The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method

The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method Full article

Journal

Journal of Inverse and Ill-Posed Problems
ISSN: 0928-0219 , E-ISSN: 1569-3945

Output data

Year: 2023, Volume: 31, Number: 2, Pages: 203-221 Pages count : 19 DOI: 10.1515/jiip-2022-0092

Tags

Continuation problem; inverse and ill-posed problem; singular values; regularization

Authors

Botchev Mikhail A. ¹ , Kabanikhin Sergey I. ^2,3,4 , Shishlenin Maxim A. ^2,3,4 , Tyrtyshnikov Eugene E. ⁵

Affiliations

1	Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow 125047, Russia
2	Novosibirsk State University, 630090 Novosibirsk, Russia
3	Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
4	Institute of Computational Mathematics and Mathematical Geophysics, 630090 Novosibirsk, Russia
5	Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia

Funding (3)

1	Russian Foundation for Basic Research	20-51-54004
2	Mathematical Center in Akademgorodok	075-15-2019-1675
3	Russian Foundation for Basic Research	19-01-00694

The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n≤2500 is significantly more efficient than Landweber iteration.

Botchev M.A. , Kabanikhin S.I. , Shishlenin M.A. , Tyrtyshnikov E.E.
The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method
Journal of Inverse and Ill-Posed Problems. 2023. V.31. N2. P.203-221. DOI: 10.1515/jiip-2022-0092 WOS Scopus РИНЦ OpenAlex

Submitted:	Dec 2, 2022
Accepted:	Dec 2, 2022
Published online:	Jan 31, 2023
Published print:	Apr 1, 2023

Web of science:	WOS:000923545500001
Scopus:	2-s2.0-85147713447
Elibrary:	60472180
OpenAlex:	W4318480906

DB	Citing
Web of science	1
Scopus	1