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Thick restarting the weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem 厚重启动加权调和Golub-Kahan-Lanczos算法求解线性响应特征值问题
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol51s529
H. Zhong, Guoliang Chen
{"title":"Thick restarting the weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem","authors":"H. Zhong, Guoliang Chen","doi":"10.1553/etna_vol51s529","DOIUrl":"https://doi.org/10.1553/etna_vol51s529","url":null,"abstract":"In this paper, we propose a weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem (LREP). Convergence properties are established for the error bounds of the approximate eigenpairs. Moreover, we consider a practical thick-restart procedure to reduce the computational and memory costs and present a weighted harmonic Golub-Kahan-Lanczos algorithm with deflated restarting. Numerical tests show the efficiency of our new algorithms.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122174525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Optimal $W^{1,infty}$-estimates for an isoparametric finite element discretization of elliptic boundary value problems 椭圆边值问题等参数有限元离散化的最优$W^{1,infty}$ -估计
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s1
Benjamin Dörich, Jan Leibold, B. Maier
{"title":"Optimal $W^{1,infty}$-estimates for an isoparametric finite element discretization of elliptic boundary value problems","authors":"Benjamin Dörich, Jan Leibold, B. Maier","doi":"10.1553/etna_vol58s1","DOIUrl":"https://doi.org/10.1553/etna_vol58s1","url":null,"abstract":". We consider an elliptic boundary value problem on a domain with regular boundary and discretize it with isoparametric finite elements of order k ≥ 1 . We show optimal order of convergence of the isoparametric finite element solution in the W 1 , ∞ -norm. As an intermediate step, we derive stability and convergence estimates of optimal order k for a (generalized) Ritz map.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124097966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Explicit deflation in Golub-Kahan-Lanczos bidiagonalization methods Golub-Kahan-Lanczos双对角化方法中的显通缩
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s164
J. Baglama, V. Perović
{"title":"Explicit deflation in Golub-Kahan-Lanczos bidiagonalization methods","authors":"J. Baglama, V. Perović","doi":"10.1553/etna_vol58s164","DOIUrl":"https://doi.org/10.1553/etna_vol58s164","url":null,"abstract":". We discuss a simple, easily overlooked, explicit deflation procedure applied to Golub-Kahan-Lanczos Bidiagonalization (GKLB)-based methods to compute the next set of the largest singular triplets of a matrix from an already computed partial singular value decomposition. Our results here complement the vast literature on this topic, provide additional insight, and highlight the simplicity and the effectiveness of this procedure. We demonstrate how existing GKLB-based routines for the computation of the largest singular triplets can be easily adapted to take advantage of explicit deflation, thus making it more appealing to a wider range of users. Numerical examples are presented including an application of singular value thresholding.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"176 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114739793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regularized functional matching pursuit for the spherical multiple-shell electro-magnetoencephalography problem 球形多壳脑电图问题的正则函数匹配追踪
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol57s153
S. Leweke, V. Michel
{"title":"Regularized functional matching pursuit for the spherical multiple-shell electro-magnetoencephalography problem","authors":"S. Leweke, V. Michel","doi":"10.1553/etna_vol57s153","DOIUrl":"https://doi.org/10.1553/etna_vol57s153","url":null,"abstract":". Reconstruction of the neuronal current inside the human brain from non-invasive measurements of the magnetic flux density via magnetoencephalography (MEG) or of electric potential differences via electroencephalography (EEG) is an invaluable tool for neuroscientific research, as it provides measures of activity in the brain. However, it is also a severely ill-posed inverse problem. Assuming spherical geometries, we consider the spherical multiple-shell model for the inverse MEG and EEG problem and apply the regularized functional matching pursuit algorithm (RFMP) for its solution. We present a new convergence proof for the RFMP for operators between two infinite-dimensional Hilbert spaces. Moreover, we utilize the complementarity of EEG and MEG data to combine inversions of simultaneous electric and magnetic measurements. Finally, we test the algorithm numerically on synthetic data using several Sobolev norms as penalty term and apply it to real data. Sobolev","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125088814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Simultaneous identification of volatility and interest rate functions-a two-parameter regularization approach 波动率和利率函数的同时识别——一种双参数正则化方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/ETNA_VOL51S99
Christopher Hofmann, B. Hofmann, A. Pichler
{"title":"Simultaneous identification of volatility and interest rate functions-a two-parameter regularization approach","authors":"Christopher Hofmann, B. Hofmann, A. Pichler","doi":"10.1553/ETNA_VOL51S99","DOIUrl":"https://doi.org/10.1553/ETNA_VOL51S99","url":null,"abstract":"This paper investigates a specific ill-posed nonlinear inverse problem that arises in financial markets. Precisely, as a benchmark problem in the context of volatility surface calibration, we consider the simultaneous recovery of implied volatility and interest rate functions over a finite time interval from corresponding calland put-price functions for idealized continuous families of European vanilla options over the same maturity interval. We prove identifiability of the pair of functions to be identified by showing injectivity of the forward operator inL2-spaces. To overcome the ill-posedness we employ a two-parameter Tikhonov regularization with heuristic parameter choice rules and demonstrate chances and limitations by means of numerical case studies using synthetic data.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128877236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On trigonometric interpolation in an even number of points 关于偶数点的三角插值
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s271
Anthony P. Austin
{"title":"On trigonometric interpolation in an even number of points","authors":"Anthony P. Austin","doi":"10.1553/etna_vol58s271","DOIUrl":"https://doi.org/10.1553/etna_vol58s271","url":null,"abstract":". In contrast to odd-length trigonometric interpolants, even-length trigonometric interpolants need not be unique; this is apparent from the representation of the interpolant in the (real or complex) Fourier basis, which possesses an extra degree of freedom in the choice of the highest-order basis function in the even case. One can eliminate this degree of freedom by imposing a constraint, but then the interpolant may cease to exist for certain choices of the interpolation points. On the other hand, the Lagrange representation developed by Gauss always produces an interpolant despite having no free parameters. We discuss the choice Gauss’s formula makes for the extra degree of freedom and show that, when the points are equispaced, its choice is optimal in the sense that it minimizes both the standard and 2-norm Lebesgue constants for the interpolation problem. For non-equispaced points, we give numerical evidence that Gauss’s formula is no longer optimal and consider interpolants of minimal 2-norm instead. We show how to modify Gauss’s formula to produce a minimal-norm interpolant and that, if the points are equispaced, no modification is necessary.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"258 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123077761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra 具有规定三谱的实非自伴三对角矩阵的构造
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol51s363
Wei-Ru Xu, N. Bebiano, Guoliang Chen
{"title":"On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra","authors":"Wei-Ru Xu, N. Bebiano, Guoliang Chen","doi":"10.1553/etna_vol51s363","DOIUrl":"https://doi.org/10.1553/etna_vol51s363","url":null,"abstract":"Non-selfadjoint tridiagonal matrices play a role in the discretization and truncation of the Schrödinger equation in some extensions of quantum mechanics, a research field particularly active in the last two decades. In this article, we consider an inverse eigenvalue problem that consists of the reconstruction of such a real nonselfadjoint matrix from its prescribed eigenvalues and those of two complementary principal submatrices. Necessary and sufficient conditions under which the problem has a solution are presented, and uniqueness is discussed. The reconstruction is performed by using a modified unsymmetric Lanczos algorithm, designed to solve the proposed inverse eigenvalue problem. Some illustrative numerical examples are given to test the efficiency and feasibility of our reconstruction algorithm.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127624369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands 解析积分中带勒让德权函数的高斯正交公式的误差界
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol55s424
D. Jandrlic, D. Krtinic, Ljubica Mihić, A. Pejcev, M. Spalević
{"title":"Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands","authors":"D. Jandrlic, D. Krtinic, Ljubica Mihić, A. Pejcev, M. Spalević","doi":"10.1553/etna_vol55s424","DOIUrl":"https://doi.org/10.1553/etna_vol55s424","url":null,"abstract":". In this paper we are concerned with a method for the numerical evaluation of the error terms in Gaussian quadrature formulae with the Legendre weight function. Inspired by the work of H. Wang and L. Zhang [J. Sci. Comput., 75 (2018), pp. 457–477] and applying the results of S. Notaris [Math. Comp., 75 (2006), pp. 1217– 1231], we determine an explicit formula for the kernel. This explicit expression is used for finding the points on ellipses where the maximum of the modulus of the kernel is attained. Effective error bounds for this quadrature formula for analytic integrands are derived.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132664657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Improved bisection eigenvalue method for band symmetric Toeplitz matrices 带对称Toeplitz矩阵的改进对分特征值法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s316
Y. Eidelman, I. Haimovici
{"title":"Improved bisection eigenvalue method for band symmetric Toeplitz matrices","authors":"Y. Eidelman, I. Haimovici","doi":"10.1553/etna_vol58s316","DOIUrl":"https://doi.org/10.1553/etna_vol58s316","url":null,"abstract":". We apply a general bisection eigenvalue algorithm, developed for Hermitian matrices with quasisep- arable representations, to the particular case of real band symmetric Toeplitz matrices. We show that every band symmetric Toeplitz matrix T q with bandwidth q admits the representation T q = A q + H q , where the eigendata of A q are obtained explicitly and the matrix H q has nonzero entries only in two diagonal blocks of size ( q − 1) × ( q − 1) . Based on this representation, one obtains an interlacing property of the eigenvalues of the matrix T q and the known eigenvalues of the matrix A q . This allows us to essentially improve the performance of the bisection eigenvalue algorithm. We also present an algorithm to compute the corresponding eigenvectors.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129772542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A multigrid frame based method for image deblurring 一种基于多网格帧的图像去模糊方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol53s283
A. Buccini, M. Donatelli
{"title":"A multigrid frame based method for image deblurring","authors":"A. Buccini, M. Donatelli","doi":"10.1553/etna_vol53s283","DOIUrl":"https://doi.org/10.1553/etna_vol53s283","url":null,"abstract":". Iterative soft thresholding algorithms combine one step of Landweber method (or accelerated vari- ants), with one step of thresholding of the wavelet (framelet) coefficients. In this paper, we improve these methods by using the framelet multilevel decomposition for defining a multigrid deconvolution with grid transfer operators given by the low-pass filter of the frame. Assuming that an estimate of the noise level is available, we combine a recently proposed iterative method for ℓ 2 - regularization with linear framelet denoising by soft-thresholding. This combination allows a fast frequency filtering in the Fourier domain and produces a sparse reconstruction in the wavelet domain. Moreover, its employment in a multigrid scheme ensures a stable convergence and a reduced noise amplification. The proposed multigrid method is independent of the imposed boundary conditions and the iterations can be easily projected in a closed and convex set, e.g., the nonnegative cone. We study the convergence of the proposed algorithm and prove that it is a regularization method. Several numerical results prove that this approach is able to provide highly accurate reconstructions in several different scenarios without requiring the setting of any parameter.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130232929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
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