A K 2016 Inverse design of anti-reflection coatings using the nonlinear approximate inverse Inverse Problems Sci. Eng.24 917-35 · Zbl 1342.65237 |

[3] | Arridge S R and Schotland J C 2009 Optical tomography: forward and inverse problems Inverse Problems25 123010 · Zbl 1188.35197 |

[4] | Arridge S R, Moskow S and Schotland J C 2012 Inverse born series for the Calderon problem Inverse Problems28 035003 · Zbl 1238.35181 |

[5] | Bakushinsky A B and Kokurin M Y 2005 Iterative Methods for Approximate Solution of Inverse Problems(Mathematics and its Applications) (Netherlands: Springer) · Zbl 1194.76259 |

[6] | Kaltenbacher B, Neubauer A and Scherzer O 2008 Iterative Regularization Methods for Nonlinear Ill-Posed Problems(Radon Series on Computational and Applied Mathematics) (Germany: de Gruyter) · Zbl 1145.65037 |

[7] | Beilina L and Klibanov M V 2012 Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems (New York: Springer) · Zbl 1255.65168 |

[8] | Comtet L 1974 Advanced Combinatorics (Dodrecht: D. Reidel Publishing Company) · Zbl 0283.05001 |

[9] | Devaney A J and Wolf E 1982 A new perturbation expansion for inverse scattering from three dimensional finite range potentials Phys. Lett. A 89 269-72 |

[10] | Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems(Mathematics and its Applications) (Dordrecht: Kluwer) · Zbl 0859.65054 |

[11] | Hofmann B 2013 Regularization for Applied Inverse and Ill-Posed Problems: a Numerical Approach(Teubner-Texte zur Mathematik) (Germany: Vieweg + Teubner) |

[12] | Hofmann B and Scherzer O 1994 Factors influencing the ill-posedness of nonlinear problems Inverse Problems10 1277 · Zbl 0809.35157 |

[13] | Hofmann B and Scherzer O 1998 Local ill-posed and source conditions of operators in Hilbert spaces Inverse Problems14 1189 |

[14] | Jost R and Kohn W 1952 Construction of a potential from a phase shift Phys. Rev.87 977-92 |

[15] | Lakhal A 2013 KAIRUAIN-algorithm applied on electromagnetic imaging Inverse Problems29 095001 |

[16] | Louis A K 1989 Inverse und Schlecht Gestellte Probleme (Germany: Teubner) · Zbl 0667.65045 |

[17] | Louis A K 1996 Approximate inverse for linear, some nonlinear problems Inverse Problems12 175-90 · Zbl 0851.65036 |

[18] | Moskow S and Schotland J C 2008 Convergence and stability of the inverse scattering series for diffuse waves Inverse Problems24 065005 · Zbl 1157.35122 |

[19] | Moskow M and Schotland J 2009 Numerical studies of the inverse born series for diffuse waves Inverse Problems25 095007 · Zbl 1173.35734 |

[20] | Moses H E 1956 Calculation of the scattering potential from reflection coefficients Phys. Rev.102 550-67 · Zbl 0070.21906 |

[21] | Natterer F and Wuebbeling F 2001 Mathematical Methods in Image Reconstruction(SIAM Monographs on Mathematical Modeling and Computation) (Philadelphia: SIAM) |

[22] | Novikov R G and Khenkin G M 1987 The ∂¯ -equation in the multidimensional inverse scattering problem convergence and stability of the inverse scattering series for diffuse waves Russ. Math. Surv.42 109-80 · Zbl 0674.35085 |

[23] | Prosser R T 1969 Formal solutions of the inverse scattering problem J. Math. Phys.10 1819-22 |

[24] | Schuster T, Kaltenbacher B, Hofmann B and Kazimierski K 2012 Regularization Methods in Banach Spaces(Radon Series on Computational and Applied Mathematics) (Berlin: De Gruyter) · Zbl 1259.65087 |

[25] | Zeidler E 1986 Nonlinear Functional Analysis and its Applications: I: Fixed-Point Theorems (New York: Springer) · Zbl 0583.47050 |

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.