M. Yueh, H. Huang, Tie-xiang Li, Wen-Wei Lin, S. Yau
Surface parameterizations have been widely applied in the computer-aided design for the geometric processing tasks of surface registration, remeshing, texture mapping, and so on. In this paper, we present an efficient balanced energy minimization (BEM) algorithm for the computation of simply connected open surface parameterizations with balanced angle and area distortions. The existence of a nontrivial accumulation function of the BEM algorithm is guaranteed under some mild conditions and the limiting function is shown to be one-to-one. Comparisons of the BEM algorithm with the angleand the area-preserving parameterizations show that the angular distortion is close to that of the angle-preserving parameterization while the area distortion is significantly improved. An application of the BEM on the Chinese virtual broadcasting technique is demonstrated thereafter, which is consisted of surface remeshing, registration, and morphing.
{"title":"Optimized surface parameterizations with applications to Chinese virtual broadcasting","authors":"M. Yueh, H. Huang, Tie-xiang Li, Wen-Wei Lin, S. Yau","doi":"10.1553/etna_vol53s383","DOIUrl":"https://doi.org/10.1553/etna_vol53s383","url":null,"abstract":"Surface parameterizations have been widely applied in the computer-aided design for the geometric processing tasks of surface registration, remeshing, texture mapping, and so on. In this paper, we present an efficient balanced energy minimization (BEM) algorithm for the computation of simply connected open surface parameterizations with balanced angle and area distortions. The existence of a nontrivial accumulation function of the BEM algorithm is guaranteed under some mild conditions and the limiting function is shown to be one-to-one. Comparisons of the BEM algorithm with the angleand the area-preserving parameterizations show that the angular distortion is close to that of the angle-preserving parameterization while the area distortion is significantly improved. An application of the BEM on the Chinese virtual broadcasting technique is demonstrated thereafter, which is consisted of surface remeshing, registration, and morphing.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67352258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider isoparametric finite element discretizations of semilinear acoustic wave equations with kinetic boundary conditions and derive a corresponding error bound as our main result. The difficulty is that such problems are stated on domains with curved boundaries and this renders the discretizations nonconforming. Our approach is to provide a unified error analysis for nonconforming space discretizations for semilinear wave equations. In particular, we introduce a general, abstract framework for nonconforming space discretizations in which we derive a-priori error bounds in terms of interpolation, data and conformity errors. The theory applies to a large class of problems and discretizations that fit into the abstract framework. The error bound for wave equations with kinetic boundary conditions is obtained from the general theory by inserting known interpolation and geometric error bounds into the abstract error result of the unified error analysis.
{"title":"Finite element discretization of semilinear acoustic wave equations with kinetic boundary conditions","authors":"M. Hochbruck, Jan Leibold","doi":"10.1553/ETNA_VOL53S522","DOIUrl":"https://doi.org/10.1553/ETNA_VOL53S522","url":null,"abstract":"We consider isoparametric finite element discretizations of semilinear acoustic wave equations with kinetic boundary conditions and derive a corresponding error bound as our main result. The difficulty is that such problems are stated on domains with curved boundaries and this renders the discretizations nonconforming. Our approach is to provide a unified error analysis for nonconforming space discretizations for semilinear wave equations. In particular, we introduce a general, abstract framework for nonconforming space discretizations in which we derive a-priori error bounds in terms of interpolation, data and conformity errors. The theory applies to a large class of problems and discretizations that fit into the abstract framework. The error bound for wave equations with kinetic boundary conditions is obtained from the general theory by inserting known interpolation and geometric error bounds into the abstract error result of the unified error analysis.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67352310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose and analyse a posteriori estimates for global-in-time, nonoverlapping domain decomposition methods for heterogeneous and anisotropic porous media diffusion problems. We consider mixed formulations, with a lowest-order Raviart–Thomas–Nedelec discretization, often used for such problems. Optimized Robin transmission conditions are employed on the space-time interface between subdomains, and different time grids are used to adapt to different time scales in the subdomains. Our estimators allow to distinguish the spatial discretization, the temporal discretization, and the domain decomposition error components. We design an adaptive space-time domain decomposition algorithm, wherein the iterations are stopped when the domain decomposition error does not affect significantly the global error. Thus, a guaranteed bound on the overall error is obtained on each iteration of the space-time domain decomposition algorithm, and simultaneously important savings in terms of the number of domain decomposition iterations can be achieved. Numerical results for two-dimensional problems with strong heterogeneities and local time stepping are presented to illustrate the performance of our adaptive domain decomposition algorithm.
{"title":"A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations","authors":"S. Hassan, C. Japhet, M. Vohralík","doi":"10.1553/ETNA_VOL49S151","DOIUrl":"https://doi.org/10.1553/ETNA_VOL49S151","url":null,"abstract":"We propose and analyse a posteriori estimates for global-in-time, nonoverlapping domain decomposition methods for heterogeneous and anisotropic porous media diffusion problems. We consider mixed formulations, with a lowest-order Raviart–Thomas–Nedelec discretization, often used for such problems. Optimized Robin transmission conditions are employed on the space-time interface between subdomains, and different time grids are used to adapt to different time scales in the subdomains. Our estimators allow to distinguish the spatial discretization, the temporal discretization, and the domain decomposition error components. We design an adaptive space-time domain decomposition algorithm, wherein the iterations are stopped when the domain decomposition error does not affect significantly the global error. Thus, a guaranteed bound on the overall error is obtained on each iteration of the space-time domain decomposition algorithm, and simultaneously important savings in terms of the number of domain decomposition iterations can be achieved. Numerical results for two-dimensional problems with strong heterogeneities and local time stepping are presented to illustrate the performance of our adaptive domain decomposition algorithm.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42301645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simon Baumstark, Georgia Kokkala, Katharina Schratz
In this paper we propose asymptotic consistent exponential-type integrators for the Klein-Gordon-Schrodinger system. This novel class of integrators allows us to solve the system from slowly varying relativistic up to challenging highly oscillatory non-relativistic regimes without any step size restriction. In particular, our first- and second-order exponential-type integrators are asymptotically consistent in the sense of asymptotically converging to the corresponding decoupled free Schrodinger limit system.
{"title":"Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes","authors":"Simon Baumstark, Georgia Kokkala, Katharina Schratz","doi":"10.1553/ETNA_VOL48S63","DOIUrl":"https://doi.org/10.1553/ETNA_VOL48S63","url":null,"abstract":"In this paper we propose asymptotic consistent exponential-type integrators for the Klein-Gordon-Schrodinger system. This novel class of integrators allows us to solve the system from slowly varying relativistic up to challenging highly oscillatory non-relativistic regimes without any step size restriction. In particular, our first- and second-order exponential-type integrators are asymptotically consistent in the sense of asymptotically converging to the corresponding decoupled free Schrodinger limit system.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2018-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44363121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Krylov subspace methods are commonly used iterative methods for solving large sparse linear systems, however they suffer from communication bottlenecks on parallel computers. Therefore, $s$-step methods have been developed where the Krylov subspace is built block by block, so that $s$ matrix-vector multiplications can be done before orthonormalizing the block. Then Communication-Avoiding algorithms can be used for both kernels. This paper introduces a new variation on $s$-step GMRES in order to reduce the number of iterations necessary to ensure convergence, with a small overhead in the number of communications. Namely, we develop a $s$-step GMRES algorithm, where the block size is variable and increases gradually. Our numerical experiments show a good agreement with our analysis of condition numbers and demonstrate the efficiency of our variable $s$-step approach.
{"title":"Varying the s in Your s-step GMRES","authors":"David Imberti, J. Erhel","doi":"10.1553/ETNA_VOL47S206","DOIUrl":"https://doi.org/10.1553/ETNA_VOL47S206","url":null,"abstract":"Krylov subspace methods are commonly used iterative methods for solving large sparse linear systems, however they suffer from communication bottlenecks on parallel computers. Therefore, $s$-step methods have been developed where the Krylov subspace is built block by block, so that $s$ matrix-vector multiplications can be done before orthonormalizing the block. Then Communication-Avoiding algorithms can be used for both kernels. This paper introduces a new variation on $s$-step GMRES in order to reduce the number of iterations necessary to ensure convergence, with a small overhead in the number of communications. Namely, we develop a $s$-step GMRES algorithm, where the block size is variable and increases gradually. Our numerical experiments show a good agreement with our analysis of condition numbers and demonstrate the efficiency of our variable $s$-step approach.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67352006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the recently introduced Tikhonov regularization filters with penalty terms having seminorms that depend on the operator itself. Exploiting the singular value decomposition of the operator, we provide optimal order conditions, smoothing properties, and a general condition (with a minor condition of the seminorm) for the saturation level. Moreover, we introduce and analyze both stationary and nonstationary iterative counterparts of the generalized Tikhonov method with operator-dependent seminorms. We establish their convergence rate under conditions affecting only the iteration parameters, proving that they overcome the saturation result. Finally, some selected numerical results confirm the effectiveness of the proposed regularization filters.
{"title":"On generalized iterated Tikhonov regularization with operator-dependent seminorms","authors":"D. Bianchi, M. Donatelli","doi":"10.1553/ETNA_VOL47S73","DOIUrl":"https://doi.org/10.1553/ETNA_VOL47S73","url":null,"abstract":"We investigate the recently introduced Tikhonov regularization filters with penalty terms having seminorms that depend on the operator itself. Exploiting the singular value decomposition of the operator, we provide optimal order conditions, smoothing properties, and a general condition (with a minor condition of the seminorm) for the saturation level. Moreover, we introduce and analyze both stationary and nonstationary iterative counterparts of the generalized Tikhonov method with operator-dependent seminorms. We establish their convergence rate under conditions affecting only the iteration parameters, proving that they overcome the saturation result. Finally, some selected numerical results confirm the effectiveness of the proposed regularization filters.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67352120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz or in the variable coefficient case Toeplitz-like structure . We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniqes can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.
{"title":"Low-Rank Solvers for Fractional Differential Equations","authors":"T. Breiten, V. Simoncini, M. Stoll","doi":"10.17617/2.2270973","DOIUrl":"https://doi.org/10.17617/2.2270973","url":null,"abstract":"Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz or in the variable coefficient case Toeplitz-like structure . We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniqes can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67654664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.15496/PUBLIKATION-27275
Bernd Brumm, Emil Kieri
We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.
{"title":"A MATRIX-FREE LEGENDRE SPECTRAL METHOD FOR INITIAL-BOUNDARY VALUE PROBLEMS","authors":"Bernd Brumm, Emil Kieri","doi":"10.15496/PUBLIKATION-27275","DOIUrl":"https://doi.org/10.15496/PUBLIKATION-27275","url":null,"abstract":"We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67154485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Low-rank versions of the alternating direction implicit (ADI) iteration are popular and well estab- lished methods for the numerical solution of large-scale Sylvester and Lyapunov equations. Probably the biggest disadvantage of these methods is their dependence on a set of shift parameters that are crucial for fast convergence. Here we firstly review existing shift generation strategies that compute a number of shifts before the actual itera- tion. These approaches come with several disadvantages such as, e.g., expensive numerical computations and the difficulty to obtain necessary spectral information or data n eeded to initially setup their generation. Secondly, we propose two novel shift selection strategies with the motivation to resolve these issues at least partially. Both ap- proaches generate shifts automatically in the course of the ADI iterations. Extensive numerical tests show that one of these new approaches, based on a Galerkin projection onto the space spanned by the current ADI data, is superior to other approaches in the majority of cases both in terms of convergence speed and required execution time.
{"title":"Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations","authors":"P. Benner, P. Kürschner, J. Saak","doi":"10.17617/2.2071065","DOIUrl":"https://doi.org/10.17617/2.2071065","url":null,"abstract":"Low-rank versions of the alternating direction implicit (ADI) iteration are popular and well estab- lished methods for the numerical solution of large-scale Sylvester and Lyapunov equations. Probably the biggest disadvantage of these methods is their dependence on a set of shift parameters that are crucial for fast convergence. Here we firstly review existing shift generation strategies that compute a number of shifts before the actual itera- tion. These approaches come with several disadvantages such as, e.g., expensive numerical computations and the difficulty to obtain necessary spectral information or data n eeded to initially setup their generation. Secondly, we propose two novel shift selection strategies with the motivation to resolve these issues at least partially. Both ap- proaches generate shifts automatically in the course of the ADI iterations. Extensive numerical tests show that one of these new approaches, based on a Galerkin projection onto the space spanned by the current ADI data, is superior to other approaches in the majority of cases both in terms of convergence speed and required execution time.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67654632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss low-rank approximation methods for large-scale symmetric Sylvester equations. Follow- ing similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank nr, we derive necessary conditions for an approximation being optimal with respect to this norm. We show that the norm minimization problem is related to an objective function based on the H2-inner product for sym- metric state space systems. This objective function leads to first-order optimality conditions that are equivalent to the ones for the norm minimization problem. We further propose an iterative procedure and demonstrate its efficiency by means of some numerical examples.
{"title":"Rational Interpolation Methods for Symmetric Sylvester Equations","authors":"P. Benner, T. Breiten","doi":"10.17617/2.2060858","DOIUrl":"https://doi.org/10.17617/2.2060858","url":null,"abstract":"We discuss low-rank approximation methods for large-scale symmetric Sylvester equations. Follow- ing similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank nr, we derive necessary conditions for an approximation being optimal with respect to this norm. We show that the norm minimization problem is related to an objective function based on the H2-inner product for sym- metric state space systems. This objective function leads to first-order optimality conditions that are equivalent to the ones for the norm minimization problem. We further propose an iterative procedure and demonstrate its efficiency by means of some numerical examples.","PeriodicalId":50536,"journal":{"name":"Electronic Transactions on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67655019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：