May  2008, 2(2): 271-290. doi: 10.3934/ipi.2008.2.271

Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise

1. 

Institut für Numerische und Angewandte Mathematik, Lotzestr. 16-18 D-37083 Göttingen, Germany, Germany

Received  April 2007 Revised  December 2007 Published  April 2008

We consider the inverse problem to identify coefficient functions in boundary value problems from noisy measurements of the solutions. Our estimators are defined as minimizers of a Tikhonov functional, which is the sum of a nonlinear data misfit term and a quadratic penalty term involving a Hilbert scale norm. In this abstract framework we derive estimates of the expected squared error under certain assumptions on the forward operator. These assumptions are shown to be satisfied for two classes of inverse elliptic boundary value problems. The theoretical results are confirmed by Monte Carlo simulations.
Citation: Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," volume 140 of "Pure and Applied Mathematics,", Elsevier Science, (2003).

[2]

N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise,, Inverse Problems, 20 (2004), 1773. doi: 10.1088/0266-5611/20/6/005.

[3]

N. Bissantz, T. Hohage, A. Munk and F. Ruymgaart, Convergence rates of general regularization methods for statistical inverse problems and applications,, SIAM J. Numer. Anal., ().

[4]

D. L. Brown and M. G. Low, Asymptotic equivalence of nonparametric regression and white noise,, Ann. Statist., 24 (1996), 2384. doi: 10.1214/aos/1032181159.

[5]

F. Colonius and K. Kunisch, Stability for parameter estimation in two-point boundary value problems,, J. Reine Angew. Math., 370 (1986), 1. doi: 10.1515/crll.1986.370.1.

[6]

F. Colonius and K. Kunisch, Output least squares stability in elliptic systems,, Appl. Math. Optim., 19 (1989), 33. doi: 10.1007/BF01448191.

[7]

H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales,, Numer. Math., 101 (2005), 643. doi: 10.1007/s00211-005-0622-5.

[8]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer Academic Publishers Group, (1996).

[9]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems,, Inverse Problems, 5 (1989), 523. doi: 10.1088/0266-5611/5/4/007.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1983).

[11]

A. Goldenshluger and S. V. Pereverzev, On adaptive inverse estimation of linear functionals in Hilbert scales,, Bernoulli, 9 (2003), 783. doi: 10.3150/bj/1066418878.

[12]

Q.-n. Jin, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales,, Inverse Problems, 16 (2000), 187. doi: 10.1088/0266-5611/16/1/315.

[13]

J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2004).

[14]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations,, J. Math. Anal. Appl., 132 (1988), 453. doi: 10.1016/0022-247X(88)90074-1.

[15]

J.-M. Loubes and C. Ludena, Penalized estimators for nonlinear inverse problems,, arXiv.org:math, 1 (2005).

[16]

B. A. Mair and F. H. Ruymgaart, Statistical inverse estimation in Hilbert scales,, SIAM J. Appl. Math., 56 (1996), 1424. doi: 10.1137/S0036139994264476.

[17]

P. Mathé and S. Pereverzev, Optimal discretization of inverse problems in Hilbert scales. regularization and self-regularization of projection methods,, SIAM J. Numer. Anal., 38 (2001), 1999. doi: 10.1137/S003614299936175X.

[18]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales,, Applicable Anal., 18 (1984), 29. doi: 10.1080/00036818408839508.

[19]

A. Neubauer, When do Sobolev spaces form a Hilbert scale?, Proc. Amer. Math. Soc., 103 (1988), 557. doi: 10.1090/S0002-9939-1988-0943084-9.

[20]

A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales,, Appl. Anal., 46 (1992), 59. doi: 10.1080/00036819208840111.

[21]

A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales,, Numer. Math., 85 (2000), 309. doi: 10.1007/s002110050487.

[22]

M. Nussbaum and S. Pereverzev, "The Degree of Ill-Posedness in Stochastic and Deterministic Noise Models,", Technical report, (1999).

[23]

F. O'Sullivan, Convergence characteristics of methods of regularization estimators for nonlinear operator equations,, SIAM J. Numer. Anal., 27 (1990), 1635. doi: 10.1137/0727096.

[24]

M. S. Pinsker, Optimal filtration of square-integrable signals in Gaussian white noise,, Probl. Inf. Transm., 16 (1980), 52.

[25]

M. Pricop, "Tikhonov Regularization in Hilbert Scales for Nonlinear Statistical Inverse Problems,", PhD thesis, (2007).

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Springer, (2004).

[27]

O. Scherzer, H. W. Engl and K. Kunisch, Optimal a-posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems,, SIAM J. Numer. Anal., 30 (1993), 1796. doi: 10.1137/0730091.

[28]

U. Tautenhahn, On a general regularization scheme for nonlinear ill-posed problems. II, Regularization in Hilbert scales,, Inverse Problems, 14 (1998), 1607. doi: 10.1088/0266-5611/14/6/016.

[29]

U. Tautenhahn and Q. nian Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems,, Inverse Problems, 19 (2003), 1. doi: 10.1088/0266-5611/19/1/301.

[30]

M. Taylor, "Partial Differential Equations: Basic Theory," volume 1,, Springer, (1996).

[31]

A. B. Tsybakov, "Introduction à L'estimation Non-Paramétrique,", Springer, (2004).

[32]

J. Wloka, "Partial Differential Equations,", Cambridge University Press, (1987).

show all references

References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," volume 140 of "Pure and Applied Mathematics,", Elsevier Science, (2003).

[2]

N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise,, Inverse Problems, 20 (2004), 1773. doi: 10.1088/0266-5611/20/6/005.

[3]

N. Bissantz, T. Hohage, A. Munk and F. Ruymgaart, Convergence rates of general regularization methods for statistical inverse problems and applications,, SIAM J. Numer. Anal., ().

[4]

D. L. Brown and M. G. Low, Asymptotic equivalence of nonparametric regression and white noise,, Ann. Statist., 24 (1996), 2384. doi: 10.1214/aos/1032181159.

[5]

F. Colonius and K. Kunisch, Stability for parameter estimation in two-point boundary value problems,, J. Reine Angew. Math., 370 (1986), 1. doi: 10.1515/crll.1986.370.1.

[6]

F. Colonius and K. Kunisch, Output least squares stability in elliptic systems,, Appl. Math. Optim., 19 (1989), 33. doi: 10.1007/BF01448191.

[7]

H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales,, Numer. Math., 101 (2005), 643. doi: 10.1007/s00211-005-0622-5.

[8]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer Academic Publishers Group, (1996).

[9]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems,, Inverse Problems, 5 (1989), 523. doi: 10.1088/0266-5611/5/4/007.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1983).

[11]

A. Goldenshluger and S. V. Pereverzev, On adaptive inverse estimation of linear functionals in Hilbert scales,, Bernoulli, 9 (2003), 783. doi: 10.3150/bj/1066418878.

[12]

Q.-n. Jin, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales,, Inverse Problems, 16 (2000), 187. doi: 10.1088/0266-5611/16/1/315.

[13]

J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2004).

[14]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations,, J. Math. Anal. Appl., 132 (1988), 453. doi: 10.1016/0022-247X(88)90074-1.

[15]

J.-M. Loubes and C. Ludena, Penalized estimators for nonlinear inverse problems,, arXiv.org:math, 1 (2005).

[16]

B. A. Mair and F. H. Ruymgaart, Statistical inverse estimation in Hilbert scales,, SIAM J. Appl. Math., 56 (1996), 1424. doi: 10.1137/S0036139994264476.

[17]

P. Mathé and S. Pereverzev, Optimal discretization of inverse problems in Hilbert scales. regularization and self-regularization of projection methods,, SIAM J. Numer. Anal., 38 (2001), 1999. doi: 10.1137/S003614299936175X.

[18]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales,, Applicable Anal., 18 (1984), 29. doi: 10.1080/00036818408839508.

[19]

A. Neubauer, When do Sobolev spaces form a Hilbert scale?, Proc. Amer. Math. Soc., 103 (1988), 557. doi: 10.1090/S0002-9939-1988-0943084-9.

[20]

A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales,, Appl. Anal., 46 (1992), 59. doi: 10.1080/00036819208840111.

[21]

A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales,, Numer. Math., 85 (2000), 309. doi: 10.1007/s002110050487.

[22]

M. Nussbaum and S. Pereverzev, "The Degree of Ill-Posedness in Stochastic and Deterministic Noise Models,", Technical report, (1999).

[23]

F. O'Sullivan, Convergence characteristics of methods of regularization estimators for nonlinear operator equations,, SIAM J. Numer. Anal., 27 (1990), 1635. doi: 10.1137/0727096.

[24]

M. S. Pinsker, Optimal filtration of square-integrable signals in Gaussian white noise,, Probl. Inf. Transm., 16 (1980), 52.

[25]

M. Pricop, "Tikhonov Regularization in Hilbert Scales for Nonlinear Statistical Inverse Problems,", PhD thesis, (2007).

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Springer, (2004).

[27]

O. Scherzer, H. W. Engl and K. Kunisch, Optimal a-posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems,, SIAM J. Numer. Anal., 30 (1993), 1796. doi: 10.1137/0730091.

[28]

U. Tautenhahn, On a general regularization scheme for nonlinear ill-posed problems. II, Regularization in Hilbert scales,, Inverse Problems, 14 (1998), 1607. doi: 10.1088/0266-5611/14/6/016.

[29]

U. Tautenhahn and Q. nian Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems,, Inverse Problems, 19 (2003), 1. doi: 10.1088/0266-5611/19/1/301.

[30]

M. Taylor, "Partial Differential Equations: Basic Theory," volume 1,, Springer, (1996).

[31]

A. B. Tsybakov, "Introduction à L'estimation Non-Paramétrique,", Springer, (2004).

[32]

J. Wloka, "Partial Differential Equations,", Cambridge University Press, (1987).

[1]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267

[2]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215

[3]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[4]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[5]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749

[6]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19

[7]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[8]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[9]

Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems & Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567

[10]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107

[11]

Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems & Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715

[12]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005

[13]

Zheng-Jian Bai, Xiao-Qing Jin, Seak-Weng Vong. On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 187-192. doi: 10.3934/naco.2012.2.187

[14]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[15]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

[16]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1

[17]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225

[18]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059

[19]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[20]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]