Pub Date : 2023-10-11DOI: 10.1007/s10543-023-00986-8
Annika Lang, Mike Pereira
Abstract A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments.
{"title":"Galerkin–Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds","authors":"Annika Lang, Mike Pereira","doi":"10.1007/s10543-023-00986-8","DOIUrl":"https://doi.org/10.1007/s10543-023-00986-8","url":null,"abstract":"Abstract A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136098688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1007/s10543-023-00991-x
Vance Faber, Jörg Liesen, Petr Tichý
Abstract Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.
{"title":"On the Forsythe conjecture","authors":"Vance Faber, Jörg Liesen, Petr Tichý","doi":"10.1007/s10543-023-00991-x","DOIUrl":"https://doi.org/10.1007/s10543-023-00991-x","url":null,"abstract":"Abstract Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135579786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1007/s10543-023-00989-5
Elena Celledoni, Helge Glöckner, Jørgen N. Riseth, Alexander Schmeding
{"title":"Deep neural networks on diffeomorphism groups for optimal shape reparametrization","authors":"Elena Celledoni, Helge Glöckner, Jørgen N. Riseth, Alexander Schmeding","doi":"10.1007/s10543-023-00989-5","DOIUrl":"https://doi.org/10.1007/s10543-023-00989-5","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135536219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-20DOI: 10.1007/s10543-023-00990-y
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang, Chun-Mei Li
{"title":"Paige’s Algorithm for solving a class of tensor least squares problem","authors":"Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang, Chun-Mei Li","doi":"10.1007/s10543-023-00990-y","DOIUrl":"https://doi.org/10.1007/s10543-023-00990-y","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-19DOI: 10.1007/s10543-023-00988-6
Zhijie Du, Huoyuan Duan
{"title":"A stabilized finite element method on nonaffine grids for time-harmonic Maxwell’s equations","authors":"Zhijie Du, Huoyuan Duan","doi":"10.1007/s10543-023-00988-6","DOIUrl":"https://doi.org/10.1007/s10543-023-00988-6","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-15DOI: 10.1007/s10543-023-00992-w
Viktor Linders, Hendrik Ranocha, Philipp Birken
Abstract We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.
{"title":"Resolving entropy growth from iterative methods","authors":"Viktor Linders, Hendrik Ranocha, Philipp Birken","doi":"10.1007/s10543-023-00992-w","DOIUrl":"https://doi.org/10.1007/s10543-023-00992-w","url":null,"abstract":"Abstract We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135436731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-09DOI: 10.1007/s10543-023-00977-9
Arne Rempke
{"title":"Parallel line identification for line-implicit-solvers","authors":"Arne Rempke","doi":"10.1007/s10543-023-00977-9","DOIUrl":"https://doi.org/10.1007/s10543-023-00977-9","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44485019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-03DOI: 10.1007/s10543-023-00982-y
Hong-lin Liao, T. Tang, Tao Zhou
{"title":"Stability and convergence of the variable-step time filtered backward Euler scheme for parabolic equations","authors":"Hong-lin Liao, T. Tang, Tao Zhou","doi":"10.1007/s10543-023-00982-y","DOIUrl":"https://doi.org/10.1007/s10543-023-00982-y","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43668320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.1007/s10543-023-00980-0
C. Zheng
{"title":"Multilevel Monte Carlo using approximate distributions of the CIR process","authors":"C. Zheng","doi":"10.1007/s10543-023-00980-0","DOIUrl":"https://doi.org/10.1007/s10543-023-00980-0","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"63 1","pages":"1-39"},"PeriodicalIF":1.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48347427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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