Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Rüde
We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.
{"title":"Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy","authors":"Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Rüde","doi":"10.1002/nla.2484","DOIUrl":"https://doi.org/10.1002/nla.2484","url":null,"abstract":"We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"150 6","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138502932","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}
A class of negative matrices including Vandermonde‐like matrices tends to be extremely ill‐conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with O(n2)$$ Oleft({n}^2right) $$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction‐free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.
{"title":"A fast and accurate algorithm for solving linear systems associated with a class of negative matrix","authors":"Zhao Yang","doi":"10.1002/nla.2483","DOIUrl":"https://doi.org/10.1002/nla.2483","url":null,"abstract":"A class of negative matrices including Vandermonde‐like matrices tends to be extremely ill‐conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with O(n2)$$ Oleft({n}^2right) $$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction‐free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783694","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}
In this paper, based on a new class of conjugate gradient methods which are proposed by Rivaie, Dai and Omer et al. we propose a class of improved conjugate gradient methods for nonconvex unconstrained optimization. Different from the above methods, our methods possess the following properties: (i) the search direction always satisfies the sufficient descent condition independent of any line search; (ii) these approaches are globally convergent with the standard Wolfe line search or standard Armijo line search without any convexity assumption. Moreover, our numerical results also demonstrated the efficiencies of the proposed methods.
{"title":"A class of improved conjugate gradient methods for nonconvex unconstrained optimization","authors":"Qingjie Hu, Hongrun Zhang, Zhijuan Zhou, Yu Chen","doi":"10.1002/nla.2482","DOIUrl":"https://doi.org/10.1002/nla.2482","url":null,"abstract":"In this paper, based on a new class of conjugate gradient methods which are proposed by Rivaie, Dai and Omer et al. we propose a class of improved conjugate gradient methods for nonconvex unconstrained optimization. Different from the above methods, our methods possess the following properties: (i) the search direction always satisfies the sufficient descent condition independent of any line search; (ii) these approaches are globally convergent with the standard Wolfe line search or standard Armijo line search without any convexity assumption. Moreover, our numerical results also demonstrated the efficiencies of the proposed methods.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42363692","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}
{"title":"Issue Information","authors":"","doi":"10.1002/nla.2449","DOIUrl":"https://doi.org/10.1002/nla.2449","url":null,"abstract":"","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46794803","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}
In this article, a parameterized extended shift‐splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the PESS$$ PESS $$ iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the PESS$$ PESS $$ method for solving nonsymmetric saddle‐point problems.
{"title":"A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems","authors":"Seryas Vakili, G. Ebadi, C. Vuik","doi":"10.1002/nla.2478","DOIUrl":"https://doi.org/10.1002/nla.2478","url":null,"abstract":"In this article, a parameterized extended shift‐splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the PESS$$ PESS $$ iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the PESS$$ PESS $$ method for solving nonsymmetric saddle‐point problems.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47934281","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}
There are many studies on the well‐known modulus‐based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second‐order cones (SOCLCPs). It also works on single second‐order cone and the non‐negative orthant. On this basis, we further propose an iteration‐independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state‐of‐the‐art solvers SCS and SuperSCS that solve the equivalent SOC programming.
{"title":"On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones","authors":"Zhizhi Li, Huai Zhang","doi":"10.1002/nla.2480","DOIUrl":"https://doi.org/10.1002/nla.2480","url":null,"abstract":"There are many studies on the well‐known modulus‐based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second‐order cones (SOCLCPs). It also works on single second‐order cone and the non‐negative orthant. On this basis, we further propose an iteration‐independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state‐of‐the‐art solvers SCS and SuperSCS that solve the equivalent SOC programming.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45285312","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}
This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$ ‐symmetric, T$$ T $$ ‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$ ‐even, T$$ T $$ ‐odd, H$$ H $$ ‐even, H$$ H $$ ‐odd, T$$ T $$ ‐palindromic, T$$ T $$ ‐anti‐palindromic, H$$ H $$ ‐palindromic, and H$$ H $$ ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.
本文研究了矩阵多项式指定特征对的非结构化和结构化后向误差分析。我们讨论的结构包括T $$ T $$对称、T $$ T $$偏对称、hermite、skew hermite、T $$ T $$偶、T $$ T $$奇、H $$ H $$偶、H $$ H $$奇、T $$ T $$回文、T $$ T $$反回文、H $$ H $$回文和H $$ H $$反回文矩阵多项式。基于Frobenius范数构造了最小结构摄动，使得指定的特征对成为适当摄动的矩阵多项式的精确特征对，并且保持了稀疏性。此外，我们已经使用我们的结果来解决实际应用中出现的各种二次型反特征值问题。
{"title":"Backward error analysis of specified eigenpairs for sparse matrix polynomials","authors":"Sk. Safique Ahmad, Prince Kanhya","doi":"10.1002/nla.2476","DOIUrl":"https://doi.org/10.1002/nla.2476","url":null,"abstract":"This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$ ‐symmetric, T$$ T $$ ‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$ ‐even, T$$ T $$ ‐odd, H$$ H $$ ‐even, H$$ H $$ ‐odd, T$$ T $$ ‐palindromic, T$$ T $$ ‐anti‐palindromic, H$$ H $$ ‐palindromic, and H$$ H $$ ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43936506","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}
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {mathbb{R}}^d $$ , d=2,3$$ d=2,3 $$ . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ Ltimes L $$ , or L×L×L$$ Ltimes Ltimes L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ ntimes ntimes n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ ntimes ntimes n $$ . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$ , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.
{"title":"Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners","authors":"B. Khoromskij, V. Khoromskaia","doi":"10.1002/nla.2477","DOIUrl":"https://doi.org/10.1002/nla.2477","url":null,"abstract":"In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {mathbb{R}}^d $$ , d=2,3$$ d=2,3 $$ . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ Ltimes L $$ , or L×L×L$$ Ltimes Ltimes L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ ntimes ntimes n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ ntimes ntimes n $$ . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$ , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46110171","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}
We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift.
{"title":"QZ algorithm with two‐sided generalized Rayleigh quotient shifts","authors":"X. Chen, Hongguo Xu","doi":"10.1002/nla.2475","DOIUrl":"https://doi.org/10.1002/nla.2475","url":null,"abstract":"We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"1 2","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41249881","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}
This article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive‐definite matrix pencil (A,M)$$ left(A,Mright) $$ . The proposed approach partitions the graph associated with (A,M)$$ left(A,Mright) $$ into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.
{"title":"Enhanced algebraic substructuring for symmetric generalized eigenvalue problems","authors":"V. Kalantzis, L. Horesh","doi":"10.1002/nla.2473","DOIUrl":"https://doi.org/10.1002/nla.2473","url":null,"abstract":"This article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive‐definite matrix pencil (A,M)$$ left(A,Mright) $$ . The proposed approach partitions the graph associated with (A,M)$$ left(A,Mright) $$ into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43396741","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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