Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils

Taras A. Dauzhenka; Igor A. Gishkeluk

doi:doi:10.11648/j.acm.20130204.11

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Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils

Taras A. Dauzhenka, Igor A. Gishkeluk

Published in Applied and Computational Mathematics (Volume 2, Issue 4)

Received: 30 May 2013 Accepted: Published: 10 August 2013

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Abstract

Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.

Published in	Applied and Computational Mathematics (Volume 2, Issue 4)
DOI	10.11648/j.acm.20130204.11
Page(s)	100-108
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Splitting Algorithms, Alternating Directions Scheme, Stefan Problem, Nonlinear Heat Equation, Consistent Approximation

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Taras A. Dauzhenka, Igor A. Gishkeluk. (2013). Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Applied and Computational Mathematics, 2(4), 100-108. https://doi.org/10.11648/j.acm.20130204.11

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Taras A. Dauzhenka; Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl. Comput. Math. 2013, 2(4), 100-108. doi: 10.11648/j.acm.20130204.11

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AMA Style

Taras A. Dauzhenka, Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl Comput Math. 2013;2(4):100-108. doi: 10.11648/j.acm.20130204.11

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@article{10.11648/j.acm.20130204.11,
  author = {Taras A. Dauzhenka and Igor A. Gishkeluk},
  title = {Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {4},
  pages = {100-108},
  doi = {10.11648/j.acm.20130204.11},
  url = {https://doi.org/10.11648/j.acm.20130204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130204.11},
  abstract = {Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.},
 year = {2013}
}

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TY  - JOUR
T1  - Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils
AU  - Taras A. Dauzhenka
AU  - Igor A. Gishkeluk
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AB  - Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.
VL  - 2
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Author Information

Taras A. Dauzhenka

R&D Dept., Simmakers Ltd., 1A-307 V. Kharuzhei str., 220005 Minsk, Belarus
Igor A. Gishkeluk

R&D Dept., Simmakers Ltd., 1A-307 V. Kharuzhei str., 220005 Minsk, Belarus

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Taras A. Dauzhenka, Igor A. Gishkeluk. (2013). Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Applied and Computational Mathematics, 2(4), 100-108. https://doi.org/10.11648/j.acm.20130204.11

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ACS Style

Taras A. Dauzhenka; Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl. Comput. Math. 2013, 2(4), 100-108. doi: 10.11648/j.acm.20130204.11

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AMA Style

Taras A. Dauzhenka, Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl Comput Math. 2013;2(4):100-108. doi: 10.11648/j.acm.20130204.11

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@article{10.11648/j.acm.20130204.11,
  author = {Taras A. Dauzhenka and Igor A. Gishkeluk},
  title = {Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {4},
  pages = {100-108},
  doi = {10.11648/j.acm.20130204.11},
  url = {https://doi.org/10.11648/j.acm.20130204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130204.11},
  abstract = {Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils
AU  - Taras A. Dauzhenka
AU  - Igor A. Gishkeluk
Y1  - 2013/08/10
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130204.11
DO  - 10.11648/j.acm.20130204.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 100
EP  - 108
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20130204.11
AB  - Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.
VL  - 2
IS  - 4
ER  -

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