Adaptive Bregman-Kaczmarz: An Approach to Solve Linear Inverse Problems with Independent Noise Exactly Lionel Tondji, Idriss Tondji, Dirk A. Lorenz Submitted, September 2023 [arxiv.org/abs/2309.06186]
Minimal error momentum Bregman-Kaczmarz Dirk A. Lorenz, Maximilian Winkler Submitted, July 2023 [arxiv.org/abs/2307.15435]
Linearly convergent adjoint free solution of least squares problems by random descent Dirk A. Lorenz, Felix Schneppe, Lionel Tondji Submitted, June 2023 [arxiv.org/abs/2306.01946]
A Bregman-Kaczmarz method for nonlinear systems of equations Robert Gower, Dirk A. Lorenz, Maximilian Winkler Submitted, March 2023 [arxiv.org/abs/2303.08549]
The Degenerate Variable Metric Proximal Point Algorithm and Adaptive Stepsizes for Primal-Dual Douglas-Rachford Dirk A. Lorenz, Jannis Marquardt, Emanuele Naldi Submitted, February 2023 [arxiv.org/abs/2302.13128]
Learning Variational Models with Unrolling and Bilevel Optimization Christoph Brauer, Niklas Breustedt, Timo de Wolff, Dirk A. Lorenz Submitted, September 2022 [arxiv.org/abs/2209.12651]
Lα -Regularization of the Beckmann Problem Dirk Lorenz, Hinrich Mahler and Christian Meyer Submitted, January 2022 [arxiv.org/abs/2201.07086]
Primal-dual residual networks Christoph Brauer and Dirk Lorenz Submitted, June 2018 [arXiv.org/abs/1806.05823].
Refereed Journal Publications
Faster Randomized Block Sparse Kaczmarz by Averaging Lionel Tondji, Dirk A Lorenz To appear in Numerical Algorithms, 2022 [doi, arxiv.org/abs/2203.10838]
Chambolle-Pock's Primal-Dual Method with Mismatched Adjoint Dirk A. Lorenz and Felix Schneppe To appear in Applied Mathematics and Optimization [arxiv.org/abs/2201.04928]
Damage Identification in Fiber Metal Laminates using Bayesian Analysis with Model Order Reduction Nanda Kishore Bellam Muralidhar, Carmen Gräßle, Natalie Rauter, Andrey Mikhaylenko, Rolf Lammering, Dirk A. Lorenz Computer Methods in Applied Mechanics and Engineering., 403, Part B, 115737, 2023 [doi, arxiv.org/abs/2206.04329]
Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition Andrea Ebner, Jürgen Frikel, Dirk Lorenz, Johannes Schwab, Markus Haltmeier Applied and Computational Harmonic Analysis, 62:66-83 2023 [doi, arxiv.org/abs/2008.06219]
Extended Randomized Kaczmarz Method for Sparse Least Squares and Impulsive Noise Problems Frank Schöpfer, Dirk A Lorenz, Lionel Tondji, and Maximilian Winkler Lineare Algebra and Applications, 652:132-154, 2022 [doi, arxiv/abs/2201.08620]
Degenerate Preconditioned Proximal Point algorithms Kristian Bredies, Enis Chenchen, Dirk A. Lorenz, Emanuele Naldi SIAM Journal on Optimization, 32(3):2376-2401, 2022 [doi, arxiv.org/abs/2109.11481]
Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction Andrey Mikhaylenko, Natalie Rauter, Nanda Kishore Bellam Muralidhar, Tilmann Barth , Dirk A. Lorenz, and Rolf Lammering Acoustics, 4(3):517-537, 2022 [doi, preprints.org/manuscript/202206.0025]
Nonconvex flexible sparsity regularization: theory and monotone numerical schemes Daria Ghilli, Dirk A. Lorenz and Elena Resmerita Optimization, 71(4), 1117-1149, 2022 [doi,arxiv]
Orlicz space regularization of continuous optimal transport problems Dirk A. Lorenz und Hinrich Mahler Applied Mathematics and Optimization, 85(14), 2022 [doi, arxiv.org/abs/2004.11574]
Entropic regularization of continuous optimal transport problems Christian Clason, Dirk A. Lorenz, Hinrich Mahler und Benedikt Wirth Journal of Mathematical Analysis and Applications, 494(1), 2021 [doi, arXiv.org/abs/1906.01333].
Quadratically regularized optimal transport Dirk A. Lorenz, Paul Manns und Christian Meyer Applied Mathematics and Optimization, 83 (3), 1919-1949, 2021 [doi, arXiv.org/abs/1903.01112].
Complexity and Applications of the Homotopy Principle for Uniformly Constrained Sparse Minimization Christoph Brauer and Dirk A. Lorenz Applied Mathematics and Optimization, 82(3), 2020 [doi]
Non-stationary Douglas-Rachford and alternating direction method of multipliers: adaptive stepsizes and convergence Dirk A. Lorenz and Quoc Tran-Dinh Computational Optimization and Applications, 74(1):67–92, 2019 [doi, arXiv.org/abs/1801.03765].
Denoising of image gradients and total generalized variation denoising Birgit Komander, Dirk A. Lorenz, and Lena Vestweber Journal of Mathematical Imaging and Vision, 61(1):21-39, 2019 [doi, arXiv.org/abs/1712.08585].
Linear convergence of the Randomized Sparse Kaczmarz method Frank Schöpfer and Dirk A. Lorenz Mathematical Programming, 173(1-2):509-536, 2019 [doi, arXiv.org/abs/1610.02889]. The is MATLAB code available as supplementary material at the Springer site. However, Springer made the files unusable, so, here is the code: MATLAB code for the randomized Kaczmarz method (zip, 64 KByte)
The randomized Kaczmarz Method with mismatched adjoint Dirk A. Lorenz, Sean Rose, and Frank Schöpfer BIT Numerical Mathematics 59(4):1079-1098, 2018 [doi, arXiv.org/abs/1803.02848].
Sarrus Rules for Matrix Determinants and Dihedral Groups Dirk A. Lorenz and Karl-Joachim Wirths The College Mathematics Journal, 49(5):333-340, 2018. Reprinted: Mathematics Newsletter, 29(2):44-48, 2018 [doi, arxiv.org/abs/1809.08948]
A Primal-Dual Homotopy Algorithm for ℓ1-Minimization with ℓ∞-Constraints Christoph Brauer, Dirk A. Lorenz, and Andreas M. Tillmann Computational Optimization and Applications, 70(2):443-478, 2018 [doi, arXiv.org/abs/1610.10022, OO].
An extended Perona-Malik model based on probabilistic models Lars M. Mescheder and Dirk A. Lorenz Journal of Mathematical Imaging and Vision, 60(1):128-144, 2018 [arXiv.org/abs/1612.06176].
How to be best in the OECD Better Life Index? Jan Lorenz, Christoph Brauer, and Dirk A. Lorenz Social Indicators Research, 134(1):75-92, 2017 [doi, arXiv.org/abs/1608.04556]
Flexible sparse regularization Dirk A. Lorenz and Elena Resmerita Inverse Problems, 33(1), 2016 [doi, arXiv.org/abs/1601.04429].
An inertial forward-backward method for monotone inclusions Dirk A. Lorenz and Thomas Pock Journal of Mathematical Imaging and Vision, 51(2):311-325, 2015 [doi, arXiv.org/abs/1403.3522].
Computing and analyzing recoverable supports for sparse reconstruction Christian Kruschel and Dirk A. Lorenz Advances in Computational Mathematics, 41(6):1119-1144, 2015 [doi, arXiv.org/abs/1309.2460].
Minimization of non-smooth, non-convex functionals by iterative thresholding Kristian Bredies, Dirk A. Lorenz and Stefan Reiterer Journal of Optimization Theory and Applications, 165(1),78-112, 2015 [doi, DFG SPP 1324 Preprint 10]
Solving Basis Pursuit: Subgradient Algorithm, Heuristic Optimality Check, and Solver Comparison Dirk A. Lorenz, Marc E. Pfetsch and Andreas M. Tillmann ACM Transactions on Mathematical Software, 41(2), 2015 [doi, OO Preprint].
Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized x-ray CT Jakob S. Jørgensen, Christian Kruschel and Dirk A. Lorenz Inverse Problems in Science and Engineering, 2014 [doi, arXiv.org/abs/1409.0214].
Data fusion of surface normals and point coordinates for deflectometric measurements Birgit Komander, Dirk A. Lorenz, Marc Fischer, Marcus Petz und Rainer Tutsch Journal of Sensors and Sensor Systems, 3:281-290, 2014 [doi].
Imaging with Kantorovich-Rubinstein discrepancy Jan Lellmann, Dirk A. Lorenz, Carola Schönlieb and Tuomo Valkonen SIAM Journal on Imaging Sciences, 7(4):2833-2859 2014 [doi, arXiv.org/abs/1407.0221].
The linearized Bregman method via split feasibility problems Dirk A. Lorenz, Frank Schöpfer and Stephan Wenger SIAM Journal in Imaging Sciences, 7(2) 2014 [doi, arXiv.org/abs/1309.2094].
Observer-independent quantification of insulin granule exocytosis and pre-exocytotic mobility by TIRF microscopy Magnus Matz, Kirstin Schumacher, Kathrin Hatlapatka, Dirk A. Lorenz, Knut Baumann, and Ingo Rustenbeck Microscopy and Microanalysis, 20(1):206-218, 2014 [doi].
An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections Dirk A. Lorenz, Marc E. Pfetsch and Andreas M. Tillmann Computational Optimization and Applications, 57(2):271-306, 2014 [doi, arXiv.org/abs/1104.5351].
Fast image-based modeling of astronomic nebulae Stephan Wenger, Dirk A. Lorenz and Marcus Magnor Computer Graphics Forum, 32(7), 2013 [doi].
Necessary conditions for variational regularization schemes Dirk A. Lorenz and Nadja Worliczek Inverse Problems, 20:075016pp, 2013 [doi, arXiv.org/abs/1204.0649].
Constructing test instances for Basis Pursuit Denoising Dirk A. Lorenz IEEE Transactions on Signal Processing, 61(5):1210-1214, 2013 [doi, arXiv.org/abs/1103.2897] Code zum Reproduzieren der Abbildungen ist hier (zip, 1 MByte).
Visualization of astronomical nebulae via distributed multi-GPU compressed sensing tomography Stephan Wenger, Marco Ament, Stefan Guthe, Dirk A. Lorenz, Andreas Tillmann, Daniel Weiskopf, and Marcus Magnor IEEE Transactions on Visualization and Computer Graphics, 18(12):2188-2197, 2012. [doi]
Gradient descent methods based on quadratic approximations of Tikhonov functionals with sparsity constraints: theory and numerical comparison of stepsize rules Dirk A. Lorenz, Peter Maass and Pham Q. Muoi Electronic Transactions on Numerical Analysis, 39:437-463, 2012.
Image Sequence Interpolation based on Optical Flow, Segmentation, and Optimal Control Kanglin Chen and Dirk A. Lorenz IEEE Transactions on Image processing, 21(3):1020-1030, 2012. [doi]
Morozov's principle for the augmented Lagrangian method applied to linear inverse problems Klaus Frick, Dirk A. Lorenz and Elena Resmerita SIAM Multiscale Modelling and Simulation, 9(4):1528-1548 2011 [doi, arXiv.org/abs/1010.5181]
Beyond convergence rates: Exact inversion with Tikhonov regularization with sparsity constraints Dirk A. Lorenz, Stefan Schiffler and Dennis Trede Inverse Problems, 27:085009, 2011 [doi, arXiv.org/abs/1001.3276]
Image sequence interpolation using optimal control Kanglin Chen and Dirk A. Lorenz Journal of Mathematical Imaging and Vision 41(3):222-238, 2011 [doi, arXiv.org/abs/1008.0548]
Heuristic parameter-choice rules for convex variational regularization based on error estimates Bangti Jin and Dirk A. Lorenz SIAM Journal on Numerical Analysis, 48(3):1208-1229, 2010 [doi, arXiv.org/abs/1001.5346]
Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems Dirk A. Lorenz und Arnd Rösch Applicable Analysis, 89(11):1679-1692, 2010 [arXiv.org/abs/0909.4648]
A projection proximal-point algorithm for l^1-minimization Dirk A. Lorenz Numerical Functional Analysis and Optimization, 31(2):172-190, 2010 [doi, arXiv.org/abs/0904.1523] m-File mit ppp_l1.m
On Conditions for Convergence to Consensus Dirk A. Lorenz and Jan Lorenz IEEE Transactions on Automatic Control, 55(7):1651-1656, 2010 [doi, arXiv.org/abs/0803.2211]
Inline hologram reconstruction with sparsity constraints Loic Denis, Dirk A. Lorenz, Eric Thiebaut, Corinne Fournier und Dennis Trede. Optics Letters, 34(22):3475-3477, 2009 [doi]
Elastic-Net Regularization: Error estimates and Active Set Methods Bangti Jin, Dirk A. Lorenz, and Stefan Schiffler Inverse Problems, 25(11):115022 (26pp), 2009 [doi, arXiv.org/abs/0905.0796]
Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion Loic Denis, Dirk A. Lorenz, and Dennis Trede Inverse Problems, 25(11):115017 (24pp), 2009 [doi, arXiv.org/abs/0904.0154]
Regularization with non-convex separable constraints Kristian Bredies, Dirk A. Lorenz Inverse Problems, 25(8):085011 (14pp), 2009 [doi,SPP 1324 Preprint 11]
Optimal convergence rates for Tikhonov regularization in Besov scales Dirk A. Lorenz and Dennis Trede Journal of Inverse and Ill Posed Problems, 17(1):69-76, 2009 [doi]
On the role of sparsity in inverse problems Dirk A. Lorenz Journal of Inverse and Ill-Posed Problems, 17(1):61-68, 2009 [doi]
A generalized conditional gradient method and its connection to an iterative shrinkage method Kristian Bredies, Dirk A. Lorenz, and Peter Maass Computational Optimization and Applications, 42(2):173-193, 2009 [doi]
On the convergence speed of iterative methods for linear inverse problems with sparsity constraints Kristian Bredies and Dirk A. Lorenz Journal of Physics: Conference Series 124:012031 (12pp), 2008 [doi]
Convergence rates and source conditions for Tikhonov regularization with sparsity constraints Dirk A. Lorenz Journal of Inverse and Ill-Posed Problems, 16(5):463-478, 2008 [doi,arXiv.org/abs/0801.1774]
Optimal Convergence Rates for Tikhonov Regularization in Besov Scales Dirk A. Lorenz and Dennis Trede Inverse Problems 24(5):055010 (14pp), 2008 [doi, arXiv.org/abs/0806.0951]
Linear convergence of iterative soft-thresholding Kristian Bredies and Dirk A. Lorenz Journal of Fourier Analysis and Applications, 14(5-6):813-837, 2008 [doi, arXiv.org/abs/0709.1598]
A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints Roland Griesse and Dirk A. Lorenz Inverse Problems 24:035007 (19pp), 2008 [doi, arXiv.org/abs/0709.3186] m-file with ssn.m
Iterated hard shrinkage for minimization problems with sparsity constraints Kristian Bredies and Dirk A. Lorenz SIAM Journal on Scientific Computing, 30(2):657-683, 2008 [doi] m-file with iter_thresh.m
The Canonical Coherent States Associated With Quotients of the Affine Weyl-Heisenberg Group Stephan Dahlke, Dirk A. Lorenz, Peter Maass, Chen Sagiv, and Gerd Teschke Journal of Applied Functional Analysis, 3(2):215-232, 2008 [.pdf]
Shrinkage versus Deconvolution Esther Klann, Michael Kuhn, Dirk A. Lorenz, Peter Maass, and Herbert Thiele Inverse Problems, 23:2231-2248, 2007 [doi]
A Generalized Conditional Gradient Method for Non-Linear Operator Equations with Sparsity Constraints Kristian Bredies, Thomas Bonesky, Dirk A. Lorenz, and Peter Maass Inverse Problems, 23:2041-2058, 2007 [doi]
Non-convex Variational Denoising of Images: Interpolation Between Hard and Soft Wavelet Shrinkage Dirk A. Lorenz Current Development in Theory and Applications of Wavelets, 1(1):31-56, 2007
Solving Variational Problems in Image Processing via Projections - A Common View on TV Denoising and Wavelet Shrinkage Dirk A. Lorenz Zeitschrift für angewandte Mathematik und Mechanik, 81(1):247-256, 2007 [doi]
Mathematical Concepts of Multiscale Smoothing Kristian Brdies, Dirk A. Lorenz, and Peter Maass Applied and Computational Harmonic Analysis, 19(2):141-161, 2005 [doi]
Refereed Conference Publications
An accelerated randomized Bregman-Kaczmarz method for strongly convex linearly constraint optimization Lionel Tondji, Dirk A. Lorenz, and Ion Necoara In 2023 European Control Conference (ECC), 2023. [IEEE]
Group equivariant networks for leakage detection in vacuum bagging Christoph Brauer, Dirk Lorenz, and Lionel Tondji 30th European Signal Processing Conference, EUSIPCO 2022 - Proceedings, European Signal Processing Conference (EU- SIPCO), pages 1437–1441. IEEE, October 2022. [IEEE]
A Model-Based Damage Identification using Guided Ultrasonic Wave Propagation in Fiber Metal Laminates Nanda Kishore Bellam Muradlidhar und Dirk A. Lorenz VI ECCOMAS Young Investigators Conference YIC2021, 2021
Orlicz-space regularization for optimal transport and algorithms for quadratic regularization Dirk A. Lorenz und Hinrich Mahler NeurIPS workshop "Optimal Transport in Machine Learning", 2020. [arXiv.org/abs/1909.06082]
Learning to Dequantize Speech Signals by Primal-dual Networks: an Approach for Acoustic Sensor Networks, Christoph Brauer, Christoph, Ziyue Zhao, Dirk Lorenz, and Tim Fingscheidt. ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019. [doi]
A Sinkhorn-Newton method for entropic optimal transport Christoph Brauer, Christian Clason, Dirk A. Lorenz und Benedikt Wirth NIPS Workshop "Optimal Transport in Machine Learning", 2017 [arXiv.org/abs/1710.06635].
Denoising of image gradients and constrained total generalized variation Birgit Komander and Dirk A. Lorenz Proceedings of Scale Space and Variational Methods 2017, Lecture notes in Computer Science, Lauze F., Dong Y., Dahl A. (Eds.), 10302:435-446. Springer, 2017. [doi].
Sparse reconstruction of quantized speech signals, Brauer, Christoph, Timo Gerkmann, and Dirk Lorenz. 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2016. [doi]
Cartoon-Texture-Noise Decomposition with Transport Norms Christoph Brauer and Dirk A. Lorenz Proceedings of Scale Space and Variational Methods 2015, Lecture notes in Computer Science, Jean-François Aujol, Mila Nikolova, Nicolas Papadakis, (Eds.), 9087:142-153. Springer, 2015. [doi].
A sparse Kaczmarz solver and a linearized Bregman method for online compressed sensing. Dirk A. Lorenz, Frank Schöpfer, Stephan Wenger, and Marcus Magnor. Proceedings of ICIP 2014, March 2014. Recognized as one of the "Top 10%" papers. [doi, arXiv.org/abs/1403.7543]
Variational methods for motion deblurring with still background Eileen Laue and Dirk A. Lorenz Proceedings of Scale Space and Variational Methods 2013, Lecture notes in Computer Science, Arjan Kuijper, Thomas Pock, Kristian Bredies, and Horst Bischof, (Eds.), 7893:74-85. Springer, 2013. [doi].
Convergence to consensus by general averaging Dirk A. Lorenz and Jan Lorenz Positive Systems, Proceedings of POSTA 2009, Lecture Notes in Control and Information Sciences, Rafeal Bru, Sergion Romero-Vivo (Eds.), 389:91-100, 2009
An active set approach to the elastic-net and its applications in mass spectrometry by Theodore Alexandrov, Oliver Keszöcze, Dirk A. Lorenz, Stefan Schiffler, and Klaus Steinhorst Proceedings of SPARS09, 2009
Greedy deconvolution of point-like objects by Dirk A. Lorenz and Dennis Trede Proceedings of SPARS09, 2009
Topology-preserving geodesic active contours for segmentation of high-content fluorescent cellular imaging by Dirk A. Lorenz, Peter Maass, Hartwig Preckel and Dennis Trede PAMM, 8(1):10941-10942, 2009, Special Issue: 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Bremen 2008 [doi]
Iterated hard-thresholding for linear inverse problems with sparsity constriants by Kristian Bredies and Dirk A. Lorenz PAMM, 7(1):2060061-2060062, 2007 [doi]
An optimal control problem in image processing by Kristian Bredies, Dirk A. Lorenz, and Peter Maass PAMM, 6(1): 859-860, 2006 [doi]
On the minimization of non-convex, non-differentiable functionals with an application to SPECT by Thomas Bonesky, Kristian Bredies, Dirk A. Lorenz, and Peter Maass In Oberwolfach Report: Mathematical Methods in Tomography34:18-22, 2006
An Optimal Control Problem in Medical Image Processing by Kristian Bredies, Dirk A. Lorenz, and Peter Maass In Systems, Control, Modeling and Optimization Proceedings of 22nd IFIP TC 7 Conference 249-26, 2006
A Partial Differential Equation for Continuous Non-Linear Shrinkage Filtering and its Application for Analyzing MMG Data by Kristian Bredies, Dirk A. Lorenz, Peter Maass, and Gerd Teschke In Photonics East, SPIE, 2003
Book chapter
On the Interplay of Subset Selection and Informed Graph Neural Networks Niklas Breustedt, Paolo Climaco, Jochen Garcke, Jan Hamaekers, Gitta Kutyniok, Dirk A. Lorenz, Rick Oerder, Chirag Varun Shukla [arxiv.org/abs/2306.10066]
Wirkstoffe, Medikamente und Mathematische Bildverarbeitung by Günther J. Bauer, Dirk A. Lorenz, Peter Maass, Hartwig Preckel, and Dennis Trede in acatech diskutiert, acatech - Deutsche Akademie der Technikwissenschaften, 2008
Wissenschaftliches Rechnen im Tandem by Dirk A. Lorenz in Jahrbuch den Universität Bremen, 2007
Multiscale Approximation by Stephan Dahlke, Peter Maass, Gerd Teschke, Karsten Koch, Dirk A. Lorenz, Stephan Müller, Stefan Schiffler, Andreas Stämpfli, Herbert Thiele, and Manuel Werner in Mathematical Methods in Time Series Analysis and Digital Image Processing, Springer, 2007
Theses
Wavelet Shrinkage in Signal and Image Processing - An Investigation of Relations and Equivalences by Dirk A. Lorenz PhD Thesis, Universität Bremen, February 2005. [.pdf, published at elib at SuUB]
Methoden der Multiskalenglättung by Dirk A. Lorenz Diploma Thesis, Universität Bremen, August 2002. [.ps]
Unpublished
Nonlinear complex and cross diffusion by Kristian Bredies, Dirk A. Lorenz, and Yehoshua Y. Zeevi, 2006
A Comparison of Denoising Methods for One Dimensional Time Series by Thoster Köhler and Dirk A. Lorenz, 2005
Variational Denoising in Besov Spaces and Interpolation of Hard and Soft Wavelet Shrinkage by Dirk A. Lorenz, 2004
