Tikhonov Regularization

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Lothar Reichel - One of the best experts on this subject based on the ideXlab platform.

  • Generalized singular value decomposition with iterated Tikhonov Regularization
    Journal of Computational and Applied Mathematics, 2020
    Co-Authors: Alessandro Buccini, Mirjeta Pasha, Lothar Reichel
    Abstract:

    Abstract Linear discrete ill-posed problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods try to reduce the sensitivity by replacing the given problem by a nearby one, whose solution is less affected by perturbations. This paper describes how generalized singular value decomposition can be combined with iterated Tikhonov Regularization and illustrates that the method so obtained determines approximate solutions of higher quality than the more commonly used approach of pairing generalized singular value decomposition with (standard) Tikhonov Regularization. The Regularization parameter is determined with the aid of the discrepancy principle. This requires the application of a zero-finder. Several zero-finders are compared.

  • Parameter determination for Tikhonov Regularization problems in general form
    Journal of Computational and Applied Mathematics, 2018
    Co-Authors: Yonggi Park, Lothar Reichel, Giuseppe Rodriguez
    Abstract:

    Abstract Tikhonov Regularization is one of the most popular methods for computing an approximate solution of linear discrete ill-posed problems with error-contaminated data. A Regularization parameter λ > 0 balances the influence of a fidelity term, which measures how well the data are approximated, and of a Regularization term, which dampens the propagation of the data error into the computed approximate solution. The value of the Regularization parameter is important for the quality of the computed solution: a too large value of λ > 0 gives an over-smoothed solution that lacks details that the desired solution may have, while a too small value yields a computed solution that is unnecessarily, and possibly severely, contaminated by propagated error. When a fairly accurate estimate of the norm of the error in the data is known, a suitable value of λ often can be determined with the aid of the discrepancy principle. This paper is concerned with the situation when the discrepancy principle cannot be applied. It then can be quite difficult to determine a suitable value of λ . We consider the situation when the Tikhonov Regularization problem is in general form, i.e., when the Regularization term is determined by a Regularization matrix different from the identity, and describe an extension of the COSE method for determining the Regularization parameter λ in this situation. This method has previously been discussed for Tikhonov Regularization in standard form, i.e., for the situation when the Regularization matrix is the identity. It is well known that Tikhonov Regularization in general form, with a suitably chosen Regularization matrix, can give a computed solution of higher quality than Tikhonov Regularization in standard form.

  • Iterated Tikhonov Regularization with a general penalty term
    Numerical Linear Algebra with Applications, 2017
    Co-Authors: Alessandro Buccini, Marco Donatelli, Lothar Reichel
    Abstract:

    Summary Tikhonov Regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice of the Regularization matrix may significantly affect the quality of the computed solution. When the Regularization matrix is the identity, iterated Tikhonov Regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov Regularization. This paper provides an analysis of iterated Tikhonov Regularization with a Regularization matrix different from the identity. Computed examples illustrate the performance of this method.

  • Modulus-based iterative methods for constrained Tikhonov Regularization
    Journal of Computational and Applied Mathematics, 2017
    Co-Authors: Zhong-zhi Bai, Alessandro Buccini, Lothar Reichel, Ken Hayami, Jun-feng Yin, Ning Zheng
    Abstract:

    Tikhonov Regularization is one of the most popular methods for the solution of linear discrete ill-posed problems. In many applications the desired solution is known to lie in the nonnegative cone. It is then natural to require that the approximate solution determined by Tikhonov Regularization also lies in this cone. The present paper describes two iterative methods, that employ modulus-based iterative methods, to compute approximate solutions in the nonnegative cone of large-scale Tikhonov Regularization problems. The first method considered consists of two steps: first the given linear discrete ill-posed problem is reduced to a small problem by a Krylov subspace method, and then the reduced Tikhonov Regularization problems so obtained is solved. The second method described explores the structure of certain image restoration problems. Computed examples illustrate the performances of these methods.

  • Fractional Tikhonov Regularization with a nonlinear penalty term
    Journal of Computational and Applied Mathematics, 2017
    Co-Authors: Serena Morigi, Lothar Reichel, Fiorella Sgallari
    Abstract:

    Tikhonov Regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix and an error-contaminated data vector (right-hand side). This Regularization method replaces the given problem by a penalized least-squares problem. It is well known that Tikhonov Regularization in standard form may yield approximate solutions that are too smooth, i.e., the computed approximate solution may lack many details that the desired solution of the associated, but unavailable, error-free problem might possess. Fractional Tikhonov Regularization methods have been introduced to remedy this shortcoming. However, the computed solution determined by fractional Tikhonov methods in standard form may display undesirable spurious oscillations. This paper proposes that fractional Tikhonov methods be equipped with a nonlinear penalty term, such as a TV-norm penalty term, to reduce unwanted oscillations. Numerical examples illustrate the benefits of this approach.

Zhousuo Zhang - One of the best experts on this subject based on the ideXlab platform.

  • Tikhonov Regularization based operational transfer path analysis
    Mechanical Systems and Signal Processing, 2016
    Co-Authors: Wei Cheng, Zhousuo Zhang
    Abstract:

    Abstract To overcome ill-posed problems in operational transfer path analysis (OTPA), and improve the stability of solutions, this paper proposes a novel OTPA based on Tikhonov Regularization, which considers both fitting degrees and stability of solutions. Firstly, fundamental theory of Tikhonov Regularization-based OTPA is presented, and comparative studies are provided to validate the effectiveness on ill-posed problems. Secondly, transfer path analysis and source contribution evaluations for numerical cases studies on spherical radiating acoustical sources are comparatively studied. Finally, transfer path analysis and source contribution evaluations for experimental case studies on a test bed with thin shell structures are provided. This study provides more accurate transfer path analysis for mechanical systems, which can benefit for vibration reduction by structural path optimization. Furthermore, with accurate evaluation of source contributions, vibration monitoring and control by active controlling vibration sources can be effectively carried out.

Fiorella Sgallari - One of the best experts on this subject based on the ideXlab platform.

  • Fractional Tikhonov Regularization with a nonlinear penalty term
    Journal of Computational and Applied Mathematics, 2017
    Co-Authors: Serena Morigi, Lothar Reichel, Fiorella Sgallari
    Abstract:

    Tikhonov Regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix and an error-contaminated data vector (right-hand side). This Regularization method replaces the given problem by a penalized least-squares problem. It is well known that Tikhonov Regularization in standard form may yield approximate solutions that are too smooth, i.e., the computed approximate solution may lack many details that the desired solution of the associated, but unavailable, error-free problem might possess. Fractional Tikhonov Regularization methods have been introduced to remedy this shortcoming. However, the computed solution determined by fractional Tikhonov methods in standard form may display undesirable spurious oscillations. This paper proposes that fractional Tikhonov methods be equipped with a nonlinear penalty term, such as a TV-norm penalty term, to reduce unwanted oscillations. Numerical examples illustrate the benefits of this approach.

  • Tikhonov Regularization based on generalized Krylov subspace methods
    Applied Numerical Mathematics, 2012
    Co-Authors: Lothar Reichel, Fiorella Sgallari
    Abstract:

    We consider Tikhonov Regularization of large linear discrete ill-posed problems with a Regularization operator of general form and present an iterative scheme based on a generalized Krylov subspace method. This method simultaneously reduces both the matrix of the linear discrete ill-posed problem and the Regularization operator. The reduced problem so obtained may be solved, e.g., with the aid of the singular value decomposition. Also, Tikhonov Regularization with several Regularization operators is discussed.

Ailin Qian - One of the best experts on this subject based on the ideXlab platform.

Yu Bin Zhou - One of the best experts on this subject based on the ideXlab platform.

  • Tikhonov Regularization method for a backward problem for the time-fractional diffusion equation
    Applied Mathematical Modelling, 2013
    Co-Authors: Jungang Wang, Ting Wei, Yu Bin Zhou
    Abstract:

    Abstract This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov Regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov Regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori Regularization parameter choice rule and an a posteriori Regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.