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 and 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, Oxford, 2nd edition, 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), 1773-1789. 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), 2384-2398. 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), 1-29. 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), 33-63. doi: 10.1007/BF01448191.

[7]

H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales, Numer. Math., 101 (2005), 643-662. doi: 10.1007/s00211-005-0622-5.

[8]

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

[9]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540. doi: 10.1088/0266-5611/5/4/007.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 2nd edition, 1983.

[11]

A. Goldenshluger and S. V. Pereverzev, On adaptive inverse estimation of linear functionals in Hilbert scales, Bernoulli, 9 (2003), 783-807. 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), 187-197. doi: 10.1088/0266-5611/16/1/315.

[13]

J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2004.

[14]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations, J. Math. Anal. Appl., 132 (1988), 453-472. 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), 1424-1444. 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), 1999-2021. doi: 10.1137/S003614299936175X.

[18]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Applicable Anal., 18 (1984), 29-37. doi: 10.1080/00036818408839508.

[19]

A. Neubauer, When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc., 103 (1988), 557-562. 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), 59-72. doi: 10.1080/00036819208840111.

[21]

A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales, Numer. Math., 85 (2000), 309-328. doi: 10.1007/s002110050487.

[22]

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

[23]

F. O'Sullivan, Convergence characteristics of methods of regularization estimators for nonlinear operator equations, SIAM J. Numer. Anal., 27 (1990), 1635-1649. doi: 10.1137/0727096.

[24]

M. S. Pinsker, Optimal filtration of square-integrable signals in Gaussian white noise, Probl. Inf. Transm., 16 (1980), 52-68.

[25]

M. Pricop, "Tikhonov Regularization in Hilbert Scales for Nonlinear Statistical Inverse Problems," PhD thesis, University of Göttingen, 2007.

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Springer, New York, 2nd edition, 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), 1796-1838. 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), 1607-1616. 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), 1-21. doi: 10.1088/0266-5611/19/1/301.

[30]

M. Taylor, "Partial Differential Equations: Basic Theory," volume 1, Springer, New York, 1996.

[31]

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

[32]

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

show all references

References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," volume 140 of "Pure and Applied Mathematics," Elsevier Science, Oxford, 2nd edition, 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), 1773-1789. 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), 2384-2398. 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), 1-29. 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), 33-63. doi: 10.1007/BF01448191.

[7]

H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales, Numer. Math., 101 (2005), 643-662. doi: 10.1007/s00211-005-0622-5.

[8]

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

[9]

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540. doi: 10.1088/0266-5611/5/4/007.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 2nd edition, 1983.

[11]

A. Goldenshluger and S. V. Pereverzev, On adaptive inverse estimation of linear functionals in Hilbert scales, Bernoulli, 9 (2003), 783-807. 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), 187-197. doi: 10.1088/0266-5611/16/1/315.

[13]

J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2004.

[14]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations, J. Math. Anal. Appl., 132 (1988), 453-472. 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), 1424-1444. 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), 1999-2021. doi: 10.1137/S003614299936175X.

[18]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Applicable Anal., 18 (1984), 29-37. doi: 10.1080/00036818408839508.

[19]

A. Neubauer, When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc., 103 (1988), 557-562. 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), 59-72. doi: 10.1080/00036819208840111.

[21]

A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales, Numer. Math., 85 (2000), 309-328. doi: 10.1007/s002110050487.

[22]

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

[23]

F. O'Sullivan, Convergence characteristics of methods of regularization estimators for nonlinear operator equations, SIAM J. Numer. Anal., 27 (1990), 1635-1649. doi: 10.1137/0727096.

[24]

M. S. Pinsker, Optimal filtration of square-integrable signals in Gaussian white noise, Probl. Inf. Transm., 16 (1980), 52-68.

[25]

M. Pricop, "Tikhonov Regularization in Hilbert Scales for Nonlinear Statistical Inverse Problems," PhD thesis, University of Göttingen, 2007.

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Springer, New York, 2nd edition, 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), 1796-1838. 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), 1607-1616. 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), 1-21. doi: 10.1088/0266-5611/19/1/301.

[30]

M. Taylor, "Partial Differential Equations: Basic Theory," volume 1, Springer, New York, 1996.

[31]

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

[32]

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

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