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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

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