Linear and Nonlinear Inverse Problems with Practical Applications

Inverse problems arise in practical applications whenever there is a need to interpret indirect measurements.

This book explains how to identify ill-posed inverse problems arising in practice and how to design computational solution methods for them; explains computational approaches in a hands-on fashion, with related codes available on a website; serves as a convenient entry point to practical inversion.

The guiding linear inversion examples are the problem of image deblurring, x-ray tomography, and backward parabolic problems, including heat transfer, and electrical impedance tomography is used as the guiding nonlinear inversion example.

The book’s nonlinear material combines the analytic-geometric research tradition and the regularization-based school of thought in a fruitful manner, paving the way to new theorems and algorithms for nonlinear inverse problems. Furthermore, it is the only mathematical textbook with a thorough treatment of electrical impedance tomography, and these sections are suitable for beginning and experienced researchers in mathematics and engineering.

A photograph of Jennifer Mueller and Samuli Siltanen's book: Linear and Nonlinear Inverse Problems With Practical Applications. The cover art features and EIT reconstruction and an ECG trace.
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Recent Publications

Journal Articles

  • Enzer, K.G., Barbosa Da Rosa, N., Keck, A. et al. Electrical impedance tomography reveals ventilation and perfusion heterogeneity in infants with bronchopulmonary dysplasia. J Perinatol (2026). https://doi.org/10.1038/s41372-026-02590-4

  • Melody Alsaker, Siiri Rautio, Fernando Silva de Moura, Juan Pablo Agnelli, Rashmi Murthy, Matti Lassas, Jennifer L. Mueller, Samuli Siltanen 2025 CT scans without X-rays: Parallel-beam imaging from nonlinear current rows, Applied Mathematics for Modern Challenges, 5, 1{35, doi: 10.3934/ammc.2025009.

  • Rocheleau CJ, Overton TD, da Rosa NB Jr, Saulnier GJ, Shishvan OR, Baker CD, Enzer KG, Mueller JL 2025 Use of an anatomical atlas in real time EIT reconstructions of ventilation and pulsatile perfusion in premature infants, Scientific Reports, Aug 13;15(1):29622. doi: 10.1038/s41598-025-15543-2. PMID: 40796647

  • Jenny E. Zablah, Catalina Vargas-Acevedo, Nilton Barbosa da Rosa Jr, Omid Rajabi Shishvan, Gary Saulnier, David Isaacson, Gareth J. Morgan, Jennifer L. Mueller 2025 Feasibility of Electric Impedance Tomography in the Assessment of Lung Perfusion and Ventilation in Congenital Pulmonary Vein Stenosis, Pediatric Cardiology, Feb;47(2):652-659. https://doi.org/10.1007/s00246-025-03816-6

  • J. L. Mueller, 2025 Non-iterative reconstruction of respiratory and cardiac activity in infants from electrical impedance tomography data Philosophical Transactions A, to appear.
  • Enzer, Katelyn; Barbosa da Rosa, Nilton; Rocheleau, Christopher; Saulnier, Gary; Mueller, Jennifer; Baker, Christopher 2025 Electrical Impedance Tomography Imaging of Ventilation and Perfusion in Bronchopulmonary Dysplasia, J Perinatol, https://doi.org/10.1038/s41372 025-02236-x
  • Jenny E. Zablah, Catalina Vargas-Acevedo, Nilton Barbosa da Rosa Jr, Omid Rajabi Shishvan, Gary Saulnier, David Isaacson, Gareth J. Morgan, Jennifer L. Mueller 2025 Feasibility of Electric Impedance Tomography in the Assessment of Lung Perfusion and Ventilation in Congenital Pulmonary Vein Stenosis, https://doi.org/10.1007/s00246-025-03816-6
  • Roberto C. Ceccato, Andre V. Pigatto, Richard C. Aster, Chi-Nan Pai, Jennifer L. Mueller, S´ergio S. Furuie. 2025 Time of Flight Transmission Mode Ultrasound Computed Tomography with Expected Gradient and Boundary Optimization, IEEE Transactions on Biomedical Engineering, https://doi.org/10.1109/TBME.2025.3550823
  • M. Alsaker, B. Bladow, S.E. Campbell, N. Linthacum, T.M. McKenzie, J. L. Mueller, and T.B.R. Santos 2025 D-bar reconstructions with nonsmooth learned spatial priors in 2D electrical impedance tomography, Journal of Computational and Applied Mathematics, 463, 116512. https://doi.org/10.1016/j.cam.2025.116512
  • Rosa, N., Kao, T-J., Brinton, J., Offner, P.J., Burnham, E.L., and Mueller, J.L 2025 Three-dimensional electrical impedance imaging during spontaneous breathing trials in patients with acute hypoxic respiratory failure: A pilot study, Critical Care Explorations, 7(1):e1198. doi: 10.1097/CCE.0000000000001198. 
  • E. Heavner, J. L. Mueller, T-J Kao, N. Barbosa da Rosa, P. Offner and E. Burnham 2025 Solution of an inverse problem to estimate airway resistance in mechanically ventilated patients from 3D electrical impedance tomography data, Applied Mathematics for Modern Challenges, 3: 90-116. doi: 10.3934/ammc.2025004. 
  • Kyler Howard, Chris Rocheleau, Trevor Overton, Joel Barraza Nava, Mason Faldet, Kristina Moen, Summer Soller, Tyler Stephens, Esther van de Lagemaat, Natalie Wijesinghe, Kaylee Wong Dolloff, Nilton Barbosa da Rosa, Jennifer L. Mueller 2025 A comparison of techniques to improve pulmonary EIT image resolution using a database of simulated EIT images, Journal of Computational and Applied Mathematics, 460, 116415. https://doi.org/10.1016/j.cam.2024.116415 
  • A. Abdelwahad, N. Rosa, C. Wilcox, O.R. Shishvan, D. Isaacson, J.C. Newell, J. L. Mueller, and G.J. Saulnier 2025 Simultaneous acquisition of EIT and ECG signals using the ACT 5 system, IEEE Transactions on Biomedical Engineering, 72(4), pp. 1498–1507, doi: 10.1109/TBME.2024.3509900 
  • E. Heavner, J. L. Mueller, K. McFann, J. Dunn, O. Alnachoukati, and C. Mohnike, 2024 Estimation of airway resistance throughout the bronchial tree from mechanical ventilation output data, Applied Mathematics for Modern Challenges, 2(3), pp. 262- 286. Doi: 10.3934/ammc.2024014 
  • A.V. Pigatto, S. Furuie, D. Card´enas, M. L. Rezende, R. Lima and J. L. Mueller 2024 Imaging of ventilation and lung injury with low-frequency ultrasound tomography, Medical Physics, pp. 117. https://doi.org/10.1002/mp.17421
  • Shin, S. Ahmad, T. B. R. Santos, N. Rosa, and J. L. Mueller 2024 Comparison of two linearization-based methods for 3-D EIT reconstructions on a simulated chest, Journal of Mathematical Imaging and Vision, 66 pp. 185–197 
  • A.V. Pigatto, N. Rosa, S. Furuie, and J.L. Mueller 2024 LUFT: A Low-Frequency Ultrasound Tomography System for Lung Imaging, IEEE Sensors, IEEE Sensors Journal, 24(7), pp. 11091-11101.
  • O.R. Shishvan, A. Abdelwahad, N. Rosa, G.J. Saulnier, J. L. Mueller, J.C. Newell, and D. Isaacson 2024 ACT 5 Electrical impedance tomography system, IEEE Transactions on Biomedical Engineering, 71(1), pp. 227–236. doi: 10.1109/TBME.2023.3295771. 
  • S. Ziegler, T. B. R. Santos, and J. L. Mueller 2024 Regularized full waveform inversion for low frequency ultrasound with a structural similarity EIT prior, Inverse Problems and Imaging, 18(1): 86–103. doi: 10.3934/ipi.2023023
  • M. Alsaker, J.L. Mueller, and A. Stahel, 2023 A Multithreaded Real-time Solution for 2D EIT Reconstruction with the D-bar Algorithm, J of Computational Science, 67, p. 101967. 
  • T. B. R. Santos, R. M. Nakanishi, T.M.M. Oleg´ario, R. G. Lima and J. L. Mueller 2023 A deep-learning approach to increasing the resolution of 2 D EIT D-bar respiratory images, Applied Mathematics for Modern Challenges, 1(1): 21-38. doi: 10.3934/ammc.2023003. 
  • T. B. R. Santos, R. M. Nakanishi, E.D.L. B. de Camargo, M.B.P. Amato, J. P. Kaipio, R. G. Lima and J. L. Mueller 2022 Improved resolution of D-Bar images of ventilation using a Schur complement property and an anatomical atlas, Medical Physics, 49(7), doi: 10.1002/mp.15669, PMID: 35411573 
  • K. Shin and J. L. Mueller 2021 Calderon’s method with a spatial prior for 2-D EIT imaging of ventilation and perfusion, Sensors, 21(16), 5635. 
  • Andre Viera Pigatto, Tzu-Jen Kao, Jennifer L. Mueller, Christopher D. Baker, Emily M. DeBoer, and Oren Kupfer 2021 Imaging ventilation before and after airway clearance therapy in spinal muscular atrophy using electrical impedance tomography, Respir Physiol Neurobiol., 294 Dec.:103773. doi: 10.1016/j.resp.2021.103773. PMID: 34400355
  • M. Alsaker, D. C´ardenas, S. Furuie, and J. L. Mueller 2021 Complementary use of priors for pulmonary imaging with electrical impedance and ultrasound computed tomography, Journal of Computational and Applied Mathematics, 395, 113591.
  • Rashmi Murthy, Yi-Hsuan Lin, Kwancheol Shin, and Jennifer L. Mueller 2020 A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calder´on’s method in the plane, Inverse Problems, 36, no. 12, 125008, doi: 10.1088/1361-6420/abbe5f.
  • K. Shin, S. Ahmad, and J. L. Mueller 2021 Three dimensional Calder´on’s method for EIT on the cylindrical geometry, IEEE Transactions on Biomedical Engineering, 68, no. 5, pp. 1487-1495, doi: 10.1109/TBME.2020.3039197.
  • T. Santos, R. M. Nakanishi, J. P. Kaipio, J. L. Mueller, and R. G. Lima 2020 Introduction of sample based prior into the D-Bar method through a Schur complement property, IEEE Transactions on Medical Imaging, 39, no. 12, doi: 10.1109/TMI.2020.3012428
  • J. L. Mueller and S. Siltanen 2020 The D-bar method for electrical impedance tomography – demystified, Inverse Problems, 36, no. 9, 093001, https://doi.org/10.1088/1361-6420/aba2f5.
  • K. Shin and J. L. Mueller 2020 A second order Calderon’s method with a correction term and a priori information, Inverse Problems, 36, no. 12, 124005, http://iopscience.iop.org/10.1088/1361-6420/abb014.
  • M. Capps and J. L. Mueller 2021 Reconstruction of Organ Boundaries with Deep
    Learning in the D-bar Method for Electrical Impedance Tomography, IEEE Transactions on Biomedical Engineering, 68, no. 3, pp. 826-833, doi: 10.1109/TBME.2020.3006175.
  • Thiago de Castro Martins, Andr´e Kubagawa Sato, Fernando Silva de Moura, Erick
    Dario Le´on Bueno de Camargo, Olavo Luppi Silva, Talles Batista Rattis Santos, Zhanqi
    Zhao, Knut M oeller, Marcelo Brito Passos Amato, Jennifer L. Mueller, Raul Gonzalez Lima, Marcos de Sales Guerra Tsuzuki 2019 A review of electrical impedance
    tomography in lung applications: Theory and algorithms for absolute images, Annual
    Reviews in Control, 48, pp. 442–471.
Referred Proceedings Publications
  • Rosa, N. Mueller, J. L., Pigatto A.V., Kao, T-J, Stoffel, N., Bromirski, C., Davenport, D., Offner, P., and Burnham, E. 2022 Ventilation Volume of Mechanically Ventilated Patients from EIT Data on a Novel Electrode Solution: A Case Study, Proceedings of the International Conference on Bioelectromagnetism, Electrical Bioimpedance, and Electrical Impedance Tomography, Seoul, South Korea
  • Andre Viera Pigatto, Luca Giacobbo, Ely Lopes Filho, Raul Gonzalez Lima, and Jennifer L. Mueller 2022 Design and calibration of a Tonpilz transducer for low frequency medical ultrasound tomography, Proceedings of the the 44th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC’22), Glasgow, United Kingdom. PMID: 36086323 DOI: 10.1109/EMBC48229.2022.9872007
  • Tzu-Jen Kao, Bruce Amm, David Isaacson, Jonathan Newell, Gary Saulnier, and Jennifer L. Mueller 2020 A 3D Reconstruction Algorithm for Real-time Simultaneous Multi-Source EIT Imaging for Lung Function Monitoring, Proceedings of the Nordic Baltic Conference on Biomedical Engineering and Medical Physics, Reykjavik, Iceland.
  • Mueller JL, C´ardenas D, and Furuie S, A preclinical simulation study of ultrasound tomography for pulmonary bedside monitoring, Proceedings of the Second International Workshop on Medical Ultrasound Tomography (MUSTII), 2019.
  • E. M. Lopes Filho, A. V. Pigatto, J. L. Mueller, and R. G. Lima, Design and performance of a Tonpilz transducer for low frequency medical ultrasound tomography, Proceedings of the Second International Workshop on Medical Ultrasound Tomography (MUSTII), 2019.
  • M. M. Mellenthin and J. L. Mueller, 2018, Physiologically Inspired Model of the Skin for Use in Electrical Impedance Tomography, Proceedings of the 2018 International Applied Computational Electromagnetics Society Symposium, Denver, CO.
  • P. A. Muller and J. L. Mueller, 2018, Reconstruction of Complex Conductivities by Caldern’s Method on Subject-Specific Domains, Proceedings of the 2018 International Applied Computational Electromagnetics Society Symposium, Denver, CO.
  •  M. Alsaker and J. L. Mueller, 2018, Spatial Priors in the D-bar Method for Human Thoracic Electrical Impedance Tomography Data, Proceedings of the 2018 International Applied Computational Electromagnetics Society Symposium, Denver, CO.
  • T. B. R. Santos, E. L. B. de Camargo, J. L. Mueller and R. G. Lima, 2018, Introduction of statistical priors into the D-bar method for electrical impedance tomography, Proceedings of the 2018 International Applied Computational Electromagnetics Society Symposium, Denver, CO.
  • J. L. Mueller, M. M. Mellenthin, P. Muller, R. R. Deterding, and S. D. Sagel, 2015, Electrical Imaging of Patients with Cystic Fibrosis. In: Proceedings of the NIH-IEEE 2015 Strategic Conference on Healthcare Innovation and Point-of-Care Technologies for Precision Medicine.
  • M. Mellenthin, J. Mueller, E. D. L. B. Camargo, F. S. de Moura, S. J. Hamilton, R. Gonzalez Lima, 2015, The ACE1 Thoracic Electrical Impedance Tomography System for Ventilation and Perfusion. In: Proceedings of the 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 4073-4076.
  • E. Murphy and J. Mueller, 2008, Effect of errors in domain shape modeling in 2-D reconstructions by the D-bar method, In: Proceedings of the Conference on Electrical Impedance Tomography, Dartmouth University.
  • E. Murphy and J. Mueller, 2008, Effect of errors in domain shape modeling in 2-D reconstructions by the D-bar method, In: Proceedings of the Conference on Electrical Impedance Tomography, Dartmouth University.
  • J. Mueller, 2004, EIT Reconstructions and Faddeev Solutions for a Numerically Simulated Phantom Chest, In: Biomedical Sciences Instrumentation, Vol. 40, ISA Volume 449, (S. James and H. Valenta, editors) pp. 213-218
  • J. Mueller and D. Isaacson, 2004, Regularization of the computed scattering transform for the D-bar method for electrical impedance tomography, In: Proceedings of the 2004 SPIE Annual Meeting Vol. 5562, (Bones, Fiddy, and Millane, editors) pp. 121-128
  • J. Bikowski and J. Mueller, 2004, Electrical impedance tomography and the fast multipole method, In: Proceedings of the 2004 SPIE Annual Meeting Vol. 5562, (Bones, Fiddy, and Millane, editors) pp. 129-140.
Plenary Talks

  • Electrical Impedance Tomography in Healthcare, Invited Plenary Lecture, Scientific Computing, Informatics, Applied Mathematics, and Healthcare Technologies Symposium, CU Anschutz School of Medicine, Aurora CO, USA, May 4, 2026.

  • Electrical Impedance Tomography in 2025, Invited Plenary Lecture, Applied Inverse Problems Conference, Rio de Janeiro, Brazil, Aug. 1, 2025.