The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by
, , , , where A is a positive self-adjoint
and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard .
We shall give a new regularization method for this problem when the operator A
is replaced by and is replaced by a nonlocal condition. We show the convergence of this
method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori
regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.
G. Alessandrini, L. Rondi, E. Rosset and S. Vessella,
The stability for the Cauchy problem for elliptic equations,
Inverse Problems 25 (2009), no. 12, Article ID 123004.
F. V. Bazan, L. S. Borges and J. B. Francisco,
On a generalization of Reginska’s parameter choice rule and its numerical realization in large-scale multi-parameter Tikhonov regularization,
Appl. Math. Comput. 219 (2012), 2100–2113.
M. Belge, M. E. Kilmer and E. L. Miller,
Effcient determination of multiple regularization parameters in a generalized L-curve framework,
Inverse Problems 18 (2002), 1161–1183.
F. Berntsson, V. A. Kozlov, L. Mpinganzimaa and B. O. Turesson,
An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation,
Comput. Math. Appl. 68 (2014), 44–60.
L. Bourgeois and J. Dard,
About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains,
Appl. Anal. 89 (2010), no. 11, 1745–1768.
N. Boussetila, S. Hamida and F. Rebbani,
Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem,
Abstr. Appl. Anal. 2013 (2013), Article ID 947379.
N. Boussetila and F. Rebbani,
The modified quasi-reversibility method for ill-posed evolution problems with two-dimensional time,
Analytic Methods of Analysis and Differential Equations: AMADE 2003 (Minsk 2003),
Cambridge Scientific Publishers, Cambridge (2005), 15–23.
N. Boussetila and F. Rebbani,
Optimal regularization method for ill-posed Cauchy problems,
Electron. J. Differential Equations 2006 (2006), Paper No. 147.
D. H. Brooks, G. F. Ahmad, R. S. MacLeod and M. G. Maratos,
Inverse electrocardiography by simultaneous imposition of multiple constraints,
IEEE Trans. Biomed. Eng. 46 (1999), 3–18.
H. Cao, S. V. Pereverzyev, I. H. Sloan and P. Tkachenko,
Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions,
Appl. Math. Comput. 273 (2016), 993–1005.
I. Cioranescu and V. Keyantuo,
Entire regularizations of strongly continuous groups and products of analytic semigroups,
Proc. Amer. Math. Soc. 128 (2000), no. 12, 3587–3593.
G. W. Clark and S. F. Oppenheimer,
Quasireversibility methods for non-well posed problems,
Electron. J. Differential Equations 8 (1994), 1–9.
N. Dunford and J. Schwartz,
Linear Operators-Part II,
John Wiley & Sons, New York, 1967.
D. Duvelmeyer and B. Hofmann,
A multi-parameter regularization approach for estimating parameters in jump diffusion process,
J. Inverse Ill-Posed Probl. 14 (2006), 861–880.
L. Elden and F. Berntsson,
Stability estimate for a Cauchy problem for an elliptic partial differential equation,
Inverse Problems 21 (2005), 1643–1653.
Q. Fan, D. Jiang and Y. Jiao,
A multi-parameter regularization model for image restoration,
Signal Process. 114 (2015), 131–142.
X. Feng and L. Eldén,
Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method,
Inverse Problems 30 (2014), Article ID 015005.
X. L. Feng, L. Eldén and C.-L. Fu,
A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data,
J. Inverse Ill-Posed Probl. 6 (2010), 617–645.
C. L. Fu, Y. J. Ma, H. Cheng and Y. X. Zhang,
The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data,
Appl. Math. Model. 37 (2013), 7764–7777.
C. L. Fu, Y. J. Ma, Y. X. Zhang and F. Yang,
A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data,
Appl. Math. Model. 39 (2015), 4103–4120.
H. Gajewski and K. Zacharias,
Zur Regularisierung einer Klasse nichtkorrekter Probleme bei Evolutionsgleichungen,
J. Math. Anal. Appl 38 (1972), 784–789.
T. Gomes, R. Prudencio, C. Soares, A. Rossi and A. Carvalho,
Combining metalearning and search techniques to select parameters for support vector machines,
Neurocomput. 75 (2012), 3–13.
Lecture Note on Cauchy’s Problem in Linear Partial Differential Equations,
Yale University Press, New Haven, 1923.
D. N. Hào, N. V. Duc and D. Lesnic,
A non-local boundary value problem method for the Cauchy problem for elliptic equations,
Inverse Problems 25 (2009), Article ID 055002.
K. Ito, B. Jin and T. Takeuchi,
Multi-parameter Tikhonov regularization,
Methods Appl. Anal. 18 (2011), no. 1, 31–46.
An Introduction to the Mathematical Theory of Inverse Problems,
Springer, New York, 1996.
M. V. Klibanov,
Carleman estimates for the regularization of ill-posed Cauchy problems,
Appl. Numer. Math. 94 (2015), 46–74.
R. Lattès and J. L. Lions,
Méthode de Quasi-Réversibilité et Applications,
Dunod, Paris, 1967.
S. Lu and S. V. Pereverzev,
Multiparameter Regularization in Downward Continuation of Satellite Data,
Handbook Geomath. 27,
Springer, Berlin, 2010.
S. Lu and S. V. Pereverzev,
Multi-parameter regularization and its numerical realization,
Numer. Math. 118 (2011), 1–31.
S. Lu, S. V. Pereverzev, Y. Shao and U. Tautenhahn,
Discrepancy curves for multiparameter regularization,
J. Inverse Ill-Posed Probl. 18 (2010), 655–676.
Semigroups of Linear Operators and Applications to Partial Differential Equations,
Appl. Math. Sci. 44,
Springer, New York, 1983.
S. V. Pereverzev and S. Lu,
Regularization Theory for Ill-posed Problems. Selected Topics,
Inverse Ill-posed Probl. Ser. 58,
De Gruyter, Berlin, 2013.
Z. Qian, X. T. Xiong and Y. J. Wu,
On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation,
J. Comput. Appl. Math. 233 (2010), 1969–1979.
Two-parameter discrepancy principle for combined projection and Tikhonov regularization of ill-posed problems,
J. Inverse Ill-Posed Probl. 21 (2013), 561–577.
N. H. Tuan, D. D. Trong and P. H. Quan,
A note on a Cauchy problem for the Laplace equation: Regularization and error estimates,
Appl. Math. Comput. 217 (2010), no. 7, 2913–2922.
N. H. Tuan, T. Q. Viet and N. V. Thinh,
Some remarks on a modified Helmholtz equation with inhomogeneous source,
Appl. Math. Model. 37 (2013), no. 3, 793–814.
Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters,
J. Comput. Appl. Math. 236 (2012), 1815–1832.
P. L. Xu, Y. Fukuda and Y. M. Liu,
Multiple parameter regularization: Numerical solutions and applications to the determination of geopotential from precise satellite orbits,
J. Geod. 80 (2006), 17–27.
F. Yang, C. Fu and X. Li,
A modified Tikhonov regularization method for the Cauchy problem of Laplace equation,
Acta Math. Sci. 35 (2015), no. 6, 1339–1348.
L. Zhang, Z. C. Li, Y. Wei and J. Y. Chiang,
Cauchy problems of Laplace’s equation by the methods of fundamental solutions and particular solutions,
Eng. Anal. Bound. Elem. 37 (2013), 765–780.
H. Zhang and T. Wei,
An improved non-local boundary value problem method for a cauchy problem of the Laplace equation,
Numer. Algorithms 59 (2012), no. 2, 249–269.
F. Zouyed and F. Rebbani,
A modified quasi-boundary value method for an ultraparabolic ill-posed problem,
J. Inverse Ill-Posed Probl. 22 (2014), no. 4, 449–466.
This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.