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Inverse Problems

Holder of the Professorship

Prof. Dr. Frank Werner

Holder of the Professorship
Professorship for Mathematics at the Chair of Mathematics IX
Emil-Fischer-Straße 30
97074 Würzburg
Building: 30 (Mathematik West)
Room: 00.008
Portrait Frank Werner

Memberships

Research Interests

My research is located at the interface between statistics and inverse problems, and I am particularly interested in the following topics:

  • (Nonlinear) statistical inverse problems and regularization theory,
  • uncertainty quantification by simultaneous (minimax) tests for inverse problems and
  • applications in biophysics (e.g. fluorescence microscopy), especially with non-Gaussian noise.

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Publications

  • Neil K. Chada, Marco A. Iglesias, Shuai Lu, Frank Werner: On a Dynamic Variant of the Iteratively Regularized Gauss-Newton Method with Sequential Data: arXiv: 2207.13499
  • Katharina Proksch, Frank Werner, Jan Keller-Findeisen, Haisen Ta, Axel Munk: Towards quantitative super-resolution microscopy: Molecular maps with statistical guarantees: arXiv: 2207.13426
  • G. Kulaitis, A. Munk, F. Werner: A minimax testing perspective on spatial statistical resolution in microscopy
  • M. Pohlmann, F. Werner, A. Munk: Minimax detection of localized signals in statistical inverse problems: arXiv: 2112.05648

  • (with T. Hohage): Error estimates for variational regularization of Inverse Problems with general noise models for data and operator: ETNA. Volume 57, pp. 127-152, 2022, DOI: 10.1553/etna_vol57s127
  • (with R. Siegmund, S. Jakobs, C. Geisler, A. Egner: isoSTED microscopy with water-immersion lenses and background reduction: Biophysical Journal. vol 120, Issue 14, 2021, DOI: 10.1016/j.bpj.2021.05.031
  • (with M. Alamo, H. Li und Axel Munk): Variational multiscale nonparametric regression: Algorithms 2020, 13(11), 296. DOI:  10.3390/a13110296  
  • (with G. Kulaitis, A. Munk ): What is resolution? A statistical minimax testing perspective on super-resolution microscopy
    arXiv: 2005.07450. Annals of Statistics. 49(4): 2292-2312 (August 2021). DOI: 10.1214/20-AOS2037
  • (mit S. Lu and P. Niu ): On the asymptotical regularization for linear inverse problems in presence of white noise.
    In: SIAM/ASA Journal on Uncertainty Quantification. 1-28, vol 9, Issue 1, 2020. DOI:  10.1137/20M1330841
  • (with F. Enikeeva, A. Munk and M. Pohlmann): Bump detection in heterogeneous Gaussian regression. Bernoulli, 26 (2020), no. 4, 3280--3310. DOI: 10.3150/20-BEJ1226
    Older version: available on arXiv.
  • (with A. Munk and T. Staudt): Statistical Foundations of Nanoscale Photonic Imaging. In: Nanoscale Photonic Imaging125-143, vol 134, Springer, 2020. DOI: 10.1007/978-3-030-34413-9_4
  • (with A. Munk, H. Li and K. Proksch): Photonic imaging with statistical guarantees. From Multiscale Testing to Multiscale Estimation. In: Nanoscale Photonic Imaging283-312 , vol 134, Springer, 2020. DOI: 10.1007/978-3-030-34413-9_11
  • (with H. Li): Empirical Risk Minimization as Parameter Choice Rule for General Linear Regularization Methods. Annales de l’Institut Henri Poincaré, 56 (2020), no. 1, 405-427, 2020. DOI: 10.1214/19-AIHP966
    Older version: available on arXiv.
  • (with C. König and A. Munk): Multidimensional multiscale scanning in Exponential Families: Limit theory and statistical consequences. The Annals of Statistics, 2020, Vol. 48, No. 2, 655-678. DOI: 10.1214/18-AOS1806
    Older version: available on arXiv.
  • (with B. Hofmann): Convergence Analysis of (Statistical) Inverse Problems under Conditional Stability Estimates. Inverse Problems 36 015004, 2020. DOI: 10.1088/1361-6420/ab4cd7
    Older version: available on arXiv.
  • (with K. Proksch and A. Munk): Multiscale Scanning in Inverse Problems. The Annals of Statistics, 46(6B), 3569-3602, 2018. DOI: 10.1214/17-AOS1669
    Older version: available on arXiv.
  • Adaptivity and Oracle Inequalities in Linear Statistical Inverse Problems: a (numerical) survey. In: New Trends in Parameter Identification for Mathematical Models, 291-316, Birkhäuser, 2018. DOI: 10.1007/978-3-319-70824-9_15
  • (with F. Enikeeva and A. Munk): Bump detection in heterogeneous Gaussian regression. Bernoulli 24(2): 1266-1306, 2018. DOI: 10.3150/16-BEJ899
    Older version: available on arXiv.
  • (with T. Hohage): Inverse Problems with Poisson Data: statistical regularization theory, applications and algorithms. Topical Review for Inverse Problems 32 093001, 2016. DOI: 10.1088/0266-5611/32/9/093001
  • (with C. König and T. Hohage): Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise. SIAM Journal on Numerical Analysis 54(1), 341-360, 2016. DOI: 10.1137/15M1022252
    Older version: available on arXiv.
  • (with A. Munk): Discussion of "Hypothesis testing by convex optimization" by A. Goldenshluger, A. Juditsky and A. Nemirovski. Electronic Journal of Statistics 9(2): 1720-1722, 2015. DOI: 10.1214/14-EJS980
  • On convergence rates for iteratively regularized Newton-type methods under a Lipschitz-type nonlinearity condition. Journal of Inverse and Ill-posed problems 23(1): 75-84, 2015. DOI: 10.1515/jiip-2013-0074
  • (with T. Hohage): Convergence rates for Inverse Problems with Impulsive Noise. SIAM Journal on Numerical Analysis 52(3), 1203-1221, 2014. DOI: 10.1137/130932661
    Older Version: available on arXiv.
  • (with T. Hohage): Iteratively regularized Newton-type methods with general data misfit functionals and applications to Poisson data. Numerische Mathematik 123(4), 745-779, 2013. DOI: 10.1007/s00211-012-0499-z
  • (with T. Hohage): Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data. Inverse Problems 28 104004, 2012. DOI: 10.1088/0266-5611/28/10/104004
    Older Version: (Preprint), also available on arXiv.

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  • (with T. Hohage): Convergence in expectation results for phase retrieval problems in x-ray imaging. In Proceedings of the 11th International Conference on Mathematical and Numerical Aspects of Waves 139-140, 2013. (online version)

  • Inverse Problems with Poisson data. In: Computational Inverse Problems, Oberwolfach Reports 9(4), 3115-3116, 2012. DOI: 10.4171/OWR/2012/51 (online version)

  • Inverse problems with Poisson data: Tikhonov-type regularization and iteratively regularized Newton methods, 2012 (PhD thesis)

  • Ein neuer numerischer Ansatz zur L^p-Regularisierung, 2008 (Diploma thesis - in German)

(link to my articles on arXiv)
(link to my MathSciNet author page)

 

Further Information

Education 

2004 - 2009 Studies of mathematics at the Georg-August-Universität Göttingen
23.01.2009: Examination (Diplom) in mathematics (passed with distinction) Thesis: "Ein neuer numerischer Ansatz zur Lp-Regularisierung" (in German)
2009 - 2012 PhD studies in mathematics at the Georg-August-Universität Göttingen Thesis: "Inverse Problems with Poisson data: Tikhonov-type regularization and iteratively regularized Newton methods"
2012 - 2014 Postdoctoral research fellow at the institute for numerical and applied mathematics
2014 - 2020 Group leader 'Statistical Inverse Problems in Biophysics' at the MPIBPC
2020 - Holder of the professorship "Inverse Problems" at the Chair of Scientific Computing in Würzburg

I am married and father of two sons (*2012 and *2015)

Upcoming Events

Academic Travels and Talks

2022

2021

2020

  • Chemnitz Symposium on Inverse Problems, integrated into the DMV-Jahrestagung, September 14-17, Chemnitz.

2019

2018

2017

2016

2015

2014

2013

2012

  • November 16-25, 2012: Research stay at the Australian National University, Canberra, also joining the Canberra Symposium On Regularisation: "Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data" (slides).
  • Oberwolfach workshop Computational Inverse Problems, Oberwolfach, Germany: "Inverse Problems with Poisson data"
  • DMV Annual Meeting, Saarland University, Saarbrücken, Germany: "Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data" (slides).
  • PhD defense January 23rd: "Inverse Probleme mit Poisson Daten: Verallgemeinerte Tikhonov-Regularisierung und iterativ regularisierte Newton Methoden" (slides, german)

2011

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