Institute for Numerical and Applied Mathematics - Research Group Inverse Problems



Short Introduction



Research Interests


  • Variational regularization theory
  • Inverse medium scattering
  • Stability estimates for scattering problems



Publications




    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



Conferences and Workshops



Teaching


  • Summer term 2018: Inverse problems II, teaching assistant

  • Winter term 2017/2018: Inverse problems I, teaching assistant

  • Summer term 2017: Functional Analysis, teaching assistant

  • Winter term 2016/2017: Numerical Analysis I, teaching assistant

  • Summer term 2016: Numerical Analysis II, teaching assistant

  • Winter term 2015/2016: Numerical Analysis I, teaching assistant

  • Winter term 2014/2015: Numerical Analysis I and Analysis I, student research assistant

  • Summer term 2014: Analysis II for teachers, student research assistant

  • Winter term 2013/2014: Analysis I, student research assistant

  • Summer term 2013: Analysis II for teachers, student research assistant

  • Winter term 2012/2013: Analysis I and Mathamatical preparatory class for mathematics, student research assistant

  • Summer term 2012: Analysis II for teachers, student research assistant

  • Winter term 2012/2011: Analysis I and Mathamatical preparatory class for forstery and agricultural science, student research assistant

  • Summer term 2011: Analysis II for teachers, student research assistant

  • Winter term 2010/2011: Analysis I, student research assistant

Research Group Inverse Problems

 
   members of the research group