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). Google Scholar

[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. Google Scholar

[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., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[32]

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

show all references

References:
[1]

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

[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. Google Scholar

[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., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[32]

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

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