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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 and 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), 268-280. doi: 10.1002/zamm.200410238. [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), 13-29. doi: 10.1515/jiip.2001.9.1.13. [3] A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570, URL http://stacks.iop.org/0266-5611/17/i=3/a=313. doi: 10.1088/0266-5611/17/3/313. [4] H. W. Engl, M. Hanke and A. Neubauer, Regularization Of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8. [5] M. Hanke, Conjugate Graident Type Methods for Ill-posed Problems, vol. 327 of Pitman Research Notes in Mathematics, Longman Scientific & Technical, 1995. [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), 199-217, URL http://imamat.oxfordjournals.org/content/65/2/199.abstract. doi: 10.1093/imamat/65.2.199. [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), 655-665, URL http://www.sciencedirect.com/science/article/pii/S0955799703001486. doi: 10.1016/j.enganabound.2003.07.002. [8] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numerical Algorithms, 21 (1999), 247-260. doi: 10.1023/A:1019134102565. [9] I. Knowles, Variational methods for ill-posed problems, in Variational methods: Open problems, recent progress, and numerical algorithms : June 5-8, 2002, Northern Arizona University, Flagstaff, Arizona (ed. J. Neuberger), Contemporary mathematics - American Mathematical Society, American Mathematical Society, 2004, URL http://books.google.com/books?id=7iPC7sXWRyUC. doi: 10.1090/conm/357/06519. [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), 1207-1228. [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), 45-52. [12] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624, URL http://www.jstor.org/stable/2372313. doi: 10.2307/2372313. [13] A. Logg and G. N. Wells, Dolfin: Automated finite element computing, ACM Transactions on Mathematical Software, 37 (2010), Art. 20, 28 pp. doi: 10.1145/1731022.1731030. [14] D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, An iterative scheme for determining glacier velocities and stresses, Journal of Glaciology, 54 (2008), 888-898, URL http://openurl.ingenta.com/content/xref?genre=article&issn=0022-1430&volume=54&issue=188&spage=888. doi: 10.3189/002214308787779889. [15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [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, Philadelphia, Pa., 1975.

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), 268-280. doi: 10.1002/zamm.200410238. [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), 13-29. doi: 10.1515/jiip.2001.9.1.13. [3] A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570, URL http://stacks.iop.org/0266-5611/17/i=3/a=313. doi: 10.1088/0266-5611/17/3/313. [4] H. W. Engl, M. Hanke and A. Neubauer, Regularization Of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8. [5] M. Hanke, Conjugate Graident Type Methods for Ill-posed Problems, vol. 327 of Pitman Research Notes in Mathematics, Longman Scientific & Technical, 1995. [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), 199-217, URL http://imamat.oxfordjournals.org/content/65/2/199.abstract. doi: 10.1093/imamat/65.2.199. [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), 655-665, URL http://www.sciencedirect.com/science/article/pii/S0955799703001486. doi: 10.1016/j.enganabound.2003.07.002. [8] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numerical Algorithms, 21 (1999), 247-260. doi: 10.1023/A:1019134102565. [9] I. Knowles, Variational methods for ill-posed problems, in Variational methods: Open problems, recent progress, and numerical algorithms : June 5-8, 2002, Northern Arizona University, Flagstaff, Arizona (ed. J. Neuberger), Contemporary mathematics - American Mathematical Society, American Mathematical Society, 2004, URL http://books.google.com/books?id=7iPC7sXWRyUC. doi: 10.1090/conm/357/06519. [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), 1207-1228. [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), 45-52. [12] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624, URL http://www.jstor.org/stable/2372313. doi: 10.2307/2372313. [13] A. Logg and G. N. Wells, Dolfin: Automated finite element computing, ACM Transactions on Mathematical Software, 37 (2010), Art. 20, 28 pp. doi: 10.1145/1731022.1731030. [14] D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, An iterative scheme for determining glacier velocities and stresses, Journal of Glaciology, 54 (2008), 888-898, URL http://openurl.ingenta.com/content/xref?genre=article&issn=0022-1430&volume=54&issue=188&spage=888. doi: 10.3189/002214308787779889. [15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [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, Philadelphia, Pa., 1975.
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