SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate regularity conditions on the nonlinearity, the total variation distance between the distributions of the numerical approximation and of the exact solution at a given time converges to 0 when the time-step size vanishes, with order of convergence [math]. Equivalently, weak error estimates with order [math] are thus obtained for bounded measurable test functions. This is an original and major improvement compared with the performance of the standard linear implicit Euler scheme or exponential Euler methods, which do not converge in the sense of total variation when the time-step size vanishes. Equivalently weak error estimates for the standard schemes require twice differentiable test functions. The proof of the total variation error bounds for the accelerated exponential Euler scheme exploits some regularization property of the associated infinite-dimensional Kolmogorov equations.
{"title":"Total Variation Error Bounds for the Accelerated Exponential Euler Scheme Approximation of Parabolic Semilinear SPDEs","authors":"Charles-Edouard Bréhier","doi":"10.1137/22m152596x","DOIUrl":"https://doi.org/10.1137/22m152596x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. <br/> Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate regularity conditions on the nonlinearity, the total variation distance between the distributions of the numerical approximation and of the exact solution at a given time converges to 0 when the time-step size vanishes, with order of convergence [math]. Equivalently, weak error estimates with order [math] are thus obtained for bounded measurable test functions. This is an original and major improvement compared with the performance of the standard linear implicit Euler scheme or exponential Euler methods, which do not converge in the sense of total variation when the time-step size vanishes. Equivalently weak error estimates for the standard schemes require twice differentiable test functions. The proof of the total variation error bounds for the accelerated exponential Euler scheme exploits some regularization property of the associated infinite-dimensional Kolmogorov equations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304419","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}
Christa Cuchiero, Christoph Reisinger, Stefan Rigger
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied by Kaushansky et al. [Ann. Appl. Probab., 33 (2023), pp. 274–298], but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.
{"title":"Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups","authors":"Christa Cuchiero, Christoph Reisinger, Stefan Rigger","doi":"10.1137/22m1509722","DOIUrl":"https://doi.org/10.1137/22m1509722","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. <br/> Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied by Kaushansky et al. [Ann. Appl. Probab., 33 (2023), pp. 274–298], but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"33 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902996","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 62, Issue 3, Page 1119-1144, June 2024. Abstract. This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal intervehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill–Richards–Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter.
{"title":"Asymptotic Compatibility of a Class of Numerical Schemes for a Nonlocal Traffic Flow Model","authors":"Kuang Huang, Qiang Du","doi":"10.1137/23m154488x","DOIUrl":"https://doi.org/10.1137/23m154488x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1119-1144, June 2024. <br/> Abstract. This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal intervehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill–Richards–Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881282","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}
Victor P. DeCaria, Cory D. Hauck, Stefan R. Schnake
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1067-1097, June 2024. Abstract. A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density [math] converges to an isotropic function [math], called the drift-diffusion limit, where [math] is a Maxwellian and the physical density [math] satisfies a second-order parabolic PDE known as the drift-diffusion equation. Numerical approximations that mirror this property are said to be asymptotic preserving. In this paper we build a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in [math], where 1/[math] is the scale of the collision frequency, and asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in [math] to an accurate [math]-approximation of the drift-diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to [math] and the spacial resolution are also included.
{"title":"An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model","authors":"Victor P. DeCaria, Cory D. Hauck, Stefan R. Schnake","doi":"10.1137/22m1485784","DOIUrl":"https://doi.org/10.1137/22m1485784","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1067-1097, June 2024. <br/> Abstract. A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density [math] converges to an isotropic function [math], called the drift-diffusion limit, where [math] is a Maxwellian and the physical density [math] satisfies a second-order parabolic PDE known as the drift-diffusion equation. Numerical approximations that mirror this property are said to be asymptotic preserving. In this paper we build a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in [math], where 1/[math] is the scale of the collision frequency, and asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in [math] to an accurate [math]-approximation of the drift-diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to [math] and the spacial resolution are also included.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845605","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 62, Issue 3, Page 1098-1118, June 2024. Abstract. Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high-dimensional data, which features bounding stochastic approximation error by the spectrum of the underlying kernel matrix. Both theoretical analysis and numerical simulations show that spectra of kernel matrices are reliable and stable barometers for gauging the performance of kernel-interpolation methods for high-dimensional data.
{"title":"Kernel Interpolation of High Dimensional Scattered Data","authors":"Shao-Bo Lin, Xiangyu Chang, Xingping Sun","doi":"10.1137/22m1519948","DOIUrl":"https://doi.org/10.1137/22m1519948","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1098-1118, June 2024. <br/> Abstract. Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high-dimensional data, which features bounding stochastic approximation error by the spectrum of the underlying kernel matrix. Both theoretical analysis and numerical simulations show that spectra of kernel matrices are reliable and stable barometers for gauging the performance of kernel-interpolation methods for high-dimensional data.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845651","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 62, Issue 3, Page 1039-1066, June 2024. Abstract. In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm.
{"title":"A Novel Mixed Spectral Method and Error Estimates for Maxwell Transmission Eigenvalue Problems","authors":"Jing An, Waixiang Cao, Zhimin Zhang","doi":"10.1137/23m1544830","DOIUrl":"https://doi.org/10.1137/23m1544830","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1039-1066, June 2024. <br/> Abstract. In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"42 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140821576","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 62, Issue 3, Page 1021-1038, June 2024. Abstract. Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of [math] with a variance that is [math] for any [math]. It also satisfies some nonasymptotic bounds where the variance is no larger than some [math] times the ordinary Monte Carlo variance. For scrambled Sobol’ points, this quantity [math] grows exponentially in [math]. For scrambled Faure points, [math] in any dimension, but those points are awkward to use for large [math]. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is [math] but not [math] for any [math] as [math]. For [math], the upper bound on the gain coefficient is never larger than [math].
{"title":"Gain Coefficients for Scrambled Halton Points","authors":"Art B. Owen, Zexin Pan","doi":"10.1137/23m1601882","DOIUrl":"https://doi.org/10.1137/23m1601882","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1021-1038, June 2024. <br/>Abstract. Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of [math] with a variance that is [math] for any [math]. It also satisfies some nonasymptotic bounds where the variance is no larger than some [math] times the ordinary Monte Carlo variance. For scrambled Sobol’ points, this quantity [math] grows exponentially in [math]. For scrambled Faure points, [math] in any dimension, but those points are awkward to use for large [math]. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is [math] but not [math] for any [math] as [math]. For [math], the upper bound on the gain coefficient is never larger than [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"380 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140821140","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 62, Issue 2, Page 998-1019, April 2024. Abstract. In this paper, we propose a two-level block preconditioned Jacobi–Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of [math]th ([math]) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several eigenpairs, including both multiple and clustered eigenvalues with corresponding eigenfunctions, particularly. The method is highly parallelizable by constructing a new and efficient preconditioner using an overlapping domain decomposition (DD). It only requires computing a couple of small scale parallel subproblems and a quite small scale eigenvalue problem per iteration. Our theoretical analysis reveals that the convergence rate of the method is bounded by [math], where [math] is the diameter of subdomains and [math] is the overlapping size among subdomains. The constant [math] is independent of the mesh size [math] and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the [math]-dependent constant [math] decreases monotonically to 1, as [math], which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.
{"title":"A Two-Level Block Preconditioned Jacobi–Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators","authors":"Qigang Liang, Wei Wang, Xuejun Xu","doi":"10.1137/23m1580711","DOIUrl":"https://doi.org/10.1137/23m1580711","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 998-1019, April 2024. <br/> Abstract. In this paper, we propose a two-level block preconditioned Jacobi–Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of [math]th ([math]) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several eigenpairs, including both multiple and clustered eigenvalues with corresponding eigenfunctions, particularly. The method is highly parallelizable by constructing a new and efficient preconditioner using an overlapping domain decomposition (DD). It only requires computing a couple of small scale parallel subproblems and a quite small scale eigenvalue problem per iteration. Our theoretical analysis reveals that the convergence rate of the method is bounded by [math], where [math] is the diameter of subdomains and [math] is the overlapping size among subdomains. The constant [math] is independent of the mesh size [math] and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the [math]-dependent constant [math] decreases monotonically to 1, as [math], which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"38 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140632364","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 62, Issue 2, Page 946-973, April 2024. Abstract. We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler–Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler–Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems.
{"title":"Sequential Discretization Schemes for a Class of Stochastic Differential Equations and their Application to Bayesian Filtering","authors":"Ö. Deniz Akyildiz, Dan Crisan, Joaquin Miguez","doi":"10.1137/23m1560124","DOIUrl":"https://doi.org/10.1137/23m1560124","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 946-973, April 2024. <br/> Abstract. We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler–Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler–Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538131","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 62, Issue 2, Page 974-997, April 2024. Abstract. Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace and Helmholtz equations show that high-order accuracy can be achieved for arbitrarily close field evaluation.
{"title":"Singularity Swapping Method for Nearly Singular Integrals Based on Trapezoidal Rule","authors":"Gang Bao, Wenmao Hua, Jun Lai, Jinrui Zhang","doi":"10.1137/23m1571666","DOIUrl":"https://doi.org/10.1137/23m1571666","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 974-997, April 2024. <br/> Abstract. Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace and Helmholtz equations show that high-order accuracy can be achieved for arbitrarily close field evaluation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538248","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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