L. Batteux, T. Gallouët, R. Herbin, J. Latché, P. Poullet
{"title":"Convergence of the MAC scheme for the incompressible Navier-Stokes equations with variable density and viscosity","authors":"L. Batteux, T. Gallouët, R. Herbin, J. Latché, P. Poullet","doi":"10.1090/mcom/3803","DOIUrl":"https://doi.org/10.1090/mcom/3803","url":null,"abstract":"The present paper addresses the convergence of the implicit Marker-and-Cell scheme for time-dependent Navier–Stokes equations with variable density and density-dependent viscosity and forcing term. A priori estimates on the unknowns are obtained, and thanks to a topological degree argument, they lead to the existence of an approximate solution at each time step. Then, by compactness arguments relying on these same estimates, we obtain the convergence (up to the extraction of a subsequence), when the space and time steps tend to zero, of the numerical solutions to a limit; this latter is shown to be a weak solution to the continuous problem by passing to the limit in the scheme.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135384033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Near-optimal approximation methods for elliptic PDEs with lognormal coefficients","authors":"Albert Cohen, Giovanni Migliorati","doi":"10.1090/mcom/3825","DOIUrl":"https://doi.org/10.1090/mcom/3825","url":null,"abstract":"This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus d i v left-parenthesis a nabla u right-parenthesis equals f\"> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>div</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>a</mml:mi> <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">-operatorname {div}(anabla u)=f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a equals exp left-parenthesis b right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mi>exp</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">a=exp (b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b\"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=\"application/x-tex\">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Gaussian random field. The approximant of the solution <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u\"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding=\"application/x-tex\">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-term polynomial expansion in the scalar Gaussian random variables that parametrize <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b\"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=\"application/x-tex\">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the rate that has been established in Bachmayr, Cohen, DeVore, and Migliorati [ESAIM Math. Model. Numer. Anal. 51 (2017), pp. 341–363] for best <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</m","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136156712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quadratic points on bielliptic modular curves","authors":"Filip Najman, Borna Vukorepa","doi":"10.1090/mcom/3805","DOIUrl":"https://doi.org/10.1090/mcom/3805","url":null,"abstract":"Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 less-than-or-equal-to g left-parenthesis upper X 0 left-parenthesis n right-parenthesis right-parenthesis less-than-or-equal-to 5\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2leq g(X_0(n)) leq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Since all the hyperelliptic curves <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 0 left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">X_0(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"less-than-or-equal-to 5\"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">leq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and as a curve can have infinitely many quadratic points only if it is either of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"less-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 0 left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">X_0(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the biellip","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136131622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The interior penalty virtual element method for the biharmonic problem","authors":"Jikun Zhao, Shipeng Mao, Bei Zhang, Fei Wang","doi":"10.1090/mcom/3828","DOIUrl":"https://doi.org/10.1090/mcom/3828","url":null,"abstract":"In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H squared\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-conforming virtual element, an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-nonconforming virtual element is obtained with the same degrees of freedom as the usual <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-conforming virtual element, such that it locally has <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H squared\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regularity on each polygon in meshes. To enforce the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136176503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interpolation operator on negative Sobolev spaces","authors":"Lars Diening, Johannes Storn, Tabea Tscherpel","doi":"10.1090/mcom/3824","DOIUrl":"https://doi.org/10.1090/mcom/3824","url":null,"abstract":"We introduce a Scott–Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136181813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation","authors":"D. Gallistl, Ngoc Tien Tran","doi":"10.1090/mcom/3794","DOIUrl":"https://doi.org/10.1090/mcom/3794","url":null,"abstract":"<p>This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and is accessible to the discretization with finite elements. This work establishes uniform convergence of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to the convex Alexandrov solution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u\">\u0000 <mml:semantics>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">u</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to the Monge–Ampère equation as the regularization parameter <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\u0000 <mml:semantics>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> approaches <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A mixed finite element method for the approximation of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inlin","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48415841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mixed methods and lower eigenvalue bounds","authors":"Dietmar Gallistl","doi":"10.1090/mcom/3820","DOIUrl":"https://doi.org/10.1090/mcom/3820","url":null,"abstract":"It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility condition (inclusion of kernels) of the mixed scheme and on local constants related to compact embeddings, which are often known explicitly. Applications include scalar second-order elliptic operators, linear elasticity, and the Steklov eigenvalue problem.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135157867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Construction of polynomial preserving cochain extensions by blending","authors":"Richard Falk, Ragnar Winther","doi":"10.1090/mcom/3819","DOIUrl":"https://doi.org/10.1090/mcom/3819","url":null,"abstract":"A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-forms defined on the boundary of the simplex to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135311193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Gearhart-Koshy acceleration for the Kaczmarz method","authors":"Janosch Rieger","doi":"10.1090/mcom/3818","DOIUrl":"https://doi.org/10.1090/mcom/3818","url":null,"abstract":"The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step. The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently. When only a single Kaczmarz cycle is considered, the proposed affine search is more effective than the Gearhart-Koshy/Tam line-search, which in turn is more effective than the underlying Kaczmarz method. A numerical experiment from the context of computerized tomography suggests that the proposed affine search has the potential to outperform the the Gearhart-Koshy/Tam line-search and the underlying Kaczmarz method in terms of the computational cost that is needed to achieve a given error tolerance.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135311196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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