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Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems
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 |
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.
References:
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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. |



Algorithm 1 | Algorithm 2 | |||||||
Example 1 | Example 2 | Example 1 | Example 2 | |||||
L2Err | L2Err | L2Err | L2Err | |||||
0.0138 | 0.0006 | 228910 | 0.0009 | 0.0037 | ||||
0.0086 | 0.0013 | 64526 | 2.0745e-5 | 0.0002 | 129082 | |||
0.0003 | 188765 | 0.0467 | 122507 | 8.8714e-5 | 0.0243 | 594791 | ||
0.0002 | 24696 | 0.0506 | 13537 | 0.0004 | 37974 | 0.0293 | 41965 | |
0.0318 | 20647 | 0.0022 | 35901 | 0.0229 | 27229 | 0.0012 | 75392 | |
0.0649 | 38976 | 0.0562 | 7626 | 0.0142 | 52076 | 0.0116 | 38853 | |
0.0002 | 79863 | 0.0074 | 13138 | 0.0003 | 56564 | 0.0016 | 67004 | |
0.0570 | 12326 | 0.0526 | 18004 | 0.0215 | 24315 | 0.0159 | 91825 |
Algorithm 1 | Algorithm 2 | |||||||
Example 1 | Example 2 | Example 1 | Example 2 | |||||
L2Err | L2Err | L2Err | L2Err | |||||
0.0138 | 0.0006 | 228910 | 0.0009 | 0.0037 | ||||
0.0086 | 0.0013 | 64526 | 2.0745e-5 | 0.0002 | 129082 | |||
0.0003 | 188765 | 0.0467 | 122507 | 8.8714e-5 | 0.0243 | 594791 | ||
0.0002 | 24696 | 0.0506 | 13537 | 0.0004 | 37974 | 0.0293 | 41965 | |
0.0318 | 20647 | 0.0022 | 35901 | 0.0229 | 27229 | 0.0012 | 75392 | |
0.0649 | 38976 | 0.0562 | 7626 | 0.0142 | 52076 | 0.0116 | 38853 | |
0.0002 | 79863 | 0.0074 | 13138 | 0.0003 | 56564 | 0.0016 | 67004 | |
0.0570 | 12326 | 0.0526 | 18004 | 0.0215 | 24315 | 0.0159 | 91825 |
Example 1 | |||||||||
Methods | L2Err | CPU | L2Err | CPU | L2Err | CPU | |||
Landweber P1 | 0.4310 | 3.6142e3 | 0.4528 | 370895 | 395.3281 | 0.5158 | 1130 | 0.0156 | |
Landweber P2 | 0.4310 | 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 | CPU | L2Err | CPU | L2Err | 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 |
Example 1 | |||||||||
Methods | L2Err | CPU | L2Err | CPU | L2Err | CPU | |||
Landweber P1 | 0.4310 | 3.6142e3 | 0.4528 | 370895 | 395.3281 | 0.5158 | 1130 | 0.0156 | |
Landweber P2 | 0.4310 | 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 | CPU | L2Err | CPU | L2Err | 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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