SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2421-2453, December 2025. Abstract. We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments, and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine series (COS) method. The classical COS method is numerically very efficient in one dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a COS. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.
{"title":"From Characteristic Functions to Multivariate Distribution Functions and European Option Prices by the (Damped) COS Method","authors":"Gero Junike, Hauke Stier","doi":"10.1137/24m1666240","DOIUrl":"https://doi.org/10.1137/24m1666240","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2421-2453, December 2025. <br/> Abstract. We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments, and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine series (COS) method. The classical COS method is numerically very efficient in one dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a COS. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"157 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771639","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 6, Page 2483-2511, December 2025. Abstract. The supremizer method enriches the reduced velocity basis for pressure recovery in incompressible flows, ensuring the inf-sup condition in the reduced space. In the full-order model, a small penalty term is often introduced to prevent spurious modes [Y. He, Math. Comp., 74 (2005), pp. 1201–1216] and is also essential for accuracy in the proper orthogonal decomposition–based reduced-order model [A.-L. Gerner and K. Veroy, Math. Models Methods Appl. Sci., 21 (2011), pp. 2103–2134]. However, coupling pressure and velocity, along with the supremizer basis, significantly increases the computational costs in both offline and online phases. We find that the primary role of supremizers is to improve stability, rather than velocity accuracy. We propose a novel method using several supremizers for the velocity basis, decoupling the penalized system to solve for velocity. The full set of supremizers is then used to recover pressure. This strategy reduces the computational cost while maintaining stability and accuracy. We derive error estimates using a supremizer-augmented projection operator, which depend on the inf-sup constant rather than on the inverse of the penalty coefficient. We also develop two new supremizer construction options satisfying the inf-sup condition, one of which avoids solving the full-order equations for obtaining supremizer basis, further reducing offline costs. Numerical experiments demonstrate the effectiveness of the proposed method. For comparable accuracy, CPU time tests show that the online computational cost is reduced by about [math], and the offline assembly cost by [math], compared to [Y. He, Math. Comp., 74 (2005), pp. 1201–1216].
SIAM数值分析杂志，第63卷，第6期，2483-2511页，2025年12月。摘要。超压器方法丰富了不可压缩流动中压力恢复的降速基础，保证了降速空间内的升压条件。在全阶模型中，通常引入一个小的惩罚项来防止伪模[Y]。他数学。Comp., 74 (2005), pp. 1201-1216]并且对于适当的基于正交分解的降阶模型的准确性也是必不可少的[A.-L.]Gerner和K. verroy，数学。模型、方法、应用。科学。， 21 (2011), pp. 2103-2134]。然而，耦合压力和速度，以及超喷器的基础，大大增加了离线和在线阶段的计算成本。我们发现，上位器的主要作用是提高稳定性，而不是速度精度。我们提出了一种新的方法，使用几个速度基的最优器，解耦惩罚系统来求解速度。然后使用全套的增压器来恢复压力。该策略在保持稳定性和准确性的同时降低了计算成本。我们使用超增广投影算子推导误差估计，它依赖于中-sup常数而不是惩罚系数的倒数。我们还开发了两种满足上料条件的新型上料器结构方案，其中一种方案避免了求解上料器基的全阶方程，进一步降低了离线成本。数值实验证明了该方法的有效性。对于类似的精度，CPU时间测试表明，与[Y]相比，在线计算成本减少了大约[math]，离线组装成本减少了[math]。他数学。《比较》，74 (2005),pp. 1201-1216。
{"title":"Fast Supremizer Method on Penalty-Based Reduced-Order Modeling for Incompressible Flows","authors":"Hui Yao, Mejdi Azaiez","doi":"10.1137/25m1746112","DOIUrl":"https://doi.org/10.1137/25m1746112","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2483-2511, December 2025. <br/> Abstract. The supremizer method enriches the reduced velocity basis for pressure recovery in incompressible flows, ensuring the inf-sup condition in the reduced space. In the full-order model, a small penalty term is often introduced to prevent spurious modes [Y. He, Math. Comp., 74 (2005), pp. 1201–1216] and is also essential for accuracy in the proper orthogonal decomposition–based reduced-order model [A.-L. Gerner and K. Veroy, Math. Models Methods Appl. Sci., 21 (2011), pp. 2103–2134]. However, coupling pressure and velocity, along with the supremizer basis, significantly increases the computational costs in both offline and online phases. We find that the primary role of supremizers is to improve stability, rather than velocity accuracy. We propose a novel method using several supremizers for the velocity basis, decoupling the penalized system to solve for velocity. The full set of supremizers is then used to recover pressure. This strategy reduces the computational cost while maintaining stability and accuracy. We derive error estimates using a supremizer-augmented projection operator, which depend on the inf-sup constant rather than on the inverse of the penalty coefficient. We also develop two new supremizer construction options satisfying the inf-sup condition, one of which avoids solving the full-order equations for obtaining supremizer basis, further reducing offline costs. Numerical experiments demonstrate the effectiveness of the proposed method. For comparable accuracy, CPU time tests show that the online computational cost is reduced by about [math], and the offline assembly cost by [math], compared to [Y. He, Math. Comp., 74 (2005), pp. 1201–1216].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771690","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 6, Page 2454-2482, December 2025. Abstract. We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE, we introduce square-squeezing—a homeomorphic multilinear hypercube-simplex transformation—reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparametrized mesh through interpolation in Chebyshev–Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the [math]-order total variation of the integrand and the surface parametrization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing [math], the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing [math]-refinements to overcome the limitations of [math]-refinements for highly varying smooth integrals.
{"title":"High-Order Integration on Regular Triangulated Manifolds Reaches Superalgebraic Approximation Rates Through Cubical Reparametrizations","authors":"Gentian Zavalani, Oliver Sander, Michael Hecht","doi":"10.1137/24m1707274","DOIUrl":"https://doi.org/10.1137/24m1707274","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2454-2482, December 2025. <br/> Abstract. We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE, we introduce square-squeezing—a homeomorphic multilinear hypercube-simplex transformation—reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparametrized mesh through interpolation in Chebyshev–Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the [math]-order total variation of the integrand and the surface parametrization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing [math], the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing [math]-refinements to overcome the limitations of [math]-refinements for highly varying smooth integrals.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"111 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771636","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 6, Page 2395-2420, December 2025. Abstract. We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schrödinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schrödinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.
{"title":"A Posteriori Error Estimates for Schrödinger Operators Discretized with Linear Combinations of Atomic Orbitals","authors":"Mi-Song Dupuy, Geneviève Dusson, Ioanna-Maria Lygatsika","doi":"10.1137/24m1700697","DOIUrl":"https://doi.org/10.1137/24m1700697","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2395-2420, December 2025. <br/> Abstract. We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schrödinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schrödinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"163 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145765628","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 6, Page 2371-2394, December 2025. Abstract. Signature kernels have become a powerful tool in kernel methods for sequential data. In “The Signature Kernel is the solution of a Goursat PDE” [], the authors introduced a kernel trick showing that, for continuously differentiable paths, the signature kernel satisfies a hyperbolic PDE of Goursat type in two independent time variables. While finite difference methods have been explored for this PDE, they suffer from accuracy and stability issues when handling highly oscillatory inputs. In this work, we propose two advanced numerical schemes that approximate the solution using polynomial representations of boundary conditions and employing either approximation or interpolation techniques. We prove the convergence of the polynomial approximation scheme and demonstrate experimentally that both methods achieve several orders of magnitude improvement in mean absolute percentage error (MAPE) over finite difference schemes without increasing computational complexity. These algorithms are implemented in a publicly available Python library: https://github.com/FrancescoPiatti/polysigkernel.
SIAM数值分析杂志，第63卷，第6期，2371-2394页，2025年12月。摘要。在序列数据核方法中，签名核已经成为一种强大的工具。在“The Signature Kernel is a Goursat PDE的解”[]中，作者介绍了一个核技巧，表明对于连续可微路径，签名核满足两个独立时间变量的Goursat型双曲PDE。虽然有限差分方法已经探索了这种PDE，但它们在处理高振荡输入时存在精度和稳定性问题。在这项工作中，我们提出了两种先进的数值方案，使用边界条件的多项式表示和采用近似或插值技术来近似解。我们证明了多项式近似格式的收敛性，并通过实验证明了这两种方法在不增加计算复杂度的情况下，比有限差分格式在平均绝对百分比误差（MAPE）方面取得了几个数量级的改进。这些算法在一个公开可用的Python库中实现：https://github.com/FrancescoPiatti/polysigkernel。
{"title":"Numerical Schemes for Signature Kernels","authors":"Thomas Cass, Francesco Piatti, Jeffrey Pei","doi":"10.1137/25m1740681","DOIUrl":"https://doi.org/10.1137/25m1740681","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2371-2394, December 2025. <br/> Abstract. Signature kernels have become a powerful tool in kernel methods for sequential data. In “The Signature Kernel is the solution of a Goursat PDE” [], the authors introduced a kernel trick showing that, for continuously differentiable paths, the signature kernel satisfies a hyperbolic PDE of Goursat type in two independent time variables. While finite difference methods have been explored for this PDE, they suffer from accuracy and stability issues when handling highly oscillatory inputs. In this work, we propose two advanced numerical schemes that approximate the solution using polynomial representations of boundary conditions and employing either approximation or interpolation techniques. We prove the convergence of the polynomial approximation scheme and demonstrate experimentally that both methods achieve several orders of magnitude improvement in mean absolute percentage error (MAPE) over finite difference schemes without increasing computational complexity. These algorithms are implemented in a publicly available Python library: https://github.com/FrancescoPiatti/polysigkernel.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145718456","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 6, Page 2343-2370, December 2025. Abstract. A geometric perspective on the gentlest ascent dynamics is presented, revealing that the dynamics is utilizing the Householder reflector—constructed via the continuous power method—to adapt the negative gradient and identify index-1 saddle points. While the adaptation appears intuitive, it is governed by a precise criterion. Building on this geometric insight, three generalized dynamical systems are introduced for locating high-index saddle points, each centered on estimating directions for constructing generalized reflectors. The first approach employs the Oja flow to evolve eigenspaces, encompassing the continuous power method as a special case. The second approach formulates a matrix Riccati differential equation for the projector operator on the Grassmann manifold, which is shown to be equivalent to a double bracket flow with inherent sorting properties. The third approach is a hybrid method based on conventional subspace iteration, incorporating [math] factorization for normalization. The equilibrium points of all three systems are classified, and convergence analyses are provided. These dynamical systems are readily solvable by using high-precision numerical ODE integrators. Numerical experiments confirm the theoretical results.
{"title":"Generalized Gentlest Ascent Dynamics Methods for High-Index Saddle Points","authors":"Moody T. Chu, Matthew M. Lin","doi":"10.1137/24m1710905","DOIUrl":"https://doi.org/10.1137/24m1710905","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2343-2370, December 2025. <br/> Abstract. A geometric perspective on the gentlest ascent dynamics is presented, revealing that the dynamics is utilizing the Householder reflector—constructed via the continuous power method—to adapt the negative gradient and identify index-1 saddle points. While the adaptation appears intuitive, it is governed by a precise criterion. Building on this geometric insight, three generalized dynamical systems are introduced for locating high-index saddle points, each centered on estimating directions for constructing generalized reflectors. The first approach employs the Oja flow to evolve eigenspaces, encompassing the continuous power method as a special case. The second approach formulates a matrix Riccati differential equation for the projector operator on the Grassmann manifold, which is shown to be equivalent to a double bracket flow with inherent sorting properties. The third approach is a hybrid method based on conventional subspace iteration, incorporating [math] factorization for normalization. The equilibrium points of all three systems are classified, and convergence analyses are provided. These dynamical systems are readily solvable by using high-precision numerical ODE integrators. Numerical experiments confirm the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145559903","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}
L. Beirão da Veiga, D. A. Di Pietro, J. Droniou, K. B. Haile, T. J. Radley
SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2317-2342, December 2025. Abstract. In this paper we propose and analyze a new finite element method for the solution of the two- and three-dimensional incompressible Navier–Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting preasymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.
{"title":"A Reynolds-Semirobust Method with Hybrid Velocity and Pressure for the Unsteady Incompressible Navier–Stokes Equations","authors":"L. Beirão da Veiga, D. A. Di Pietro, J. Droniou, K. B. Haile, T. J. Radley","doi":"10.1137/25m1736104","DOIUrl":"https://doi.org/10.1137/25m1736104","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2317-2342, December 2025. <br/> Abstract. In this paper we propose and analyze a new finite element method for the solution of the two- and three-dimensional incompressible Navier–Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting preasymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145536093","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 6, Page 2296-2316, December 2025. Abstract. We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous, preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.
{"title":"Boundary-Value Problems of Functional Differential Equations with State-Dependent Delays","authors":"Alessia Andò, Jan Sieber","doi":"10.1137/24m1711182","DOIUrl":"https://doi.org/10.1137/24m1711182","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2296-2316, December 2025. <br/> Abstract. We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous, preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145485828","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 6, Page 2272-2295, December 2025. Abstract. The quad-curl problem is a critical issue in magnetohydrodynamics and inverse electromagnetic scattering theory. It has traditionally been addressed by most existing numerical schemes through the formation of saddle-point systems, thereby introducing substantial challenges for both theoretical analysis and practical numerical implementations. This study introduces a novel regularization-based approach that diverges from these conventional methods, specifically designed to avoid the saddle-point issue. The challenge of addressing the divergence-free constraint in finite element methods is tackled in a unique way. Moreover, it ensures a consistent well-posedness, leading to a symmetric, positive-definite system in finite element discretization, which simplifies the implementation process. The regularized problem is addressed using the conforming finite element method, employing [math]-conforming element, and the discontinuous Galerkin method, utilizing Nédélec’s element, both of which achieve quasi-optimal error bounds in relevant norms. The efficiency of our proposed methods is further demonstrated through a series of numerical experiments in both two and three dimensions.
{"title":"An Efficient Finite Element Method for the Quad-Curl Problem","authors":"Jingzhi Li, Shipeng Mao, Chao Wang, Zhimin Zhang","doi":"10.1137/24m166022x","DOIUrl":"https://doi.org/10.1137/24m166022x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2272-2295, December 2025. <br/> Abstract. The quad-curl problem is a critical issue in magnetohydrodynamics and inverse electromagnetic scattering theory. It has traditionally been addressed by most existing numerical schemes through the formation of saddle-point systems, thereby introducing substantial challenges for both theoretical analysis and practical numerical implementations. This study introduces a novel regularization-based approach that diverges from these conventional methods, specifically designed to avoid the saddle-point issue. The challenge of addressing the divergence-free constraint in finite element methods is tackled in a unique way. Moreover, it ensures a consistent well-posedness, leading to a symmetric, positive-definite system in finite element discretization, which simplifies the implementation process. The regularized problem is addressed using the conforming finite element method, employing [math]-conforming element, and the discontinuous Galerkin method, utilizing Nédélec’s element, both of which achieve quasi-optimal error bounds in relevant norms. The efficiency of our proposed methods is further demonstrated through a series of numerical experiments in both two and three dimensions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"114 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145462322","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 6, Page 2249-2271, December 2025. Abstract. Lawson’s iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson’s iteration for the rational minimax approximation problem with both computationally high efficiency and theoretical guarantee is challenging. The recent work [L.-H. Zhang et al., Math. Comp., 94 (2025), pp. 2457–2494] reveals that Lawson’s iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and the work proposes a new type of Lawson’s iteration, namely, d-Lawson, which reduces to the classical Lawson’s iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan’s sufficient condition, and, numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, [math] is a near-optimal Lawson exponent in Lawson’s iteration; and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small [math], and the limiting approximant satisfies the complementary slackness condition which states that any node associated with positive weight is either an interpolation point or has a constant error.
SIAM数值分析杂志，第63卷，第6期，2249-2271页，2025年12月。摘要。Lawson迭代法是求解复平面线性（多项式）极大极小逼近问题的一种经典而有效的方法。对有理极大极小逼近问题的劳森迭代法的推广具有较高的计算效率和理论保证。最近的工作[l . h .]Zhang et al.，数学。Comp., 94 (2025), pp. 2457-2494]揭示了Lawson迭代可以看作是解决原有理极大极小逼近问题对偶问题的一种方法，并提出了一种新型的Lawson迭代，即d-Lawson，将其简化为线性极大极小逼近问题的经典Lawson迭代。对于有理情况，在Ruttan的充分条件下，保证了该对偶问题得到原始的极大极小解，并且在数值上观察到d-Lawson对对偶目标函数的单调收敛。本文对线性极大极小逼近问题和有理极大极小逼近问题给出了d-Lawson的理论收敛性分析。特别地，我们证明了(i)对于线性极大极小逼近问题，[math]是Lawson迭代中的近最优Lawson指数；（ii）对于有理极大极小逼近问题，在一定条件下，对于任何足够小的对偶目标函数d-Lawson单调收敛[math]，并且极限逼近满足互补松弛条件，即任何与正权相关的节点要么是插值点，要么具有恒定误差。
{"title":"A Convergence Analysis Of Lawson’s Iteration For Computing Polynomial And Rational Minimax Approximations","authors":"Lei-Hong Zhang, Shanheng Han","doi":"10.1137/24m1708814","DOIUrl":"https://doi.org/10.1137/24m1708814","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2249-2271, December 2025. <br/> Abstract. Lawson’s iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson’s iteration for the rational minimax approximation problem with both computationally high efficiency and theoretical guarantee is challenging. The recent work [L.-H. Zhang et al., Math. Comp., 94 (2025), pp. 2457–2494] reveals that Lawson’s iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and the work proposes a new type of Lawson’s iteration, namely, d-Lawson, which reduces to the classical Lawson’s iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan’s sufficient condition, and, numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, [math] is a near-optimal Lawson exponent in Lawson’s iteration; and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small [math], and the limiting approximant satisfies the complementary slackness condition which states that any node associated with positive weight is either an interpolation point or has a constant error.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455188","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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