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August  2017, 11(4): 703-720. doi: 10.3934/ipi.2017033

Convergence of the gradient method for ill-posed problems

Stefan Kindermann ,

Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstrasse 69,4040 Linz, Austria

Received  June 2016 Revised  May 2017 Published  June 2017

We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for nonlinear operator equations is a special instance of this framework, and the gradient method then reduces to Landweber iteration. The main result of this article is a proof of weak and strong convergence under new nonlinearity conditions that generalize the classical tangential cone conditions.

Keywords: Inverse problems, iterative Regularization, Landweber iteration, tangential cone condition, quasiconvexity.
Mathematics Subject Classification: Primary: 65J20, 65J15; Secondary: 47J06.
Citation: Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
References:
[1]

K. J. Arrow and A. C. Enthoven, Quasi-concave programming, Econometrica, 29 (1961), 779-800.  doi: 10.2307/1911819.  Google Scholar

[2]

M. Avriel, W. E. Diewert, S. Schaible and I. Zang, Generalized Concavity, vol. 63 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898719437.ch1.  Google Scholar

[3]

A. Bakushinsky and A. Goncharsky, Ill-posed Problems: Theory and Applications, vol. 301 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1994. doi: 10.1007/978-94-011-1026-6.  Google Scholar

[4]

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, vol. 577 of Mathematics and Its Applications (New York), Springer, Dordrecht, 2004.  Google Scholar

[5]

A. Bakushinsky and A. Smirnova, A posteriori stopping rule for regularized fixed point iterations, Nonlinear Anal., 64 (2006), 1255-1261.  doi: 10.1016/j.na.2005.06.031.  Google Scholar

[6]

A. Bakushinsky and A. Smirnova, Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim., 28 (2007), 13-25.  doi: 10.1080/01630560701190315.  Google Scholar

[7]

A. B. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-Posed Problems, vol. 54 of Inverse and Ill-posed Problems Series, Walter de Gruyter GmbH & Co. KG, Berlin, 2011.  Google Scholar

[8]

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.  doi: 10.1016/0022-247X(67)90085-6.  Google Scholar

[9]

A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73.  Google Scholar

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[11]

M. Haltmeier, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. Ⅰ. Convergence analysis, Inverse Probl. Imaging, 1 (2007), 289-298.  doi: 10.3934/ipi.2007.1.289.  Google Scholar

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M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.  Google Scholar

[13]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar

[14]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces Inverse Problems, 26 (2002), Art. 055002, 17 pages. doi: 10.1088/0266-5611/26/5/055002.  Google Scholar

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N. S. Hoang and A. G. Ramm, Dynamical systems gradient method for solving nonlinear equations with monotone operators, Acta Appl. Math., 106 (2009), 473-499.  doi: 10.1007/s10440-008-9308-1.  Google Scholar

[16]

N. S. Hoang and A. G. Ramm, The dynamical systems method for solving nonlinear equations with monotone operators, Asian-Eur. J. Math., 3 (2010), 57-105.  doi: 10.1142/S1793557110000064.  Google Scholar

[17]

Q. Jin, A general convergence analysis of some Newton-type methods for nonlinear inverse problems, SIAM J. Numer. Anal., 49 (2011), 549-573.  doi: 10.1137/100804231.  Google Scholar

[18]

B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13 (1997), 729-753.  doi: 10.1088/0266-5611/13/3/012.  Google Scholar

[19]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, vol. 6 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.  Google Scholar

[20]

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

[21]

A. Neubauer, Some generalizations for Landweber iteration for nonlinear ill-posed problems, Journal Inverse and Ill-posed Problems, 24 (2016), 393-406.  doi: 10.1515/jiip-2015-0086.  Google Scholar

[22]

S. S. Pereverzyev, R. Pinnau and N. Siedow, Regularized fixed-point iterations for nonlinear inverse problems, Inverse Problems, 22 (2006), 1-22.  doi: 10.1088/0266-5611/22/1/001.  Google Scholar

[23]

A. G. Ramm, Dynamical systems method for ill-posed equations with monotone operators, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 935-940.  doi: 10.1016/j.cnsns.2003.07.002.  Google Scholar

[24]

O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl., 194 (1995), 911-933.  doi: 10.1006/jmaa.1995.1335.  Google Scholar

[25]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, vol. 10 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.  Google Scholar

[26]

V. V. Vasin, Convergence of gradient-type methods for nonlinear equations, Dokl. Akad. Nauk, 359 (1998), 7-9.   Google Scholar

show all references

References:
[1]

K. J. Arrow and A. C. Enthoven, Quasi-concave programming, Econometrica, 29 (1961), 779-800.  doi: 10.2307/1911819.  Google Scholar

[2]

M. Avriel, W. E. Diewert, S. Schaible and I. Zang, Generalized Concavity, vol. 63 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898719437.ch1.  Google Scholar

[3]

A. Bakushinsky and A. Goncharsky, Ill-posed Problems: Theory and Applications, vol. 301 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1994. doi: 10.1007/978-94-011-1026-6.  Google Scholar

[4]

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, vol. 577 of Mathematics and Its Applications (New York), Springer, Dordrecht, 2004.  Google Scholar

[5]

A. Bakushinsky and A. Smirnova, A posteriori stopping rule for regularized fixed point iterations, Nonlinear Anal., 64 (2006), 1255-1261.  doi: 10.1016/j.na.2005.06.031.  Google Scholar

[6]

A. Bakushinsky and A. Smirnova, Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim., 28 (2007), 13-25.  doi: 10.1080/01630560701190315.  Google Scholar

[7]

A. B. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-Posed Problems, vol. 54 of Inverse and Ill-posed Problems Series, Walter de Gruyter GmbH & Co. KG, Berlin, 2011.  Google Scholar

[8]

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.  doi: 10.1016/0022-247X(67)90085-6.  Google Scholar

[9]

A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73.  Google Scholar

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[11]

M. Haltmeier, A. Leitão and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. Ⅰ. Convergence analysis, Inverse Probl. Imaging, 1 (2007), 289-298.  doi: 10.3934/ipi.2007.1.289.  Google Scholar

[12]

M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.  Google Scholar

[13]

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.  Google Scholar

[14]

T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces Inverse Problems, 26 (2002), Art. 055002, 17 pages. doi: 10.1088/0266-5611/26/5/055002.  Google Scholar

[15]

N. S. Hoang and A. G. Ramm, Dynamical systems gradient method for solving nonlinear equations with monotone operators, Acta Appl. Math., 106 (2009), 473-499.  doi: 10.1007/s10440-008-9308-1.  Google Scholar

[16]

N. S. Hoang and A. G. Ramm, The dynamical systems method for solving nonlinear equations with monotone operators, Asian-Eur. J. Math., 3 (2010), 57-105.  doi: 10.1142/S1793557110000064.  Google Scholar

[17]

Q. Jin, A general convergence analysis of some Newton-type methods for nonlinear inverse problems, SIAM J. Numer. Anal., 49 (2011), 549-573.  doi: 10.1137/100804231.  Google Scholar

[18]

B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13 (1997), 729-753.  doi: 10.1088/0266-5611/13/3/012.  Google Scholar

[19]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, vol. 6 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.  Google Scholar

[20]

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

[21]

A. Neubauer, Some generalizations for Landweber iteration for nonlinear ill-posed problems, Journal Inverse and Ill-posed Problems, 24 (2016), 393-406.  doi: 10.1515/jiip-2015-0086.  Google Scholar

[22]

S. S. Pereverzyev, R. Pinnau and N. Siedow, Regularized fixed-point iterations for nonlinear inverse problems, Inverse Problems, 22 (2006), 1-22.  doi: 10.1088/0266-5611/22/1/001.  Google Scholar

[23]

A. G. Ramm, Dynamical systems method for ill-posed equations with monotone operators, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 935-940.  doi: 10.1016/j.cnsns.2003.07.002.  Google Scholar

[24]

O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl., 194 (1995), 911-933.  doi: 10.1006/jmaa.1995.1335.  Google Scholar

[25]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, vol. 10 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.  Google Scholar

[26]

V. V. Vasin, Convergence of gradient-type methods for nonlinear equations, Dokl. Akad. Nauk, 359 (1998), 7-9.   Google Scholar

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