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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[22]

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

[23]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[22]

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

[23]

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

[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.

[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.

[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.

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
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Figure 2.  The estimated rate constant distribution by Algorithm 1
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Figure 3.  The measured individual responses and the simulated responses by Algorithm 1
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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
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$ \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
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$ (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
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