On the randomised stability constant for inverse problems

Giovanni S. Alberti; Yves Capdeboscq; Yannick Privat; Giovanni S. Alberti; Yves Capdeboscq; Yannick Privat

doi:10.3934/mine.2020013

Mathematics in Engineering

2020, Volume 2, Issue 2: 264-286. doi: 10.3934/mine.2020013

Previous Article Next Article

Research article Special Issues

On the randomised stability constant for inverse problems

1.
MaLGa Center, Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genova, Italy
2.
Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75013 Paris, France
3.
Sorbonne Université, CNRS, LJLL, F-75005 Paris, France
4.
IRMA, Université de Strasbourg, CNRS UMR 7501, 7 rue René Descartes, 67084 Strasbourg, France

Received: 23 March 2019 Accepted: 17 December 2019 Published: 12 February 2020

In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.
- inverse problems,
- observability constant,
- compressed sensing,
- passive imaging,
- regularisation,
- randomisation,
- deep learning,
- electrical impedance tomography
Citation: Giovanni S. Alberti, Yves Capdeboscq, Yannick Privat. On the randomised stability constant for inverse problems[J]. Mathematics in Engineering, 2020, 2(2): 264-286. doi: 10.3934/mine.2020013

Related Papers:

Abstract

In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.

References

[1]	Akhtar N, Mian A (2018) Threat of adversarial attacks on deep learning in computer vision: A survey. IEEE Access 6: 14410-14430. doi: 10.1109/ACCESS.2018.2807385
[2]	Alberti GS, Capdeboscq Y (2018) Lectures on Elliptic Methods for Hybrid Inverse Problems, Paris: Société Mathématique de France.
[3]	Ammari H (2008) An Introduction to Mathematics of Emerging Biomedical Imaging, Berlin: Springer.
[4]	Ammari H, Garnier J, Kang H, et al. (2017) Multi-Wave Medical Imaging: Mathematical Modelling & Imaging Reconstruction, World Scientific.
[5]	Antholzer S, Haltmeier M, Schwab J (2018) Deep learning for photoacoustic tomography from sparse data. Inverse Probl Sci Eng 27: 987-1005.
[6]	Antun V, Renna F, Poon C, et al. (2019) On instabilities of deep learning in image reconstruction-Does AI come at a cost?. arXiv preprint arXiv:1902.05300.
[7]	Arridge S, Hauptmann A (2019) Networks for nonlinear diffusion problems in imaging. J Math Imaging Vis DOI: https://xs.scihub.ltd/https://doi.org/10.1007/s10851-019-00901-3.
[8]	Bardos C, Lebeau G, Rauch J (1992) Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J Control Optim 30: 1024-1065. doi: 10.1137/0330055
[9]	Berg J, Nyström K (2017) Neural network augmented inverse problems for PDEs. arXiv preprint arXiv:1712.09685.
[10]	Bubba TA, Kutyniok G, Lassas M, et al. (2019) Learning the invisible: A hybrid deep learning-shearlet framework for limited angle computed tomography. Inverse Probl 35: 064002. doi: 10.1088/1361-6420/ab10ca
[11]	Burq N (2010) Random data Cauchy theory for dispersive partial differential equations. In: Proceedings of the International Congress of Mathematicians, New Delhi: Hindustan Book Agency, 3: 1862-1883.
[12]	Burq N, Tzvetkov N (2008) Random data Cauchy theory for supercritical wave equations II: A global existence result. Invent Math 173: 477-496. doi: 10.1007/s00222-008-0123-0
[13]	Burq N, Tzvetkov N (2014) Probabilistic well-posedness for the cubic wave equation. J Eur Math Soc 16: 1-30. doi: 10.4171/JEMS/426
[14]	Candès EJ, Romberg J (2007) Sparsity and incoherence in compressive sampling. Inverse Probl 23: 969-985. doi: 10.1088/0266-5611/23/3/008
[15]	Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inform Theory 52: 489-509. doi: 10.1109/TIT.2005.862083
[16]	Donoho DL (2006) Compressed sensing. IEEE Trans Inform Theory 52: 1289-1306. doi: 10.1109/TIT.2006.871582
[17]	Engl HW, Hanke M, Neubauer A (1996) Regularization of Inverse Problems, Dordrecht: Kluwer Academic Publishers Group.
[18]	Feng J, Sun Q, Li Z, et al. (2018) Back-propagation neural network-based reconstruction algorithm for diffuse optical tomography. J Biomed Opt 24: 051407.
[19]	Foucart S, Rauhut H (2013) A Mathematical Introduction to Compressive Sensing, New York: Birkhäuser/Springer.
[20]	Garnier J, Papanicolaou G (2016) Passive Imaging with Ambient Noise, Cambridge: Cambridge University Press.
[21]	Goodfellow I, Bengio Y, Courville A (2016) Deep Learning, MIT Press.
[22]	Haltmeier M, Antholzer S, Schwab J (2019) Deep Learning for Image Reconstruction, World Scientific.
[23]	Hamilton SJ, Hauptmann A (2018) Deep d-bar: Real-time electrical impedance tomography imaging with deep neural networks. IEEE Trans Med Imaging 37: 2367-2377. doi: 10.1109/TMI.2018.2828303
[24]	Hasanoğlu AH, Romanov VG (2017) Introduction to Inverse Problems for Differential Equations, Cham: Springer.
[25]	Hauptmann A, Lucka F, Betcke M, et al. (2018) Model-based learning for accelerated, limitedview 3-d photoacoustic tomography. IEEE Trans Med Imaging 37: 1382-1393. doi: 10.1109/TMI.2018.2820382
[26]	Humbert E, Privat Y, Trélat E (2019) Observability properties of the homogeneous wave equation on a closed manifold. Commun Part Diff Eq, 44: 749-772. doi: 10.1080/03605302.2019.1581799
[27]	Isakov V (2006) Inverse Problems for Partial Differential Equations, New York: Springer.
[28]	Jin KH, McCann MT, Froustey E, et al. (2017) Deep convolutional neural network for inverse problems in imaging. IEEE Trans Image Process 26: 4509-4522. doi: 10.1109/TIP.2017.2713099
[29]	Kaltenbacher B, Neubauer A, Scherzer O (2008) Iterative Regularization Methods for Nonlinear ill-posed Problems, Berlin: Walter de Gruyter.
[30]	Kirsch A (2011) An Introduction to the Mathematical Theory of Inverse Problems, New York: Springer.
[31]	Laurent C, Léautaud M (2019) Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves. J Eur Math Soc 21: 957-1069.
[32]	Maass P (2019) Deep learning for trivial inverse problems. In: Compressed Sensing and Its Applications, Cham: Springer International Publishing, 195-209.
[33]	Martin S, Choi CTM (2017) A post-processing method for three-dimensional electrical impedance tomography. Sci Rep 7: 7212. doi: 10.1038/s41598-017-07727-2
[34]	Modin K, Nachman A, Rondi L (2019) A multiscale theory for image registration and nonlinear inverse problems. Adv Math 346: 1009-1066. doi: 10.1016/j.aim.2019.02.014
[35]	Paley R (1930) On some series of functions. Math Proc Cambridge 26: 458-474. doi: 10.1017/S0305004100016212
[36]	Paley R, Zygmund A (1930) On some series of functions, (1). Math Proc Cambridge, 26: 337-357. doi: 10.1017/S0305004100016078
[37]	Paley R, Zygmund A (1932) On some series of functions, (3). Math Proc Cambridge, 28: 190-205. doi: 10.1017/S0305004100010860
[38]	Privat Y, Trélat E, Zuazua E (2013) Optimal observation of the one-dimensional wave equation. J Fourier Anal Appl 19: 514-544. doi: 10.1007/s00041-013-9267-4
[39]	Privat Y, Trélat E, Zuazua E (2015) Optimal shape and location of sensors for parabolic equations with random initial data. Arch Ration Mech An 216: 921-981. doi: 10.1007/s00205-014-0823-0
[40]	Privat Y, Trélat E, Zuazua E (2016) Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains. J Eur Math Soc 18: 1043-1111. doi: 10.4171/JEMS/608
[41]	Privat Y, Trélat E, Zuazua E (2016) Randomised observation, control and stabilization of waves [Based on the plenary lecture presented at the 86th Annual GAMM Conference, Lecce, Italy, March 24, 2015]. ZAMM Z Angew Math Mech 96: 538-549. doi: 10.1002/zamm.201500181
[42]	Privat Y, Trélat E, Zuazua E (2019) Spectral shape optimization for the neumann traces of the dirichlet-laplacian eigenfunctions. Calc Var Partial Dif 58: 64. doi: 10.1007/s00526-019-1522-3
[43]	Rellich F (1940) Darstellung der eigenwerte von δu+λu= 0 durch ein randintegral. Math Z 46: 635-636. doi: 10.1007/BF01181459
[44]	Otmar Scherzer (2015) Handbook of Mathematical Methods in Imaging. Vol. 1, 2, 3. New York: Springer.
[45]	Szegedy C, Zaremba W, Sutskever I, et al. (2013) Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199.
[46]	Tadmor E, Nezzar S, Vese L (2004) A multiscale image representation using hierarchical (BV, L2) decompositions. Multiscale Model Simul 2: 554-579. doi: 10.1137/030600448
[47]	Tarantola A (2005) Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia: Society for Industrial and Applied Mathematics.
[48]	Wei Z, Liu D, Chen X (2019) Dominant-current deep learning scheme for electrical impedance tomography. IEEE Trans Biomedical Eng 66: 2546-2555. doi: 10.1109/TBME.2019.2891676
[49]	Yang G, Yu S, Dong H, et al. (2018) Dagan: Deep de-aliasing generative adversarial networks for fast compressed sensing mri reconstruction. IEEE Trans Med Imaging 37: 1310-1321. doi: 10.1109/TMI.2017.2785879
[50]	Zhu B, Liu JZ, Cauley SF, et al. (2018) Image reconstruction by domain-transform manifold learning. Nature 555: 487-492. doi: 10.1038/nature25988

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)