SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2048-2071, October 2025. Abstract. Convergence and compactness properties of approximate solutions to elliptic partial differential equations computed with the hybridized discontinuous Galerkin (HDG) scheme of Cockburn, Gopalakrishnan, and Sayas (Math. Comp., 79 (2010), pp. 1351–1367) are established. While it is known that solutions computed using this scheme converge at optimal rates to smooth solutions, this does not establish the stability of the method or convergence to solutions with minimal regularity. The compactness and convergence results show that the HDG scheme can be utilized for the solution of nonlinear problems and linear problems with nonsmooth coefficients on domains with reentrant corners.
SIAM数值分析杂志，第63卷，第5期，2048-2071页，2025年10月。摘要。用Cockburn， Gopalakrishnan和Sayas的杂交不连续Galerkin （HDG）格式计算椭圆型偏微分方程近似解的收敛性和紧性。Comp., 79 (2010), pp. 1351-1367)建立。虽然已知使用该格式计算的解以最优速率收敛到光滑解，但这并不能确定该方法的稳定性或收敛到具有最小规则性的解。紧凑性和收敛性结果表明，HDG格式可用于求解具有可重入角域上的非线性和非光滑系数线性问题。
{"title":"Stability and Convergence of HDG Schemes under Minimal Regularity","authors":"Jiannan Jiang, Noel J. Walkington, Yukun Yue","doi":"10.1137/23m1612846","DOIUrl":"https://doi.org/10.1137/23m1612846","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2048-2071, October 2025. <br/> Abstract. Convergence and compactness properties of approximate solutions to elliptic partial differential equations computed with the hybridized discontinuous Galerkin (HDG) scheme of Cockburn, Gopalakrishnan, and Sayas (Math. Comp., 79 (2010), pp. 1351–1367) are established. While it is known that solutions computed using this scheme converge at optimal rates to smooth solutions, this does not establish the stability of the method or convergence to solutions with minimal regularity. The compactness and convergence results show that the HDG scheme can be utilized for the solution of nonlinear problems and linear problems with nonsmooth coefficients on domains with reentrant corners.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145141191","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2026-2047, October 2025. Abstract. We quantify the accuracy of the approximate shape gradient for a shape optimization problem constrained by parabolic PDEs. The focus is on the volume form of the shape gradient, which is discretized using the finite element method and the implicit Euler scheme. Our estimate goes beyond previous work done in the elliptic setting and considers the error introduced by polygonal approximation of curved domains. Numerical experiments support the theoretical findings, and the code is made publicly available.
{"title":"Approximating Volumetric Shape Gradients for Shape Optimization with Curved Boundaries Constrained by Parabolic PDEs","authors":"Leonardo Mutti, Michael Ulbrich","doi":"10.1137/24m1681938","DOIUrl":"https://doi.org/10.1137/24m1681938","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2026-2047, October 2025. <br/> Abstract. We quantify the accuracy of the approximate shape gradient for a shape optimization problem constrained by parabolic PDEs. The focus is on the volume form of the shape gradient, which is discretized using the finite element method and the implicit Euler scheme. Our estimate goes beyond previous work done in the elliptic setting and considers the error introduced by polygonal approximation of curved domains. Numerical experiments support the theoretical findings, and the code is made publicly available.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"156 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145127788","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2009-2025, October 2025. Abstract. The formulation of finite difference approximations is a classical problem in numerical analysis. In this article, we consider difference approximations that are based on a series expansion in powers of the second undivided difference. Each additional term in the series increases the order of accuracy by two. These expansions are useful in a variety of contexts such as in the development of modified equation schemes, the design of high-order accurate energy stable discretizations, and error analysis of certain finite element or finite difference schemes. Here, we provide closed form expressions for the coefficients in the series expansions for derivatives of all orders. We also provide some short recursions defining the series coefficients, and formulae for the stencil coefficients in standard difference approximations. The series expansions are used to show some useful properties of the Fourier symbols of difference approximations and to derive rules of thumb for the number of points-per-wavelength needed to achieve a given error tolerance when solving wave propagation problems involving higher spatial derivatives.
{"title":"On the Coefficients in Finite Difference Series Expansions of Derivatives","authors":"J. W. Banks, W. D. Henshaw","doi":"10.1137/25m1731782","DOIUrl":"https://doi.org/10.1137/25m1731782","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2009-2025, October 2025. <br/> Abstract. The formulation of finite difference approximations is a classical problem in numerical analysis. In this article, we consider difference approximations that are based on a series expansion in powers of the second undivided difference. Each additional term in the series increases the order of accuracy by two. These expansions are useful in a variety of contexts such as in the development of modified equation schemes, the design of high-order accurate energy stable discretizations, and error analysis of certain finite element or finite difference schemes. Here, we provide closed form expressions for the coefficients in the series expansions for derivatives of all orders. We also provide some short recursions defining the series coefficients, and formulae for the stencil coefficients in standard difference approximations. The series expansions are used to show some useful properties of the Fourier symbols of difference approximations and to derive rules of thumb for the number of points-per-wavelength needed to achieve a given error tolerance when solving wave propagation problems involving higher spatial derivatives.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"79 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145083733","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1986-2008, October 2025. Abstract. We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global [math]-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilized finite element method.
{"title":"Computational Unique Continuation with Finite Dimensional Neumann Trace","authors":"Erik Burman, Lauri Oksanen, Ziyao Zhao","doi":"10.1137/24m164080x","DOIUrl":"https://doi.org/10.1137/24m164080x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1986-2008, October 2025. <br/> Abstract. We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global [math]-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilized finite element method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145067796","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1962-1985, October 2025. Abstract. In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multilevel block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the polynomial accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.
{"title":"Projection Method for Quasiperiodic Elliptic Equations and Application to Quasiperiodic Homogenization","authors":"Kai Jiang, Meng Li, Juan Zhang, Lei Zhang","doi":"10.1137/24m1697797","DOIUrl":"https://doi.org/10.1137/24m1697797","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1962-1985, October 2025. <br/> Abstract. In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multilevel block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the polynomial accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145089599","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1909-1932, October 2025. Abstract. We develop couplings of a recent space-time first-order system least-squares method for parabolic problems and space-time boundary element methods for the heat equation to numerically solve a parabolic transmission problem on the full space and a finite time interval. In particular, we demonstrate coercivity of the couplings under certain restrictions and validate our theoretical findings by numerical experiments.
{"title":"Space-Time FEM-BEM Couplings for Parabolic Transmission Problems","authors":"Thomas Führer, Gregor Gantner, Michael Karkulik","doi":"10.1137/24m1695646","DOIUrl":"https://doi.org/10.1137/24m1695646","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 1909-1932, October 2025. <br/> Abstract. We develop couplings of a recent space-time first-order system least-squares method for parabolic problems and space-time boundary element methods for the heat equation to numerically solve a parabolic transmission problem on the full space and a finite time interval. In particular, we demonstrate coercivity of the couplings under certain restrictions and validate our theoretical findings by numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003458","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1886-1908, August 2025. Abstract. The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure calculations. In this paper, based on the investigation of a quasi-orthogonality, we present the numerical analysis of the parallel orbital-updating approach for linear eigenvalue problems, including convergence and error estimates of the numerical approximations.
{"title":"Numerical Analysis of the Parallel Orbital-Updating Approach for Eigenvalue Problems","authors":"Xiaoying Dai, Yan Li, Bin Yang, Aihui Zhou","doi":"10.1137/24m1690084","DOIUrl":"https://doi.org/10.1137/24m1690084","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1886-1908, August 2025. <br/> Abstract. The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure calculations. In this paper, based on the investigation of a quasi-orthogonality, we present the numerical analysis of the parallel orbital-updating approach for linear eigenvalue problems, including convergence and error estimates of the numerical approximations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900116","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1808-1832, August 2025. Abstract. The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge–Kutta methods, including the additive implicit-explicit Runge–Kutta, explicit exponential Runge–Kutta, and corrected integrating factor Runge–Kutta methods, for the Swift–Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge–Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge–Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.
{"title":"A Unified Framework on the Original Energy Laws of Three Effective Classes of Runge–Kutta Methods for Phase Field Crystal Type Models","authors":"Xuping Wang, Xuan Zhao, Hong-lin Liao","doi":"10.1137/24m1701770","DOIUrl":"https://doi.org/10.1137/24m1701770","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1808-1832, August 2025. <br/> Abstract. The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge–Kutta methods, including the additive implicit-explicit Runge–Kutta, explicit exponential Runge–Kutta, and corrected integrating factor Runge–Kutta methods, for the Swift–Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge–Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge–Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900120","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1729-1756, August 2025. Abstract. We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the time step size ([math]-method), we achieve accuracy by increasing the order of the method ([math]-method). We base this method on discontinuous Galerkin time stepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observe a (root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the finding.
{"title":"A P-Version of Convolution Quadrature in Wave Propagation","authors":"Alexander Rieder","doi":"10.1137/24m1642524","DOIUrl":"https://doi.org/10.1137/24m1642524","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1729-1756, August 2025. <br/> Abstract. We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the time step size ([math]-method), we achieve accuracy by increasing the order of the method ([math]-method). We base this method on discontinuous Galerkin time stepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observe a (root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the finding.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"105 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840295","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}
Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1757-1775, August 2025. Abstract. We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD and then prove its stable and nonlocal convergence from stationary points that may not be close to the target saddle point, which reduces the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD for implementation. Furthermore, based on Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Ideally, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.
{"title":"Improved High-Index Saddle Dynamics for Finding Saddle Points and Solution Landscape","authors":"Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng","doi":"10.1137/25m173212x","DOIUrl":"https://doi.org/10.1137/25m173212x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1757-1775, August 2025. <br/> Abstract. We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD and then prove its stable and nonlocal convergence from stationary points that may not be close to the target saddle point, which reduces the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD for implementation. Furthermore, based on Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Ideally, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840296","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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