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Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials 用移位拉普拉斯多项式和费伯多项式预处理亥姆霍兹方程
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol54s534
L. G. Ramos, O. Sète, R. Nabben
{"title":"Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials","authors":"L. G. Ramos, O. Sète, R. Nabben","doi":"10.1553/etna_vol54s534","DOIUrl":"https://doi.org/10.1553/etna_vol54s534","url":null,"abstract":"We introduce a new polynomial preconditioner for solving the discretized Helmholtz equation preconditioned with the complex shifted Laplace (CSL) operator. We exploit the localization of the spectrum of the CSL-preconditioned system to approximately enclose the eigenvalues by a non-convex ‘bratwurst’ set. On this set, we expand the function 1/z into a Faber series. Truncating the series gives a polynomial, which we apply to the Helmholtz matrix preconditioned by the shifted Laplacian to obtain a new preconditioner, the Faber preconditioner. We prove that the Faber preconditioner is nonsingular for degrees one and two of the truncated series. Our numerical experiments (for problems with constant and varying wavenumber) show that the Faber preconditioner reduces the number of GMRES iterations.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"360 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":"126697485","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 posteriori error estimates for stabilised mixed finite element methods for a nonlinear elliptic problem 一类非线性椭圆型问题的稳定混合有限元方法的后验误差估计
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol55s706
María González, Hiram Varela
{"title":"A posteriori error estimates for stabilised mixed finite element methods for a nonlinear elliptic problem","authors":"María González, Hiram Varela","doi":"10.1553/etna_vol55s706","DOIUrl":"https://doi.org/10.1553/etna_vol55s706","url":null,"abstract":". In this paper we propose new adaptive stabilised mixed finite element methods for a nonlinear elliptic boundary value problem of second order in divergence form that appears, among other applications, in magnetostatics. The method is based on a three-field formulation that is augmented with suitable residual least-squares terms arising from the constitutive and equilibrium equations and from the equation that defines the gradient as an additional unknown. We show that the resulting scheme is well posed and obtain optimal error estimates. We also develop an a posteriori error analysis of residual type and derive a simple a posteriori error indicator which is reliable and locally efficient. Finally, we include several numerical experiments that confirm the theoretical results.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"13 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":"123559595","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
Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides 鞍点问题的预处理全局Krylov子空间解法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol51s495
A. Badahmane, A. Bentbib, H. Sadok
{"title":"Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides","authors":"A. Badahmane, A. Bentbib, H. Sadok","doi":"10.1553/etna_vol51s495","DOIUrl":"https://doi.org/10.1553/etna_vol51s495","url":null,"abstract":"In the present paper, we propose a preconditioned global approach as a new strategy to solve linear systems with several right-hand sides coming from saddle point problems. The preconditioner is obtained by replacing a (2,2)-block in the original saddle-point matrix A by another well-chosen block. We apply the global GMRES method to solve this new problem with several right-hand sides and give some convergence results. Moreover, we analyze the eigenvalue distribution and the eigenvectors of the proposed preconditioner when the first block is positive definite. We also compare different preconditioned global Krylov subspace algorithms (CG, MINRES, FGMRES, GMRES) with preconditioned block (CG, GMRES) algorithms. Numerical results show that our preconditioned global GMRES method is competitive with other preconditioned global Krylov subspace and preconditioned block Krylov subspace methods for solving saddle point problems with several right-hand sides.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"60 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":"126333585","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
Error estimates for variational regularization of inverse problems with general noise models for data and operator 具有数据和算子一般噪声模型的反问题变分正则化的误差估计
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol57s127
T. Hohage, Frank Werner
{"title":"Error estimates for variational regularization of inverse problems with general noise models for data and operator","authors":"T. Hohage, Frank Werner","doi":"10.1553/etna_vol57s127","DOIUrl":"https://doi.org/10.1553/etna_vol57s127","url":null,"abstract":". This paper is concerned with variational regularization of inverse problems where both the data and the forward operator are given only approximately. We propose a general approach to derive error estimates which separates the analysis of smoothness of the exact solution from the analysis of the effect of errors in the data and the operator. Our abstract error bounds are applied to both discrete and continuous data, random and deterministic types of error, as well as Huber data fidelity terms for impulsive noise.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"1 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":"126487487","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
Range restricted iterative methods for linear discrete ill-posed problems 线性离散不适定问题的限域迭代方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s348
A. Buccini, Lucas Onisk, L. Reichel
{"title":"Range restricted iterative methods for linear discrete ill-posed problems","authors":"A. Buccini, Lucas Onisk, L. Reichel","doi":"10.1553/etna_vol58s348","DOIUrl":"https://doi.org/10.1553/etna_vol58s348","url":null,"abstract":". Linear systems of equations with a matrix whose singular values decay to zero with increasing index number, and without a significant gap, are commonly referred to as linear discrete ill-posed problems. Such systems arise, e.g., when discretizing a Fredholm integral equation of the first kind. The right-hand side vectors of linear discrete ill-posed problems that arise in science and engineering often represent an experimental measurement that is contaminated by measurement error. The solution to these problems typically is very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace often is quite small, its choice is important for the quality of the computed solution. This work describes algorithms for three iterative methods that modify the GMRES, block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. We contribute to the work already available on this topic by introducing two block variants for the solution of linear systems of equations with multiple right-hand side vectors. The dominant computational aspects are discussed, and software for each method is provided. Additionally, we illustrate the utility of these iterative subspace methods through numerical examples focusing on image reconstruction. This paper is accompanied by software.","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":"120939443","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
Reuben Louis Rosenberg (1909–1986) and the Stein-Rosenberg theorem Reuben Louis Rosenberg(1909-1986)和Stein-Rosenberg定理
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58sa1
C. Brezinski, M. Redivo-Zaglia
{"title":"Reuben Louis Rosenberg (1909–1986) and the Stein-Rosenberg theorem","authors":"C. Brezinski, M. Redivo-Zaglia","doi":"10.1553/etna_vol58sa1","DOIUrl":"https://doi.org/10.1553/etna_vol58sa1","url":null,"abstract":"","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"13 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":"134115662","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
The extended global Lanczos method for matrix function approximation 矩阵函数逼近的扩展全局Lanczos方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/ETNA_VOL50S144
A. H. Bentbib, M. Ghomari, Carl Jagels, K. Jbilou, L. Reichel
{"title":"The extended global Lanczos method for matrix function approximation","authors":"A. H. Bentbib, M. Ghomari, Carl Jagels, K. Jbilou, L. Reichel","doi":"10.1553/ETNA_VOL50S144","DOIUrl":"https://doi.org/10.1553/ETNA_VOL50S144","url":null,"abstract":"The need to compute the trace of a large matrix that is not explicitly known, such as the matrix exp(A), where A is a large symmetric matrix, arises in various applications including in network analysis. The global Lanczos method is a block method that can be applied to compute an approximation of the trace. When the block size is one, this method simplifies to the standard Lanczos method. It is known that for some matrix functions and matrices, the extended Lanczos method, which uses subspaces with both positive and negative powers of A, can give faster convergence than the standard Lanczos method, which uses subspaces with nonnegative powers of A only. This suggests that it may be beneficial to use an extended global Lanczos method instead of the (standard) global Lanczos method. This paper describes an extended global Lanczos method and discusses properties of the associated Gauss-Laurent quadrature rules. Computed examples that illustrate the performance of the extended global Lanczos method are presented.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"1 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":"132933211","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: 8
Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh bakhvalov型网格截断误差分析及障位函数技术
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol51s315
T. Nhan, R. Vulanović
{"title":"Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh","authors":"T. Nhan, R. Vulanović","doi":"10.1553/etna_vol51s315","DOIUrl":"https://doi.org/10.1553/etna_vol51s315","url":null,"abstract":"We use a barrier-function technique to prove the parameter-uniform convergence for singularly perturbed convection-diffusion problems discretized on a Bakhvalov-type mesh. This is the first proof of this kind in the research literature, the barrier-function approach having only been applied so far to Shishkin-type meshes.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"16 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114018574","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: 12
On Cvetković-Kostić-Varga type matrices
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol58s244
Lei Gao, Chaoqian Li
{"title":"On Cvetković-Kostić-Varga type matrices","authors":"Lei Gao, Chaoqian Li","doi":"10.1553/etna_vol58s244","DOIUrl":"https://doi.org/10.1553/etna_vol58s244","url":null,"abstract":". Cvetkovi´c-Kosti´c-Varga (CKV)-type matrices play a significant role in numerical linear algebra. However, verifying whether a given matrix is a CKV-type matrix is complicated because it involves choosing a suitable subset of { 1 , 2 ,...,n } . In this paper, we give some easily computable and verifiable equivalent conditions for a CKV-type matrix, and based on these conditions, two direct algorithms with less computational cost for identifying CKV-type matrices are put forward. Moreover, by considering the matrix sparsity pattern, two classes of matrices called S -Sparse Ostrowski-Brauer type-I and type-II matrices are proposed and then proved to be subclasses of CKV-type matrices. The relationships with other subclasses of H -matrices are also discussed. Besides, a new eigenvalue localization set involving the sparsity pattern for matrices is presented, which requires less computational cost than that provided by Cvetkovi´c et al. [Linear Algebra Appl., 608 (2021), pp.158–184].","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"95 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":"116718538","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
Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains 矩形域泊松方程的同步和异步优化Schwarz方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 1900-01-01 DOI: 10.1553/etna_vol55s744
J. C. Garay, F. Magoulès, D. Szyld
{"title":"Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains","authors":"J. C. Garay, F. Magoulès, D. Szyld","doi":"10.1553/etna_vol55s744","DOIUrl":"https://doi.org/10.1553/etna_vol55s744","url":null,"abstract":". Convergence results for optimized Schwarz methods (OSM) applied as solvers for Poisson’s equation in a bounded rectangular domain with Dirichlet (physical) boundary conditions and zeroth-order (Robin) artificial transmission conditions between subdomains are presented. The analysis presented applies to a continuous formulation on an arbitrary number of subdomains with cross points. Both synchronous and asynchronous versions of OSM are discussed. Convergence theorems are presented, and it is shown numerically that the hypotheses of these theorems are satisfied for certain configurations of the subdomains. Additional numerical experiments illustrate the practical behavior of the methods discussed.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"56 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":"116831305","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: 9
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