American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex
  • IPI Home
  • This Issue
  • Next Article
    Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption
April  2021, 15(2): 229-256. doi: 10.3934/ipi.2020062

Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

Ye Zhang 1,2, and  Bernd Hofmann 3,,

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China
2. 

Shenzhen MSU-BIT University, 518172 Shenzhen, China
3. 

Faculty of Mathematics, Chemnitz University of Technology, Reichenhainer Str. 39/41, 09107 Chemnitz, Germany

* Corresponding author: hofmannb@mathematik.tu-chemnitz.de

Received  February 2020 Revised  July 2020 Published  April 2021 Early access  October 2020

Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods exhibit strong convergence. There are presented convergence results, even for a combination of noisy right-hand side and imperfect forward operators, and for one of the approaches there are also convergence rates results. Specifically adapted discrepancy principles are used as a posteriori stopping rules of the established iterative regularization algorithms. For an application of the suggested new approaches, we consider a biosensor problem, which is modelled as a two dimensional linear Fredholm integral equation of the first kind. Several numerical examples, as well as a comparison with the projected Landweber method, are presented to show the accuracy and the acceleration effect of the novel methods. Case studies of a real data problem indicate that the developed methods can produce meaningful featured regularized solutions.

Keywords: Ill-posed inverse problems, non-negativity constraint, regularization, iterative scheme, convergence rate.
Mathematics Subject Classification: 47A52, 65J20, 65F22, 65R30.
Citation: Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems & Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062
References:
[1]

V. Albani, P. Elbau, M. de Hoop and O. Scherzer, Optimal convergence rates results for linear inverse problems in Hilbert spaces, Numerical Functional Analysis and Optimization, 37 (2016), 521-540.  doi: 10.1080/01630563.2016.1144070.  Google Scholar

[2]

K. Atkinson and W. Han, Theoreitcal Numerical Analysis: A Functional Analysis Framework. Third Edition, Springer: New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

[3]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer: Cham, 2017. doi: 10.1007/978-3-319-48311-5.  Google Scholar

[4]

C. Clason, B. Kaltenbacher and E. Resmerita, Regularization of ill-posed problems with non-negative solutions, Splitting Algorithms, Modern Operator Theory and Applications, H. Bauschke, R. Burachik, R. Luke (eds.), 2019,113–135.  Google Scholar

[5]

A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society: Series B, 39 (1977), 1-38.  doi: 10.1111/j.2517-6161.1977.tb01600.x.  Google Scholar

[6]

B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numerical Functional Analysis and Optimization, 13 (1992), 413-429.  doi: 10.1080/01630569208816489.  Google Scholar

[7]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.  Google Scholar

[8]

J. Flemming and B. Hofmann, Convergence rates in constrained Tikhonov regularization: Equivalence of projected source conditions and variational inequalities, Inverse Problems, 27 (2011), 085001, 11pp. doi: 10.1088/0266-5611/27/8/085001.  Google Scholar

[9]

M. Haltmeier, A. Leitao and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008, 17pp. doi: 10.1088/0266-5611/25/7/075008.  Google Scholar

[10]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar

[11]

G. Helmberg, Introduction to Spectral Theory in Hilbert Spaces, North Holland: Amsterdam, 1969.  Google Scholar

[12]

B. Hofmann and R. Plato, On ill-posedness concepts, stable solvability and saturation, J. Inverse Ill-Posed Probl., 26 (2018), 287-297.  doi: 10.1515/jiip-2017-0090.  Google Scholar

[13]

Y. Korolev, Making use of a partial order in solving inverse problems: II, Inverse Problems, 30 (2014), 085003, 9pp. doi: 10.1088/0266-5611/30/8/085003.  Google Scholar

[14]

R. Lagendijk, J. Biemond and D. Boekee, Regularized iterative image restoration with ringing reduction, IEEE Transactions on Acoustics Speech and Signal Processing, 36 (1988), 1874-1888.  doi: 10.1109/29.9032.  Google Scholar

[15]

P. Lions, Approximation de points fixes de contractions, Comptes rendus de l'Académie des sciences, Série A-B Paris, 284 (1977), 1357-1359.   Google Scholar

[16]

P. Mathé and S. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803.  doi: 10.1088/0266-5611/19/3/319.  Google Scholar

[17]

A. Neubauer, Tikhonov-regularization of ill-posed linear operator equations on closed convex sets, Journal of Approximation Theory, 53 (1988), 304-320.  doi: 10.1016/0021-9045(88)90025-1.  Google Scholar

[18]

A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM Journal on Numerical Analysis, 34 (1997), 517-527.  doi: 10.1137/S0036142993253928.  Google Scholar

[19]

M. Piana and M. Bertero, Projected Landweber method and preconditioning, Inverse Problems, 13 (1997), 441-463.  doi: 10.1088/0266-5611/13/2/016.  Google Scholar

[20]

E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence, Constructive Methods for the Practical Treatment of Integral Equations, 73 (1985), 234-243.   Google Scholar

[21]

A. Tikhonov, A. Goncharsky, V. Stepanov and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer: Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7.  Google Scholar

[22]

G. Vainikko and A. Veretennikov, Iteration Procedures in Ill-Posed Problems, Moscow: Nauka (In Russian), 1986.  Google Scholar

[23]

R. Wittmann, Approximation of fixed points of non-expansive mappings, Arch. Math., 58 (1992), 486-491.  doi: 10.1007/BF01190119.  Google Scholar

[24]

Y. Zhang, P. Forssén, T. Fornstedt, M. Gulliksson and X. Dai, An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data, Inverse Problems in Science & Engineering, 26 (2018), 1464-1489.  doi: 10.1080/17415977.2017.1411912.  Google Scholar

[25]

Y. Zhang and B. Hofmann, On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces, Fractional Calculus and Applied Analysis, 22 (2019), 699-721.  doi: 10.1515/fca-2019-0039.  Google Scholar

[26]

Y. Zhang and B. Hofmann, On the second order asymptotical regularization of linear ill-posed inverse problems, Applicable Analysis, 99 (2020), 1000-1025.  doi: 10.1080/00036811.2018.1517412.  Google Scholar

show all references

References:
[1]

V. Albani, P. Elbau, M. de Hoop and O. Scherzer, Optimal convergence rates results for linear inverse problems in Hilbert spaces, Numerical Functional Analysis and Optimization, 37 (2016), 521-540.  doi: 10.1080/01630563.2016.1144070.  Google Scholar

[2]

K. Atkinson and W. Han, Theoreitcal Numerical Analysis: A Functional Analysis Framework. Third Edition, Springer: New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

[3]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer: Cham, 2017. doi: 10.1007/978-3-319-48311-5.  Google Scholar

[4]

C. Clason, B. Kaltenbacher and E. Resmerita, Regularization of ill-posed problems with non-negative solutions, Splitting Algorithms, Modern Operator Theory and Applications, H. Bauschke, R. Burachik, R. Luke (eds.), 2019,113–135.  Google Scholar

[5]

A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society: Series B, 39 (1977), 1-38.  doi: 10.1111/j.2517-6161.1977.tb01600.x.  Google Scholar

[6]

B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numerical Functional Analysis and Optimization, 13 (1992), 413-429.  doi: 10.1080/01630569208816489.  Google Scholar

[7]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.  Google Scholar

[8]

J. Flemming and B. Hofmann, Convergence rates in constrained Tikhonov regularization: Equivalence of projected source conditions and variational inequalities, Inverse Problems, 27 (2011), 085001, 11pp. doi: 10.1088/0266-5611/27/8/085001.  Google Scholar

[9]

M. Haltmeier, A. Leitao and E. Resmerita, On regularization methods of EM-Kaczmarz type, Inverse Problems, 25 (2009), 075008, 17pp. doi: 10.1088/0266-5611/25/7/075008.  Google Scholar

[10]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numerische Mathematik, 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar

[11]

G. Helmberg, Introduction to Spectral Theory in Hilbert Spaces, North Holland: Amsterdam, 1969.  Google Scholar

[12]

B. Hofmann and R. Plato, On ill-posedness concepts, stable solvability and saturation, J. Inverse Ill-Posed Probl., 26 (2018), 287-297.  doi: 10.1515/jiip-2017-0090.  Google Scholar

[13]

Y. Korolev, Making use of a partial order in solving inverse problems: II, Inverse Problems, 30 (2014), 085003, 9pp. doi: 10.1088/0266-5611/30/8/085003.  Google Scholar

[14]

R. Lagendijk, J. Biemond and D. Boekee, Regularized iterative image restoration with ringing reduction, IEEE Transactions on Acoustics Speech and Signal Processing, 36 (1988), 1874-1888.  doi: 10.1109/29.9032.  Google Scholar

[15]

P. Lions, Approximation de points fixes de contractions, Comptes rendus de l'Académie des sciences, Série A-B Paris, 284 (1977), 1357-1359.   Google Scholar

[16]

P. Mathé and S. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803.  doi: 10.1088/0266-5611/19/3/319.  Google Scholar

[17]

A. Neubauer, Tikhonov-regularization of ill-posed linear operator equations on closed convex sets, Journal of Approximation Theory, 53 (1988), 304-320.  doi: 10.1016/0021-9045(88)90025-1.  Google Scholar

[18]

A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM Journal on Numerical Analysis, 34 (1997), 517-527.  doi: 10.1137/S0036142993253928.  Google Scholar

[19]

M. Piana and M. Bertero, Projected Landweber method and preconditioning, Inverse Problems, 13 (1997), 441-463.  doi: 10.1088/0266-5611/13/2/016.  Google Scholar

[20]

E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence, Constructive Methods for the Practical Treatment of Integral Equations, 73 (1985), 234-243.   Google Scholar

[21]

A. Tikhonov, A. Goncharsky, V. Stepanov and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer: Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7.  Google Scholar

[22]

G. Vainikko and A. Veretennikov, Iteration Procedures in Ill-Posed Problems, Moscow: Nauka (In Russian), 1986.  Google Scholar

[23]

R. Wittmann, Approximation of fixed points of non-expansive mappings, Arch. Math., 58 (1992), 486-491.  doi: 10.1007/BF01190119.  Google Scholar

[24]

Y. Zhang, P. Forssén, T. Fornstedt, M. Gulliksson and X. Dai, An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data, Inverse Problems in Science & Engineering, 26 (2018), 1464-1489.  doi: 10.1080/17415977.2017.1411912.  Google Scholar

[25]

Y. Zhang and B. Hofmann, On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces, Fractional Calculus and Applied Analysis, 22 (2019), 699-721.  doi: 10.1515/fca-2019-0039.  Google Scholar

[26]

Y. Zhang and B. Hofmann, On the second order asymptotical regularization of linear ill-posed inverse problems, Applicable Analysis, 99 (2020), 1000-1025.  doi: 10.1080/00036811.2018.1517412.  Google Scholar

Figure 1.  The evolution of L2-norm relative errors 'L2Err' for different methods for Example 1 with noise levels $ h' = \delta' = 5\% $. Upper (left): Algorithm 2; Upper (right): Algorithm 1; Lower (left): Landweber P1; Lower (right): Landweber P2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The estimated rate constant distribution by Algorithm 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The measured individual responses and the simulated responses by Algorithm 1
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  The iterative number $ k^* $ and the corresponding relative error L2Err vs $ \mathbf{G} $. $ h' = \delta' = 0.1\% $. $ C^\dagger = 1.1, \tau_0 = 1.1 $ in Algorithms 1 and 2, and $ \alpha_k = 1/k $ in Algorithm 2
$ \mathbf{G} $ Algorithm 1 Algorithm 2
Example 1 Example 2 Example 1 Example 2
L2Err $ k^* $ L2Err $ k^* $ L2Err $ k^* $ L2Err $ k^* $
$ \mathbf{G}_1 $ 0.0138 $ N_{\max}=10^6 $ 0.0006 228910 0.0009 $ N_{\max}=10^6 $ 0.0037 $ N_{\max}=10^6 $
$ \mathbf{G}_2 $ 0.0086 $ N_{\max}=10^6 $ 0.0013 64526 2.0745e-5 $ N_{\max}=10^6 $ 0.0002 129082
$ \mathbf{G}_3 $ 0.0003 188765 0.0467 122507 8.8714e-5 $ N_{\max}=10^6 $ 0.0243 594791
$ \mathbf{G}_4 $ 0.0002 24696 0.0506 13537 0.0004 37974 0.0293 41965
$ \mathbf{G}_5 $ 0.0318 20647 0.0022 35901 0.0229 27229 0.0012 75392
$ \mathbf{G}_6 $ 0.0649 38976 0.0562 7626 0.0142 52076 0.0116 38853
$ \mathbf{G}_7 $ 0.0002 79863 0.0074 13138 0.0003 56564 0.0016 67004
$ \mathbf{G}_8 $ 0.0570 12326 0.0526 18004 0.0215 24315 0.0159 91825
Table Options

Download as excel

$ \mathbf{G} $ Algorithm 1 Algorithm 2
Example 1 Example 2 Example 1 Example 2
L2Err $ k^* $ L2Err $ k^* $ L2Err $ k^* $ L2Err $ k^* $
$ \mathbf{G}_1 $ 0.0138 $ N_{\max}=10^6 $ 0.0006 228910 0.0009 $ N_{\max}=10^6 $ 0.0037 $ N_{\max}=10^6 $
$ \mathbf{G}_2 $ 0.0086 $ N_{\max}=10^6 $ 0.0013 64526 2.0745e-5 $ N_{\max}=10^6 $ 0.0002 129082
$ \mathbf{G}_3 $ 0.0003 188765 0.0467 122507 8.8714e-5 $ N_{\max}=10^6 $ 0.0243 594791
$ \mathbf{G}_4 $ 0.0002 24696 0.0506 13537 0.0004 37974 0.0293 41965
$ \mathbf{G}_5 $ 0.0318 20647 0.0022 35901 0.0229 27229 0.0012 75392
$ \mathbf{G}_6 $ 0.0649 38976 0.0562 7626 0.0142 52076 0.0116 38853
$ \mathbf{G}_7 $ 0.0002 79863 0.0074 13138 0.0003 56564 0.0016 67004
$ \mathbf{G}_8 $ 0.0570 12326 0.0526 18004 0.0215 24315 0.0159 91825
Table 2.  Comparison with the projected Landweber methods. The CPU time is measured in seconds
$ (h', \delta') $ $ (0.1\%, 0.1\%) $ $ (1\%, 1\%) $ $ (5\%, 5\%) $
Example 1
Methods L2Err $ k^* $ CPU L2Err $ k^* $ CPU L2Err $ k^* $ CPU
Landweber P1 0.4310 $ N_{\max}=10^6 $ 3.6142e3 0.4528 370895 395.3281 0.5158 1130 0.0156
Landweber P2 0.4310 $ N_{\max}=10^6 $ 3.6257e3 0.4905 63599 2.3281 0.4964 43438 1.2813
Algorithm 1 0.0002 79863 44.7344 0.0008 63602 34.7969 0.0053 43438 19.5625
Algorithm 2 0.0003 56235 43.6212 0.0005 62941 47.3762 0.0021 60257 42.8194
Example 2
Methods L2Err $ k^* $ CPU L2Err $ k^* $ CPU L2Err $ k^* $ CPU
Landweber P1 0.9285 229498 1.0150e3 0.9360 57647 44.6563 0.9630 13 0.1719
Landweber P2 0.9611 1989 1.0313 0.9615 1573 0.7656 0.9619 1055 0.5469
Algorithm 1 0.0007 1999 4.4219 0.0030 1575 3.4063 0.0195 1059 2.4375
Algorithm 2 0.0002 3432 5.0292 0.0016 2162 5.0594 0.0025 4284 5.6638
Table Options

Download as excel

$ (h', \delta') $ $ (0.1\%, 0.1\%) $ $ (1\%, 1\%) $ $ (5\%, 5\%) $
Example 1
Methods L2Err $ k^* $ CPU L2Err $ k^* $ CPU L2Err $ k^* $ CPU
Landweber P1 0.4310 $ N_{\max}=10^6 $ 3.6142e3 0.4528 370895 395.3281 0.5158 1130 0.0156
Landweber P2 0.4310 $ N_{\max}=10^6 $ 3.6257e3 0.4905 63599 2.3281 0.4964 43438 1.2813
Algorithm 1 0.0002 79863 44.7344 0.0008 63602 34.7969 0.0053 43438 19.5625
Algorithm 2 0.0003 56235 43.6212 0.0005 62941 47.3762 0.0021 60257 42.8194
Example 2
Methods L2Err $ k^* $ CPU L2Err $ k^* $ CPU L2Err $ k^* $ CPU
Landweber P1 0.9285 229498 1.0150e3 0.9360 57647 44.6563 0.9630 13 0.1719
Landweber P2 0.9611 1989 1.0313 0.9615 1573 0.7656 0.9619 1055 0.5469
Algorithm 1 0.0007 1999 4.4219 0.0030 1575 3.4063 0.0195 1059 2.4375
Algorithm 2 0.0002 3432 5.0292 0.0016 2162 5.0594 0.0025 4284 5.6638
[1]

Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083
[2]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
[3]

Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259
[4]

Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011
[5]

Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems & Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047
[6]

Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467
[7]

Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155
[8]

Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479
[9]

Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems & Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409
[10]

Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2010, 4 (3) : 335-350. doi: 10.3934/ipi.2010.4.335
[11]

Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems & Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289
[12]

Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems & Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971
[13]

Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications. Inverse Problems & Imaging, 2007, 1 (3) : 507-523. doi: 10.3934/ipi.2007.1.507
[14]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[15]

Zonghao Li, Caibin Zeng. Center manifolds for ill-posed stochastic evolution equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021142
[16]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[17]

Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems & Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395
[18]

Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009
[19]

Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293
[20]

Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & Control Theory, 2014, 3 (3) : 499-524. doi: 10.3934/eect.2014.3.499

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (246)
  • HTML views (225)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy