Institut für Numerische und Angewandte Mathematik - Arbeitsgruppe Inverse Probleme



Kurzvorstellung



Forschungsinteressen ´


  • Variationelle Regularisierungstheorie
  • Inverse Medium Streuprobleme
  • Stabilitätsresultate für Streuprobleme



Publikationen




    Articles


    • Frederic Weidling, Benjamin Sprung, Thorsten Hohage. 2020. Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces. SIAM Journal Numer. Analysis 58: 21-47. https://doi.org/10.1137/18M1178098 (also available on arxiv).  

      Abstract: This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.
    • Hohage, Thorsten, Weidling, Frederic. 2017. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems and Imaging 11(1): American Institute of Mathematical Sciences (AIMS. 203-220. http://dx.doi.org/10.3934/ipi.2017010 (also available on arxiv).  

      Abstract: This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.
    • Hohage, Thorsten, Weidling, Frederic. 2017. Characterizations of Variational Source Conditions, Converse Results, and Maxisets of Spectral Regularization Methods. SIAM Journal on Numerical Analysis 55(2): Society for Industrial & Applied Mathematics (SIAM. 598-620. http://dx.doi.org/10.1137/16M1067445 (also available on arxiv).  

      Abstract: We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of ill-posedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. If the forward operator can be written in terms of the functional calculus of a Laplace-Beltrami operator, variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.
    • Hohage, Thorsten, Weidling, Frederic. 2015. Verification of a variational source condition for acoustic inverse medium scattering problems. Inverse Problems 31(7): IOP Publishing. 075006. http://dx.doi.org/10.1088/0266-5611/31/7/075006 (also available on arxiv).  

      Abstract: This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory.


    • Weidling, F., Hohage, T.. 2017. Tikhonov Regularization for Inverse Medium Scattering in Banach Spaces. In Oberwolfach Reports. 14: 1478-1480. Workshop: Computational Inverse Problems for Partial Differential Equations, 14-20 May 2017. https://www.mfo.de/occasion/1720



    All publications as BibTeX-file: bibtex



Konferenzen und Workshops



Teaching


  • Sommersemester 2018: Inverse Probleme II, Vorlesungsassistenz

  • Wintersemester 2017/18: Inverse Probleme I, Vorlesungsassistenz

  • Sommersemester 2017: Funktionalanalysis, Vorlesungsassistenz

  • Wintersemester 2016/2017: Numerik I, Vorlesungsassistenz

  • Sommersemester 2016: Numerik II, Vorlesungsassistenz

  • Wintersemester 2015/2016: Numerik I, Vorlesungsassistenz

  • Wintersemester 2014/2015: Numerik I and Differential- und Integralrechnung I, Studentische Hilfskraft

  • Sommersemester 2014: Differential- und Integralrechnung II für Lehramtskanidaten, Studentische Hilfskraft

  • Wintersemester 2013/2014: Differential- und Integralrechnung I, Studentische Hilfskraft

  • Sommersemester 2013: Differential- und Integralrechnung II für Lehramtskanidaten, Studentische Hilfskraft

  • Wintersemester 2012/2013: Differential- und Integralrechnung I und Mathematischer Vorkurs für Mathematiker, Studentische Hilfskraft

  • Sommersemester 2012: Differential- und Integralrechnung II für Lehramtskanidaten, Studentische Hilfskraft

  • Wintersemester 2012/2011: Differential- und Integralrechnung I und Mathematischer Vorkurs für Agrar- und Forstwissenschaften, Studentische Hilfskraft

  • Sommersemester 2011: Differential- und Integralrechnung II für Lehramtskanidaten, Studentische Hilfskraft

  • Wintersemester 2010/2011: Differential- und Integralrechnung I, Studentische Hilfskraft

Arbeitsgruppe Inverse Probleme

 
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