---
_id: '52089'
abstract:
- lang: eng
  text: |-
    <jats:title>Abstract</jats:title><jats:p>Image restoration via alternating direction method of multipliers (ADMM) has gained large interest within the last decade. Solving standard problems of Gaussian and Poisson noise, the set of “Total Variation” (TV)-based regularizers proved to be efficient and versatile. In the last few years, the “Total Generalized Variation” (TGV) approach combined TV regularizers of different orders adaptively to better suit local regions in the image. This improved the technique significantly. The approach solved the staircase problem inherent of the first-order TV while keeping the beneficial edge preservation. The iterative minimization for the augmented Lagrangian of TGV problems requires four important parameters: two penalty parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$${\rho }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:mi>ρ</mml:mi>
                  </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\eta }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:mi>η</mml:mi>
                  </mml:math></jats:alternatives></jats:inline-formula> and two regularization parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{0}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:msub>
                      <mml:mi>λ</mml:mi>
                      <mml:mn>0</mml:mn>
                    </mml:msub>
                  </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{1}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:msub>
                      <mml:mi>λ</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msub>
                  </mml:math></jats:alternatives></jats:inline-formula>. The choice of penalty parameters decides on the convergence speed, and the regularization parameters decide on the impact of the respective regularizer and are determined by the noise level in the image. For scientific applications of such algorithms, an automated and thus objective method to determine these parameters is essential to receive unbiased results independent of the user. Obviously, both sets of parameters are to be well chosen to achieve optimal results, too. In this paper, a method is proposed to adaptively choose optimal <jats:inline-formula><jats:alternatives><jats:tex-math>$${\rho }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:mi>ρ</mml:mi>
                  </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\eta }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:mi>η</mml:mi>
                  </mml:math></jats:alternatives></jats:inline-formula> values for the iteration to converge faster, based on the primal and dual residuals arising from the optimality conditions of the augmented Lagrangian. Further, we show how to choose <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{0}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:msub>
                      <mml:mi>λ</mml:mi>
                      <mml:mn>0</mml:mn>
                    </mml:msub>
                  </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{1}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                    <mml:msub>
                      <mml:mi>λ</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msub>
                  </mml:math></jats:alternatives></jats:inline-formula> based on the inherent noise in the image.</jats:p>
author:
- first_name: Christian
  full_name: Zietlow, Christian
  last_name: Zietlow
- first_name: Jörg K. N.
  full_name: Lindner, Jörg K. N.
  last_name: Lindner
citation:
  ama: Zietlow C, Lindner JKN. ADMM-TGV image restoration for scientific applications
    with unbiased parameter choice. <i>Numerical Algorithms</i>. Published online
    2024. doi:<a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>
  apa: Zietlow, C., &#38; Lindner, J. K. N. (2024). ADMM-TGV image restoration for
    scientific applications with unbiased parameter choice. <i>Numerical Algorithms</i>.
    <a href="https://doi.org/10.1007/s11075-024-01759-2">https://doi.org/10.1007/s11075-024-01759-2</a>
  bibtex: '@article{Zietlow_Lindner_2024, title={ADMM-TGV image restoration for scientific
    applications with unbiased parameter choice}, DOI={<a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>},
    journal={Numerical Algorithms}, publisher={Springer Science and Business Media
    LLC}, author={Zietlow, Christian and Lindner, Jörg K. N.}, year={2024} }'
  chicago: Zietlow, Christian, and Jörg K. N. Lindner. “ADMM-TGV Image Restoration
    for Scientific Applications with Unbiased Parameter Choice.” <i>Numerical Algorithms</i>,
    2024. <a href="https://doi.org/10.1007/s11075-024-01759-2">https://doi.org/10.1007/s11075-024-01759-2</a>.
  ieee: 'C. Zietlow and J. K. N. Lindner, “ADMM-TGV image restoration for scientific
    applications with unbiased parameter choice,” <i>Numerical Algorithms</i>, 2024,
    doi: <a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>.'
  mla: Zietlow, Christian, and Jörg K. N. Lindner. “ADMM-TGV Image Restoration for
    Scientific Applications with Unbiased Parameter Choice.” <i>Numerical Algorithms</i>,
    Springer Science and Business Media LLC, 2024, doi:<a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>.
  short: C. Zietlow, J.K.N. Lindner, Numerical Algorithms (2024).
date_created: 2024-02-27T07:35:36Z
date_updated: 2025-01-22T09:06:50Z
department:
- _id: '286'
- _id: '15'
doi: 10.1007/s11075-024-01759-2
keyword:
- Applied Mathematics
language:
- iso: eng
publication: Numerical Algorithms
publication_identifier:
  issn:
  - 1017-1398
  - 1572-9265
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: ADMM-TGV image restoration for scientific applications with unbiased parameter
  choice
type: journal_article
user_id: '77496'
year: '2024'
...
---
_id: '59179'
article_type: original
author:
- first_name: Christian
  full_name: Zietlow, Christian
  id: '77368'
  last_name: Zietlow
  orcid: https://orcid.org/0000-0001-9696-619X
- first_name: Jörg
  full_name: Lindner, Jörg
  id: '20797'
  last_name: Lindner
citation:
  ama: Zietlow C, Lindner J. ADMM-TGV image restoration for scientific applications
    with unbiased parameter choice. <i>Numerical Algorithms</i>. 2024;97(4):1481-1512.
    doi:<a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>
  apa: Zietlow, C., &#38; Lindner, J. (2024). ADMM-TGV image restoration for scientific
    applications with unbiased parameter choice. <i>Numerical Algorithms</i>, <i>97</i>(4),
    1481–1512. <a href="https://doi.org/10.1007/s11075-024-01759-2">https://doi.org/10.1007/s11075-024-01759-2</a>
  bibtex: '@article{Zietlow_Lindner_2024, title={ADMM-TGV image restoration for scientific
    applications with unbiased parameter choice}, volume={97}, DOI={<a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>},
    number={4}, journal={Numerical Algorithms}, publisher={Springer Science and Business
    Media LLC}, author={Zietlow, Christian and Lindner, Jörg}, year={2024}, pages={1481–1512}
    }'
  chicago: 'Zietlow, Christian, and Jörg Lindner. “ADMM-TGV Image Restoration for
    Scientific Applications with Unbiased Parameter Choice.” <i>Numerical Algorithms</i>
    97, no. 4 (2024): 1481–1512. <a href="https://doi.org/10.1007/s11075-024-01759-2">https://doi.org/10.1007/s11075-024-01759-2</a>.'
  ieee: 'C. Zietlow and J. Lindner, “ADMM-TGV image restoration for scientific applications
    with unbiased parameter choice,” <i>Numerical Algorithms</i>, vol. 97, no. 4,
    pp. 1481–1512, 2024, doi: <a href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>.'
  mla: Zietlow, Christian, and Jörg Lindner. “ADMM-TGV Image Restoration for Scientific
    Applications with Unbiased Parameter Choice.” <i>Numerical Algorithms</i>, vol.
    97, no. 4, Springer Science and Business Media LLC, 2024, pp. 1481–512, doi:<a
    href="https://doi.org/10.1007/s11075-024-01759-2">10.1007/s11075-024-01759-2</a>.
  short: C. Zietlow, J. Lindner, Numerical Algorithms 97 (2024) 1481–1512.
date_created: 2025-03-28T07:10:05Z
date_updated: 2026-02-04T07:54:06Z
doi: 10.1007/s11075-024-01759-2
intvolume: '        97'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
oa: '1'
page: 1481-1512
publication: Numerical Algorithms
publication_identifier:
  issn:
  - 1017-1398
  - 1572-9265
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: ADMM-TGV image restoration for scientific applications with unbiased parameter
  choice
type: journal_article
user_id: '77368'
volume: 97
year: '2024'
...
---
_id: '19937'
abstract:
- lang: eng
  text: Symplectic integrators can be excellent for Hamiltonian initial value problems.
    Reasons for this include their preservation of invariant sets like tori, good
    energy behaviour, nonexistence of attractors, and good behaviour of statistical
    properties. These all refer to {\em long-time} behaviour. They are directly connected
    to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under
    iteration. Boundary value problems, in contrast, are posed for fixed (and often
    quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is
    not iterated. Is there any point, therefore, for a symplectic integrator to be
    used on a Hamiltonian boundary value problem? In this paper we announce results
    that symplectic integrators preserve bifurcations of Hamiltonian boundary value
    problems and that nonsymplectic integrators do not.
article_type: original
author:
- first_name: Robert I
  full_name: McLachlan, Robert I
  last_name: McLachlan
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: https://orcid.org/0000-0002-5940-8057
citation:
  ama: McLachlan RI, Offen C. Symplectic integration of boundary value problems. <i>Numerical
    Algorithms</i>. 2018:1219-1233. doi:<a href="https://doi.org/10.1007/s11075-018-0599-7">10.1007/s11075-018-0599-7</a>
  apa: McLachlan, R. I., &#38; Offen, C. (2018). Symplectic integration of boundary
    value problems. <i>Numerical Algorithms</i>, 1219–1233. <a href="https://doi.org/10.1007/s11075-018-0599-7">https://doi.org/10.1007/s11075-018-0599-7</a>
  bibtex: '@article{McLachlan_Offen_2018, title={Symplectic integration of boundary
    value problems}, DOI={<a href="https://doi.org/10.1007/s11075-018-0599-7">10.1007/s11075-018-0599-7</a>},
    journal={Numerical Algorithms}, author={McLachlan, Robert I and Offen, Christian},
    year={2018}, pages={1219–1233} }'
  chicago: McLachlan, Robert I, and Christian Offen. “Symplectic Integration of Boundary
    Value Problems.” <i>Numerical Algorithms</i>, 2018, 1219–33. <a href="https://doi.org/10.1007/s11075-018-0599-7">https://doi.org/10.1007/s11075-018-0599-7</a>.
  ieee: R. I. McLachlan and C. Offen, “Symplectic integration of boundary value problems,”
    <i>Numerical Algorithms</i>, pp. 1219–1233, 2018.
  mla: McLachlan, Robert I., and Christian Offen. “Symplectic Integration of Boundary
    Value Problems.” <i>Numerical Algorithms</i>, 2018, pp. 1219–33, doi:<a href="https://doi.org/10.1007/s11075-018-0599-7">10.1007/s11075-018-0599-7</a>.
  short: R.I. McLachlan, C. Offen, Numerical Algorithms (2018) 1219–1233.
date_created: 2020-10-06T16:29:14Z
date_updated: 2022-01-06T06:54:14Z
department:
- _id: '636'
doi: 10.1007/s11075-018-0599-7
extern: '1'
language:
- iso: eng
main_file_link:
- url: https://rdcu.be/b79ap
page: 1219-1233
publication: Numerical Algorithms
publication_identifier:
  issn:
  - 1017-1398
  - 1572-9265
publication_status: published
status: public
title: Symplectic integration of boundary value problems
type: journal_article
user_id: '85279'
year: '2018'
...