TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Hybrid ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Monthly Journal: Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Inverse problem with an operator that depends on unknown parameter 

* , ,

* Corresponding author · click or hover a name for details

pp. 841–859Vol. 29Issue 4April 2026DOI: 10.47974/JIM-2280XML
Received:
11 Dec 2024
Published Online:
06 Feb 2026
Article type:
Research Article
Language:
EN
Article no.:
JIM-2280
Pages:
841–859

Abstract

In this work, we propose a method for solving a linear inverse problem where the operator depends on an unknown parameter. Our approach involves approximating the exact solution using a stochastic iterative Landweber-type scheme, while the optimal parameter value is determined by minimizing a least squares functional.  An exponential inequality will be established to demonstrate the almost complete convergence of the method, specifying both the rate of convergence and the confidence interval within which the solution resides.  Finally, we present a numerical application involving a Fredholm integral equation of the first kind, where the kernel depends on an unknown parameter. 

Keywords

Subject Classifications

45Q0565N21

References

[1] R.C. Allen, W.R. Boland, V. Faber and G.M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” Journal of mathematical analysis, Vol. 109, no. 2, pp. 564-590, (1985). 
[2] G. Arfken, F.E. Harris and H. Weber, Mathematical Methods for Physicists. Academic Press, (2012). 
[3] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind. Cambridge Univ. Press, Cambridge, (1997). 
[4] S.E. Blanke, B.N. Hahn and A. Wald, “Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging,” Inverse Problems, vol. 36, (2020). 
[5] I.R. Bleyer and R.Ramlau, “A double regularization approach for inverse problems with noisy data and inexact operator,” Inverse Problems, vol. 29, (2013). 
[6] L. Bungert, M. Burger, Y. Korolev and C.B. Schönlieb, “Variational regularisation for inverse problems with imperfect forward operators and general noise models,” Inverse Problems, vol. 36, (2020). 
[7] M. Burger, Y. Korolev and J. Rasch, “Convergence rates and structure of solutions of inverse problems with imperfect forward models,” Inverse Problems, vol. 35, (2019). 
[8] L. Cavalier and N.W. Hengartner, “Adaptive estimation for inverse problems with noisy operators,” Inverse Problems, vol. 21, no. 4, pp. 1345-1361, (2005). 
[9] X. Chen and D. Pouzo, “On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and op-tion pricing,” Science in China Series A : Mathematics, vol. 52, no. 6, (2009). 
[10] J.B. Conway, A Course in Functional Analysis. 2nd ed. Springer-Verlag, (1990). 
[11] A. Dahmani and F. Bouhmila, “Consistency of Landweber algorithm in ill-posed problem with random data,” C. R. Acad. Sci. Paris, vol. 343, pp. 487-491, (2006). 
[12] E.H. Doha, A.H. Bhrawy and S.S. Ezz-Eldien, “A new Jacobi operational matrix : an application for solving fractional differential equations,” Appl. Math. Model, vol. 36, pp. 4931-4943, (2012). 
[13] S. Efromovich and V. Koltchinskii, “On inverse problems with unknown operators,” IEEE Trans. Inform. Theory, vol. 47, no. 7, pp. 2876-2894, (2001). 
[14] G.L. Frontini and E.M. Fernandez Berdaguer, “Inversion of elastic light scattering measurements to determine refractive index and particle size distribution of polymeric emulsions,” Inverse Problems in Engineering, vol. 11, no. 4, pp. 329-340, (2003). 
[15] B.Y. Guo and Z.Q. Wang, “Legendre-Gauss collocation methods for ordinary differential equations,” Adv. Comput. Math, vol. 30, pp. 249-280, (2009). 
[16] C.W. Groetsch, “Integral equations of the first kind, inverse problems and regularization : A crash course,” Journal of Physics : Conference Series, 73 012001, (2007). 
[17] J. Hadamard, “Lectures on Chauchy’s Problem in Linear Partial Differential Equations,” Yale University Press, New Haven, (1923). 
[18] P.C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numerical Algorithms, vol. 29, no. 4, pp. 323-378, (2002). 
[19] W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, vol. 58, pp. 13-30, (1963). 
[20] M. Hoffmann and M. Reiß, “Nonlinear estimation for linear inverse problems with error in the operator,” Ann. Statist, vol. 36, no. 1, pp. 310-336, (2008). 
[21] M.B. Kadhem, M.R. Nasif, “Quadratic non-polynomial spline approximation for non-linear Volterra-Fredholm integral equations of the second kind,” Journal of Interdisciplinary Mathematics, vol. 25, no. 5, 1383-1390, (2022). 
[22] A. Kirsch, An introduction to the Mathematical Theory of Inverse Problems. Applied mathematical sciences, Springer, vol. 120, New-York, (1996). 
[23] R. Kress, Linear Integral Equations. Springer-Verlag, vol. 82, New York, (1989). 
[24] L. Kronecker, “Quelques remarques sur la détermination des valeurs moyennes,” C. R. Acad. Sci. Paris., vol. 103, pp. 980-987, (1886). 
[25] L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Amer. J. Math., vol. 73, pp. 615-624, (1951). 
[26] M.M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics. Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, English Transl, Springer-Verlag Tracts in Natural Philosophy, Springer-Verlag, vol. 2, Berlin, (1967). 
[27] F. Maouche, “An iterative stochastic procedure with a general step to a linear regular inverse problem,” Filomat, vol. 37, no. 6, pp. 1723-1732, (2023). 
[28] C. Marteau, “Regularization of inverse problems with unknown operator,” Math. Methods. Statist., vol. 15, no. 4, pp. 415-443, (2007). 
[29] N. Sarkar, M. Sen and D. Saha, “Solution of non linear Fredholm integral equation involving constant delay by BEM with piecewise linear approximation,” Journal of Interdisciplinary Mathematics, vol. 23, no. 2, 537-544, (2020). 
[30] G. Szego, “Orthogonal polynomials,” American Mathematical Society. Colloquium Publications., vol. 23, (1967). 
[31] M. Trabs, “Bayesian inverse problems with unknown operators,” Inverse Problems, vol. 34, no. 8, (2018).

Views: 323Downloads: 88Citations: 0