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The Milstein scheme for singular SDEs with Hölder continuous drift 具有Hölder连续漂移的奇异SDEs的Milstein格式
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-12-14 DOI: 10.1093/imanum/drae083
Máté Gerencsér, Gerald Lampl, Chengcheng Ling
{"title":"The Milstein scheme for singular SDEs with Hölder continuous drift","authors":"Máté Gerencsér, Gerald Lampl, Chengcheng Ling","doi":"10.1093/imanum/drae083","DOIUrl":"https://doi.org/10.1093/imanum/drae083","url":null,"abstract":"We study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823191","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}
引用次数: 0
A conforming multi-domain Legendre spectral method for solving diffusive-viscous wave equations in the exterior domain with separated star-shaped obstacles 求解星形分离障碍物外域扩散粘性波方程的符合多域 Legendre 频谱方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-12-14 DOI: 10.1093/imanum/drae085
Guoqing Yao, Zicheng Wang, Zhongqing Wang
{"title":"A conforming multi-domain Legendre spectral method for solving diffusive-viscous wave equations in the exterior domain with separated star-shaped obstacles","authors":"Guoqing Yao, Zicheng Wang, Zhongqing Wang","doi":"10.1093/imanum/drae085","DOIUrl":"https://doi.org/10.1093/imanum/drae085","url":null,"abstract":"In this paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate transformations along radial direction to each sub-domain, we map them into eight regular sub-blocks. Subsequently, we perform numerical simulations using classical spectral methods on these regular sub-blocks. Our analysis focuses on the optimal convergence of this approach. The numerical results demonstrate the high-order accuracy of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823307","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}
引用次数: 0
Asymptotic consistency of the WSINDy algorithm in the limit of continuum data 连续体数据极限下WSINDy算法的渐近一致性
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-12-13 DOI: 10.1093/imanum/drae086
Daniel A Messenger, David M Bortz
{"title":"Asymptotic consistency of the WSINDy algorithm in the limit of continuum data","authors":"Daniel A Messenger, David M Bortz","doi":"10.1093/imanum/drae086","DOIUrl":"https://doi.org/10.1093/imanum/drae086","url":null,"abstract":"In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $sigma _{c}$. We provide explicit bounds on $sigma _{c}$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142820569","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}
引用次数: 0
A spectral collocation method for functional and delay differential equations 泛函与时滞微分方程的谱配置方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-11-29 DOI: 10.1093/imanum/drae079
Nicholas Hale
{"title":"A spectral collocation method for functional and delay differential equations","authors":"Nicholas Hale","doi":"10.1093/imanum/drae079","DOIUrl":"https://doi.org/10.1093/imanum/drae079","url":null,"abstract":"A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142752867","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}
引用次数: 0
Error analysis for a finite element approximation of the steady p·-Navier–Stokes equations 纳维尔-斯托克斯稳定方程的有限元近似误差分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-11-25 DOI: 10.1093/imanum/drae082
Luigi C Berselli, Alex Kaltenbach
{"title":"Error analysis for a finite element approximation of the steady p·-Navier–Stokes equations","authors":"Luigi C Berselli, Alex Kaltenbach","doi":"10.1093/imanum/drae082","DOIUrl":"https://doi.org/10.1093/imanum/drae082","url":null,"abstract":"In this paper, we examine a finite element approximation of the steady $p(cdot )$-Navier–Stokes equations ($p(cdot )$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law index $p(cdot )$. Numerical experiments confirm the quasi-optimality of the a priori error estimates (for the velocity) with respect to fractional regularity assumptions on the velocity vector field and the kinematic pressure.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696578","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}
引用次数: 0
A unified framework for the error analysis of physics-informed neural networks 物理信息神经网络误差分析的统一框架
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-11-20 DOI: 10.1093/imanum/drae081
Marius Zeinhofer, Rami Masri, Kent–André Mardal
{"title":"A unified framework for the error analysis of physics-informed neural networks","authors":"Marius Zeinhofer, Rami Masri, Kent–André Mardal","doi":"10.1093/imanum/drae081","DOIUrl":"https://doi.org/10.1093/imanum/drae081","url":null,"abstract":"We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^{2}$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678475","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}
引用次数: 0
Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions 尼斯特伦离散化亥姆霍兹分解边界积分方程公式的稳定性估计，用于解决二维纳维散射问题（带德里赫特边界条件）
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-11-10 DOI: 10.1093/imanum/drae078
Víctor Domínguez, Catalin Turc
{"title":"Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions","authors":"Víctor Domínguez, Catalin Turc","doi":"10.1093/imanum/drae078","DOIUrl":"https://doi.org/10.1093/imanum/drae078","url":null,"abstract":"Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"95 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142597484","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}
引用次数: 0
Positive definite functions on a regular domain 规则域上的正定函数
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-11-06 DOI: 10.1093/imanum/drae074
Martin Buhmann, Yuan Xu
{"title":"Positive definite functions on a regular domain","authors":"Martin Buhmann, Yuan Xu","doi":"10.1093/imanum/drae074","DOIUrl":"https://doi.org/10.1093/imanum/drae074","url":null,"abstract":"We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142588647","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}
引用次数: 0
An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation 将近似分量模式合成法扩展至异质亥姆霍兹方程
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-10-26 DOI: 10.1093/imanum/drae076
Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom
{"title":"An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation","authors":"Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom","doi":"10.1093/imanum/drae076","DOIUrl":"https://doi.org/10.1093/imanum/drae076","url":null,"abstract":"In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490355","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}
引用次数: 0
Time-dependent electromagnetic scattering from dispersive materials 色散材料随时间变化的电磁散射
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-10-25 DOI: 10.1093/imanum/drae071
Jörg Nick, Selina Burkhard, Christian Lubich
{"title":"Time-dependent electromagnetic scattering from dispersive materials","authors":"Jörg Nick, Selina Burkhard, Christian Lubich","doi":"10.1093/imanum/drae071","DOIUrl":"https://doi.org/10.1093/imanum/drae071","url":null,"abstract":"This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490356","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}
引用次数: 0
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