SIAM Journal on Numerical Analysis最新文献_第10页

A Universal Median Quasi-Monte Carlo Integration 通用中值准蒙特卡罗积分法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-16 DOI: 10.1137/22m1525077

Takashi Goda, Kosuke Suzuki, Makoto Matsumoto

{"title":"A Universal Median Quasi-Monte Carlo Integration","authors":"Takashi Goda, Kosuke Suzuki, Makoto Matsumoto","doi":"10.1137/22m1525077","DOIUrl":"https://doi.org/10.1137/22m1525077","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024. Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750213","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

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition 满足换元条件的 SDE 的高阶分裂方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-15 DOI: 10.1137/23m155147x

James M. Foster, Gonçalo dos Reis, Calum Strange

{"title":"High Order Splitting Methods for SDEs Satisfying a Commutativity Condition","authors":"James M. Foster, Gonçalo dos Reis, Calum Strange","doi":"10.1137/23m155147x","DOIUrl":"https://doi.org/10.1137/23m155147x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 500-532, February 2024. Abstract. In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal, and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely, Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs—which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox–Ingersoll–Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh–Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of [math] and outperform schemes previously proposed in the literature.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139739343","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

Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation 用于波方程分布式优化控制的时空有限元方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1532962

Richard Löscher, Olaf Steinbach

{"title":"Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation","authors":"Richard Löscher, Olaf Steinbach","doi":"10.1137/22m1532962","DOIUrl":"https://doi.org/10.1137/22m1532962","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024. Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139700968","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

On Uncertainty Quantification of Eigenvalues and Eigenspaces with Higher Multiplicity 论高倍性特征值和特征空间的不确定性量化

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1529324

Jürgen Dölz, David Ebert

引用次数: 0

Convergence Analysis for Bregman Iterations in Minimizing a Class of Landau Free Energy Functionals 最小化一类朗道自由能函数的布雷格曼迭代收敛分析

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1517664

Chenglong Bao, Chang Chen, Kai Jiang, Lingyun Qiu

引用次数: 0

Frequency-Explicit A Posteriori Error Estimates for Discontinuous Galerkin Discretizations of Maxwell’s Equations 麦克斯韦方程非连续伽勒金离散化的频率显式后验误差估计值

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-06 DOI: 10.1137/22m1516348

Théophile Chaumont-Frelet, Patrick Vega

引用次数: 0

Structure Preserving Primal Dual Methods for Gradient Flows with Nonlinear Mobility Transport Distances 非线性流动传输距离梯度流的结构保持原点二元法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-05 DOI: 10.1137/23m1562068

José A. Carrillo, Li Wang, Chaozhen Wei

引用次数: 0

Numerical Methods and Analysis of Computing Quasiperiodic Systems 计算准周期系统的数值方法与分析

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-01 DOI: 10.1137/22m1524783

Kai Jiang, Shifeng Li, Pingwen Zhang

{"title":"Numerical Methods and Analysis of Computing Quasiperiodic Systems","authors":"Kai Jiang, Shifeng Li, Pingwen Zhang","doi":"10.1137/22m1524783","DOIUrl":"https://doi.org/10.1137/22m1524783","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024. Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660060","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

Numerical Integration of Schrödinger Maps via the Hasimoto Transform 通过 Hasimoto 变换对薛定谔图进行数值积分

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-01-31 DOI: 10.1137/22m1531555

Valeria Banica, Georg Maierhofer, Katharina Schratz

{"title":"Numerical Integration of Schrödinger Maps via the Hasimoto Transform","authors":"Valeria Banica, Georg Maierhofer, Katharina Schratz","doi":"10.1137/22m1531555","DOIUrl":"https://doi.org/10.1137/22m1531555","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024. Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realization based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e., under lower-regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"121 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139655653","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 Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost 与频率相关成本的振荡二阶线性 ODE 的自适应谱方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-01-29 DOI: 10.1137/23m1546609

Fruzsina J. Agocs, Alex H. Barnett

{"title":"An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost","authors":"Fruzsina J. Agocs, Alex H. Barnett","doi":"10.1137/23m1546609","DOIUrl":"https://doi.org/10.1137/23m1546609","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 295-321, February 2024. Abstract. We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to [math] times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"343 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139573530","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