• Previous Article
    Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography
  • IPI Home
  • This Issue
  • Next Article
    A stable method solving the total variation dictionary model with $L^\infty$ constraints
May  2014, 8(2): 537-560. doi: 10.3934/ipi.2014.8.537

Kozlov-Maz'ya iteration as a form of Landweber iteration

1. 

Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99557-6660, United States

Received  July 2011 Revised  November 2012 Published  May 2014

We consider the alternating method of Kozlov and Maz'ya for solving the Cauchy problem for elliptic boundary-value problems. Considering the case of the Laplacian, we show that this method can be recast as a form of Landweber iteration. In addition to conceptual advantages, this observation leads to some practical improvements. We show how to accelerate Kozlov-Maz'ya iteration using the conjugate gradient algorithm, and we show how to modify the method to obtain a more practical stopping criterion.
Citation: David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537
References:
[1]

G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system,, Zeitschrift für Angewandte Mathematik und Mechanik, 86 (2005), 268. doi: 10.1002/zamm.200410238. Google Scholar

[2]

J. Baumeister and A. Leitao, On iterative methods for solving ill-posed problems modeled by partial differential equations,, J. Inverse Ill-Posed Probl., 9 (2001), 13. doi: 10.1515/jiip.2001.9.1.13. Google Scholar

[3]

A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313. Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, Regularization Of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar

[5]

M. Hanke, Conjugate Graident Type Methods for Ill-posed Problems, vol. 327 of Pitman Research Notes in Mathematics,, Longman Scientific & Technical, (1995). Google Scholar

[6]

D. N. Hào and D. Lesnic, The Cauchy problem for Laplace's equation via the conjugate gradient method,, IMA Journal of Applied Mathematics, 65 (2000), 199. doi: 10.1093/imamat/65.2.199. Google Scholar

[7]

M. Jourhmane, D. Lesnic and N. S. Mera, Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation,, Engineering Analysis with Boundary Elements, 28 (2004), 655. doi: 10.1016/j.enganabound.2003.07.002. Google Scholar

[8]

M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem,, Numerical Algorithms, 21 (1999), 247. doi: 10.1023/A:1019134102565. Google Scholar

[9]

I. Knowles, Variational methods for ill-posed problems,, in Variational methods: Open problems, (2002), 5. doi: 10.1090/conm/357/06519. Google Scholar

[10]

V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations,, Lenningrad Mathematics Journal, 1 (1990), 1207. Google Scholar

[11]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic eqations,, U.S.S.R. Computational Mathematics and Mathematical Physics, 31 (1991), 45. Google Scholar

[12]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind,, American Journal of Mathematics, 73 (1951), 615. doi: 10.2307/2372313. Google Scholar

[13]

A. Logg and G. N. Wells, Dolfin: Automated finite element computing,, ACM Transactions on Mathematical Software, 37 (2010). doi: 10.1145/1731022.1731030. Google Scholar

[14]

D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, An iterative scheme for determining glacier velocities and stresses,, Journal of Glaciology, 54 (2008), 888. doi: 10.3189/002214308787779889. Google Scholar

[15]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[16]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations, vol. 22 of CBMS-NSF Regional Conference Series in Applied Mathematics,, Society for Industrial Mathematics and Applied Mathematics, (1975). Google Scholar

show all references

References:
[1]

G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system,, Zeitschrift für Angewandte Mathematik und Mechanik, 86 (2005), 268. doi: 10.1002/zamm.200410238. Google Scholar

[2]

J. Baumeister and A. Leitao, On iterative methods for solving ill-posed problems modeled by partial differential equations,, J. Inverse Ill-Posed Probl., 9 (2001), 13. doi: 10.1515/jiip.2001.9.1.13. Google Scholar

[3]

A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313. Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, Regularization Of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar

[5]

M. Hanke, Conjugate Graident Type Methods for Ill-posed Problems, vol. 327 of Pitman Research Notes in Mathematics,, Longman Scientific & Technical, (1995). Google Scholar

[6]

D. N. Hào and D. Lesnic, The Cauchy problem for Laplace's equation via the conjugate gradient method,, IMA Journal of Applied Mathematics, 65 (2000), 199. doi: 10.1093/imamat/65.2.199. Google Scholar

[7]

M. Jourhmane, D. Lesnic and N. S. Mera, Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation,, Engineering Analysis with Boundary Elements, 28 (2004), 655. doi: 10.1016/j.enganabound.2003.07.002. Google Scholar

[8]

M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem,, Numerical Algorithms, 21 (1999), 247. doi: 10.1023/A:1019134102565. Google Scholar

[9]

I. Knowles, Variational methods for ill-posed problems,, in Variational methods: Open problems, (2002), 5. doi: 10.1090/conm/357/06519. Google Scholar

[10]

V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations,, Lenningrad Mathematics Journal, 1 (1990), 1207. Google Scholar

[11]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic eqations,, U.S.S.R. Computational Mathematics and Mathematical Physics, 31 (1991), 45. Google Scholar

[12]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind,, American Journal of Mathematics, 73 (1951), 615. doi: 10.2307/2372313. Google Scholar

[13]

A. Logg and G. N. Wells, Dolfin: Automated finite element computing,, ACM Transactions on Mathematical Software, 37 (2010). doi: 10.1145/1731022.1731030. Google Scholar

[14]

D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, An iterative scheme for determining glacier velocities and stresses,, Journal of Glaciology, 54 (2008), 888. doi: 10.3189/002214308787779889. Google Scholar

[15]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[16]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations, vol. 22 of CBMS-NSF Regional Conference Series in Applied Mathematics,, Society for Industrial Mathematics and Applied Mathematics, (1975). Google Scholar

[1]

Moez Kallel, Maher Moakher, Anis Theljani. The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting. Inverse Problems & Imaging, 2015, 9 (3) : 853-874. doi: 10.3934/ipi.2015.9.853

[2]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-23. doi: 10.3934/dcds.2019227

[3]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[4]

Mauro Garavello, Paola Goatin. The Cauchy problem at a node with buffer. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915

[5]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012

[6]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043

[7]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863

[8]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043

[9]

Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753

[10]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149

[11]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[12]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417

[13]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[14]

Hernan R. Henriquez. Generalized solutions for the abstract singular Cauchy problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 955-976. doi: 10.3934/cpaa.2009.8.955

[15]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337

[16]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401

[17]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431

[18]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[19]

Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084

[20]

Todor Gramchev, Nicola Orrú. Cauchy problem for a class of nondiagonalizable hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 533-542. doi: 10.3934/proc.2011.2011.533

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]