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

Ye Zhang; Bernd Hofmann

doi:10.3934/ipi.2020062

Article Contents

2021, Volume 15, Issue 2: 229-256. Doi: 10.3934/ipi.2020062

This issue Previous Article Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption Next Article Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex

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

Received: February 2020

Revised: July 2020

Early access: October 2020

Published: April 2021

Abstract / Introduction Full Text(HTML) Figure(3) / Table(2) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 47A52, 65J20, 65F22, 65R30.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Figure 2. The estimated rate constant distribution by Algorithm 1

Download: Full-size image PowerPoint slide

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

Download: Full-size image 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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.