This paper introduces a very fast method for the computation of the resolvent of fractional powers of operators. The analysis is kept in the continuous setting of (potentially unbounded) self-adjoint positive operators in Hilbert spaces. The method is based on the Gauss-Laguerre rule, exploiting a particular integral representation of the resolvent. We provide sharp error estimates that can be used to a priori select the number of nodes to achieve a prescribed tolerance.
{"title":"A Gauss-Laguerre approach for the resolvent of fractional powers","authors":"Eleonora Denich, Laura Grazia Dolce, Paolo Novati","doi":"10.1553/etna_vol58s517","DOIUrl":"https://doi.org/10.1553/etna_vol58s517","url":null,"abstract":"This paper introduces a very fast method for the computation of the resolvent of fractional powers of operators. The analysis is kept in the continuous setting of (potentially unbounded) self-adjoint positive operators in Hilbert spaces. The method is based on the Gauss-Laguerre rule, exploiting a particular integral representation of the resolvent. We provide sharp error estimates that can be used to a priori select the number of nodes to achieve a prescribed tolerance.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136217944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a novel strategy for the automatic estimation of the two regularization parameters arising in the image decomposition variational model employed for the restoration task when the underlying corrupting noise is known to be additive white Gaussian. In the model of interest, the target image is decomposed in its piecewise constant and smooth components, with a total variation term penalizing the former and a Tikhonov term acting on the latter. The proposed criterion, which relies on the whiteness property of the noise, extends the residual whiteness principle, originally introduced in the case of a single regularization parameter. The structure of the considered decomposition model allows for an efficient estimation of the pair of unknown parameters, that can be automatically adjusted along the iterations with the alternating direction method of multipliers employed for the numerical solution. The proposed multi-parameter residual whiteness principle is tested on different images with different levels of corruption. The performed tests highlight that the whiteness criterion is particularly effective and robust when moving from a single-parameter to a multi-parameter scenario.
{"title":"Parameter-free restoration of piecewise smooth images","authors":"Alessandro Lanza, Monica Pragliola, Fiorella Sgallari","doi":"10.1553/etna_vol59s202","DOIUrl":"https://doi.org/10.1553/etna_vol59s202","url":null,"abstract":"We propose a novel strategy for the automatic estimation of the two regularization parameters arising in the image decomposition variational model employed for the restoration task when the underlying corrupting noise is known to be additive white Gaussian. In the model of interest, the target image is decomposed in its piecewise constant and smooth components, with a total variation term penalizing the former and a Tikhonov term acting on the latter. The proposed criterion, which relies on the whiteness property of the noise, extends the residual whiteness principle, originally introduced in the case of a single regularization parameter. The structure of the considered decomposition model allows for an efficient estimation of the pair of unknown parameters, that can be automatically adjusted along the iterations with the alternating direction method of multipliers employed for the numerical solution. The proposed multi-parameter residual whiteness principle is tested on different images with different levels of corruption. The performed tests highlight that the whiteness criterion is particularly effective and robust when moving from a single-parameter to a multi-parameter scenario.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval $[0,1] subset mathbb{R}$, but little is known about the multidimensional situation. This article tries to fill this gap with analytical and numerical studies on the reconstruction of a real function of two real variables over the unit square from observations of its autoconvolution on $[0,2]^2 subset mathbb{R}^2$ (full data case) or on $[0,1]^2$ (limited data case). In an $L^2$-setting, twofoldness and uniqueness assertions are proven for the deautoconvolution problem in 2D. Moreover, its ill-posedness is characterized and illustrated. Extensive numericalcase studies give an overview of the behaviour of stable approximate solutions to the two-dimensional deautoconvolution problem obtained by Tikhonov-type regularization with different penalties and the iteratively regularized Gauss–Newton method.
{"title":"Deautoconvolution in the two-dimensional case","authors":"Yu Deng, Bernd Hofmann, Frank Werner","doi":"10.1553/etna_vol59s24","DOIUrl":"https://doi.org/10.1553/etna_vol59s24","url":null,"abstract":"There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval $[0,1] subset mathbb{R}$, but little is known about the multidimensional situation. This article tries to fill this gap with analytical and numerical studies on the reconstruction of a real function of two real variables over the unit square from observations of its autoconvolution on $[0,2]^2 subset mathbb{R}^2$ (full data case) or on $[0,1]^2$ (limited data case). In an $L^2$-setting, twofoldness and uniqueness assertions are proven for the deautoconvolution problem in 2D. Moreover, its ill-posedness is characterized and illustrated. Extensive numericalcase studies give an overview of the behaviour of stable approximate solutions to the two-dimensional deautoconvolution problem obtained by Tikhonov-type regularization with different penalties and the iteratively regularized Gauss–Newton method.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135637414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In paper D. Lj. ÄjukiÄ, R. M. MutavdžiÄ ÄjukiÄ, A. V. PejÄev, and M. M. SpaleviÄ, Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses â a survey of recent results, Electron. Trans. Numer. Anal., 53 (2020), pp. 352â382, Lemma 4.1 can be applied to show the asymptotic behaviour of the modulus of the complex kernel in the remainder term of all the quadrature formulas in the recent papers that are concerned with error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses. However, in the paper D. R. JandrliÄ, Dj. M. KrtiniÄ, Lj. V. MihiÄ, A. V. PejÄev, M. M. SpaleviÄ, Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands, Electron. Trans. Anal. 55 (2022), pp. 424â437, which this note is concerned with, there is a kernel whose numerator contains an infinite series, and in this case the mentioned lemma cannot be applied. This note shows that the modulus of the latter kernel attains its maximum as conjectured in the latter paper.
论文中d.l j。Ä´jukiÄ°，R. M. MutavdžiÄ°Ä´jukiÄ°，a . V. PejÄ´ev, and M. M. SpaleviÄ°，椭圆上解析函数的高斯型正交公式的误差估计，最近结果的综述，电子。反式。号码。分析的在最近关于椭圆上解析函数的高斯型正交公式的误差估计的论文中，引理4.1可用于显示所有正交公式的余项中复核模的渐近行为。然而，在论文中d.r. JandrliÄ;M. KrtiniÄ;V. MihiÄ°n, A. V. PejÄ°n, M. M. SpaleviÄ°n，解析积分高斯正交公式与Legendre权函数的误差界，电子。反式。在本注注所讨论的(Anal. 55 (2022)， pp. 424 - ' 437)中，存在一个核，其分子包含一个无穷级数，在这种情况下，上述引理不能应用。这说明后一核的模达到了后一篇文章所推测的最大值。
{"title":"A note on “Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands” by M. M. Spalević et al.","authors":"Aleksandar V. Pejčev","doi":"10.1553/etna_vol59s89","DOIUrl":"https://doi.org/10.1553/etna_vol59s89","url":null,"abstract":"In paper D. Lj. ÄjukiÄ, R. M. MutavdžiÄ ÄjukiÄ, A. V. PejÄev, and M. M. SpaleviÄ, Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses â a survey of recent results, Electron. Trans. Numer. Anal., 53 (2020), pp. 352â382, Lemma 4.1 can be applied to show the asymptotic behaviour of the modulus of the complex kernel in the remainder term of all the quadrature formulas in the recent papers that are concerned with error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses. However, in the paper D. R. JandrliÄ, Dj. M. KrtiniÄ, Lj. V. MihiÄ, A. V. PejÄev, M. M. SpaleviÄ, Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands, Electron. Trans. Anal. 55 (2022), pp. 424â437, which this note is concerned with, there is a kernel whose numerator contains an infinite series, and in this case the mentioned lemma cannot be applied. This note shows that the modulus of the latter kernel attains its maximum as conjectured in the latter paper.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135685801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider a class of non-standard elliptic transmission problems in disjoint domains. As a model example, we consider an area consisting of two non-adjacent rectangles. In each subarea, a boundary-value problem of elliptic type is considered, where the interaction between their solutions is described by nonlocal integral conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, up to a slowly increasing logarithmic factor of the mesh size, is obtained.
{"title":"On the numerical solution of an elliptic problem with nonlocal boundary conditions","authors":"Zorica Milovanović Jeknić, Bratislav Sredojević, Dejan Bojović","doi":"10.1553/etna_vol59s179","DOIUrl":"https://doi.org/10.1553/etna_vol59s179","url":null,"abstract":"In this paper we consider a class of non-standard elliptic transmission problems in disjoint domains. As a model example, we consider an area consisting of two non-adjacent rectangles. In each subarea, a boundary-value problem of elliptic type is considered, where the interaction between their solutions is described by nonlocal integral conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, up to a slowly increasing logarithmic factor of the mesh size, is obtained.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns the theory and development of inexact rational Krylovsubspace methods for approximating the action of a function of a matrix $f(A)$to a column vector $b$. At each step of the rational Krylov subspace methods, ashifted linear system of equations needs to be solved to enlarge the subspace.For large-scale problems, such a linear system is usually solved approximatelyby an iterative method. The main question is how to relax the accuracy of theselinear solves without negatively affecting the convergence of the approximationof $f(A)b$. Our insight into this issue is obtained by exploring the residualbounds for the rational Krylov subspace approximations of $f(A)b$, based on thedecaying behavior of the entries in the first column of certain matrices of $A$restricted to the rational Krylov subspaces. The decay bounds for these entriesfor both analytic functions and Markov functions can be efficiently andaccurately evaluated by appropriate quadrature rules. A heuristic based on thesebounds is proposed to relax the tolerances of the linear solves arising in eachstep of the rational Krylov subspace methods. As the algorithm progresses towardconvergence, the linear solves can be performed with increasingly lower accuracyand computational cost. Numerical experiments for large nonsymmetric matricesshow the effectiveness of the tolerance relaxation strategy for the inexactlinear solves of rational Krylov subspace methods.
{"title":"Inexact rational Krylov subspace methods for approximating the action of functions of matrices","authors":"Shengjie Xu, Fei Xue","doi":"10.1553/etna_vol58s538","DOIUrl":"https://doi.org/10.1553/etna_vol58s538","url":null,"abstract":"This paper concerns the theory and development of inexact rational Krylovsubspace methods for approximating the action of a function of a matrix $f(A)$to a column vector $b$. At each step of the rational Krylov subspace methods, ashifted linear system of equations needs to be solved to enlarge the subspace.For large-scale problems, such a linear system is usually solved approximatelyby an iterative method. The main question is how to relax the accuracy of theselinear solves without negatively affecting the convergence of the approximationof $f(A)b$. Our insight into this issue is obtained by exploring the residualbounds for the rational Krylov subspace approximations of $f(A)b$, based on thedecaying behavior of the entries in the first column of certain matrices of $A$restricted to the rational Krylov subspaces. The decay bounds for these entriesfor both analytic functions and Markov functions can be efficiently andaccurately evaluated by appropriate quadrature rules. A heuristic based on thesebounds is proposed to relax the tolerances of the linear solves arising in eachstep of the rational Krylov subspace methods. As the algorithm progresses towardconvergence, the linear solves can be performed with increasingly lower accuracyand computational cost. Numerical experiments for large nonsymmetric matricesshow the effectiveness of the tolerance relaxation strategy for the inexactlinear solves of rational Krylov subspace methods.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents some preconditioning techniques that enhance the performance of iterative regularization methods applied to image deblurring problems determined by a wide variety of point spread functions (PSFs) and boundary conditions. We first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, we show that the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. Extensive numerical experiments show the effectiveness of the new preconditioning techniques, including when the deblurring of sparse images is considered.
{"title":"Symmetrization techniques in image deblurring","authors":"Marco Donatelli, Paola Ferrari, Silvia Gazzola","doi":"10.1553/etna_vol59s157","DOIUrl":"https://doi.org/10.1553/etna_vol59s157","url":null,"abstract":"This paper presents some preconditioning techniques that enhance the performance of iterative regularization methods applied to image deblurring problems determined by a wide variety of point spread functions (PSFs) and boundary conditions. We first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, we show that the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. Extensive numerical experiments show the effectiveness of the new preconditioning techniques, including when the deblurring of sparse images is considered.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136003785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we propose a Gauss-type quadrature rule into which the external zeros of the integrand (the zeros of the integrand outside the integration interval) are incorporated. This new formula with $n$ nodes, denoted by $mathcal G_n$, proves to be exact for certain polynomials of degree greater than $2n-1$ (while the Gauss quadrature formula with the same number of nodes is exact for all polynomials of degree less than or equal to $2n-1$). It turns out that $mathcal G_n$ has several good properties: all its nodes are pairwise distinct and belong to the interior of the integration interval, all its weights are positive, it converges, and it is applicable both when the external zeros of the integrand are known exactly and when they are known approximately. In order to economically estimate the error of $mathcal G_n$, we construct its extensions that inherit the $n$ nodes of $mathcal G_n$ and that are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Further, we show that $mathcal G_n$ with respect to the pairwise distinct external zeros of the integrand represents a special case of the (slightly modified) Gauss quadrature formula with preassigned nodes. The accuracy of $mathcal G_n$ and its extensions is confirmed by numerical experiments.
{"title":"Gauss-type quadrature rules with respect to external zeros of the integrand","authors":"Jelena Tomanović","doi":"10.1553/etna_vol59s230","DOIUrl":"https://doi.org/10.1553/etna_vol59s230","url":null,"abstract":"In the present paper, we propose a Gauss-type quadrature rule into which the external zeros of the integrand (the zeros of the integrand outside the integration interval) are incorporated. This new formula with $n$ nodes, denoted by $mathcal G_n$, proves to be exact for certain polynomials of degree greater than $2n-1$ (while the Gauss quadrature formula with the same number of nodes is exact for all polynomials of degree less than or equal to $2n-1$). It turns out that $mathcal G_n$ has several good properties: all its nodes are pairwise distinct and belong to the interior of the integration interval, all its weights are positive, it converges, and it is applicable both when the external zeros of the integrand are known exactly and when they are known approximately. In order to economically estimate the error of $mathcal G_n$, we construct its extensions that inherit the $n$ nodes of $mathcal G_n$ and that are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Further, we show that $mathcal G_n$ with respect to the pairwise distinct external zeros of the integrand represents a special case of the (slightly modified) Gauss quadrature formula with preassigned nodes. The accuracy of $mathcal G_n$ and its extensions is confirmed by numerical experiments.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present the concept of perturbation bounds for the right eigenvalues of a quaternionic matrix. In particular, a Bauer-Fike-type theorem for the right eigenvalues of a diagonalizable quaternionic matrix is derived. In addition, perturbations of a quaternionic matrix are discussed via a block-diagonal decomposition and the Jordan canonical form of a quaternionic matrix. The location of the standard right eigenvalues of a quaternionic matrix and a sufficient condition for the stability of a perturbed quaternionic matrix are given. As an application, perturbation bounds for the zeros of quaternionic polynomials are derived. Finally, we give numerical examples to illustrate our results.
{"title":"Perturbation analysis of matrices over a quaternion division algebra","authors":"Sk. Safique Ahmad, I. Ali, I. Slapničar","doi":"10.1553/ETNA_VOL54S128","DOIUrl":"https://doi.org/10.1553/ETNA_VOL54S128","url":null,"abstract":"In this paper, we present the concept of perturbation bounds for the right eigenvalues of a quaternionic matrix. In particular, a Bauer-Fike-type theorem for the right eigenvalues of a diagonalizable quaternionic matrix is derived. In addition, perturbations of a quaternionic matrix are discussed via a block-diagonal decomposition and the Jordan canonical form of a quaternionic matrix. The location of the standard right eigenvalues of a quaternionic matrix and a sufficient condition for the stability of a perturbed quaternionic matrix are given. As an application, perturbation bounds for the zeros of quaternionic polynomials are derived. Finally, we give numerical examples to illustrate our results.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67352033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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