IMA Journal of Numerical Analysis最新文献_第4页

Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation 随机 Landau-Lifshitz-Bloch 方程的数值方法和误差估计

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-08-10 DOI: 10.1093/imanum/drae046

Beniamin Goldys, Chunxi Jiao, Kim-Ngan Le

引用次数: 0

On the fast convergence of minibatch heavy ball momentum 论小批重球动量的快速收敛

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-08-09 DOI: 10.1093/imanum/drae033

Raghu Bollapragada, Tyler Chen, Rachel Ward

引用次数: 0

Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm 以随机纳维-斯托克斯方程为范例的 SPDE 数值近似的长期准确性

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-07-16 DOI: 10.1093/imanum/drae043

Nathan E Glatt-Holtz, Cecilia F Mondaini

{"title":"Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm","authors":"Nathan E Glatt-Holtz, Cecilia F Mondaini","doi":"10.1093/imanum/drae043","DOIUrl":"https://doi.org/10.1093/imanum/drae043","url":null,"abstract":"This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141631548","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

The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures 切比雪夫量纲的一次合理修正的高斯四则运算误差边界

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-07-09 DOI: 10.1093/imanum/drae039

Rada M Mutavdžić Djukić

{"title":"The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures","authors":"Rada M Mutavdžić Djukić","doi":"10.1093/imanum/drae039","DOIUrl":"https://doi.org/10.1093/imanum/drae039","url":null,"abstract":"For an analytic integrand, the error term in the Gaussian quadrature can be represented as a contour integral, where the contour is commonly taken to be an ellipse. Thus, finding its upper bound can be reduced to finding the maximum of the modulus of the kernel on the ellipse. The location of this maximum was investigated in many special cases, particularly, for the Gaussian quadrature with respect to the Chebyshev measures modified by a quadratic divisor (known as the Bernstein–Szeg̋ measures). Here, for the Gaussian quadratures with respect to the Chebyshev measures modified by a linear over linear rational factor, we examine the kernel and describe sufficient conditions for the maximum to occur on the real axis. Furthermore, an assessment of the kernel is made in each case, since in some cases the true maximum is hard to reach. Hence, we derive the error bounds for these quadrature formulas. The results are illustrated by the numerical examples. An alternative approach for estimating the error of the Gaussian quadrature with respect to the same measure can be found in [Djukić, D. L., Djukić, R. M. M., Reichel, L. & Spalević, M. M. (2023, Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. Appl. Numer. Math., ISSN 0168-9274)].","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"51 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566247","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

Barycentric rational interpolation of exponentially clustered poles 指数集群极点的巴利心理性插值法

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-07-06 DOI: 10.1093/imanum/drae040

Kelong Zhao, Shuhuang Xiang

引用次数: 0

First-Order Perturbation Theory of Trust-Region Subproblem 信任区域子问题的一阶扰动理论

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-07-06 DOI: 10.1093/imanum/drae042

Bo Feng, Gang Wu

{"title":"First-Order Perturbation Theory of Trust-Region Subproblem","authors":"Bo Feng, Gang Wu","doi":"10.1093/imanum/drae042","DOIUrl":"https://doi.org/10.1093/imanum/drae042","url":null,"abstract":"Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in easy case, we give a sufficient condition under which the perturbed TRS is still in easy case. Secondly, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define their condition numbers. Thirdly, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in nearly hard case. The established results are computable, and are helpful to evaluate ill-conditioning of the large-scale TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553449","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 geometric integration approach to smooth optimization: foundations of the discrete gradient method 平滑优化的几何整合方法：离散梯度法的基础

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-07-01 DOI: 10.1093/imanum/drae037

Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, Carola-Bibiane Schönlieb

{"title":"A geometric integration approach to smooth optimization: foundations of the discrete gradient method","authors":"Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, Carola-Bibiane Schönlieb","doi":"10.1093/imanum/drae037","DOIUrl":"https://doi.org/10.1093/imanum/drae037","url":null,"abstract":"Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489284","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

Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model 时间分数卡恩-希利亚德模型的变步分数 BDF2 方案的渐近兼容能量

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-06-25 DOI: 10.1093/imanum/drae034

Hong-lin Liao, Nan Liu, Xuan Zhao

{"title":"Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model","authors":"Hong-lin Liao, Nan Liu, Xuan Zhao","doi":"10.1093/imanum/drae034","DOIUrl":"https://doi.org/10.1093/imanum/drae034","url":null,"abstract":"A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $alpha in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960le tau _{k}/tau _{k-1}&lt;r^{*}(alpha )$, where $tau _{k}$ is the $k$th time-step size and $r^{*}(alpha )ge 4.660$ for $alpha in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"334 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141453029","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

Interpolation of set-valued functions 定值函数插值

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-06-25 DOI: 10.1093/imanum/drae031

Nira Dyn, David Levin, Qusay Muzaffar

{"title":"Interpolation of set-valued functions","authors":"Nira Dyn, David Levin, Qusay Muzaffar","doi":"10.1093/imanum/drae031","DOIUrl":"https://doi.org/10.1093/imanum/drae031","url":null,"abstract":"Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators. Imperial College Press). By this theory, a ‘metric polynomial interpolant’ is a collection of polynomial interpolants to all the ‘metric chains’ of the given samples of $F$. For set-valued functions whose graphs have nonempty interior, the collection of these ‘metric chains’ can be infinite. Our algorithm computes a small finite subset of ‘significant metric chains’, which is sufficient for approximating $F$. For the class of Lipschitz continuous functions with samples at the roots of the Chebyshev polynomials of the first kind, we prove that the error incurred by our computed interpolant decays with increasing number of interpolation points in the same rate as in the case of interpolation by the metric polynomial interpolant. This is also demonstrated by our numerical examples. For the class of set-valued functions whose graphs have smooth boundaries, we extend our algorithm to achieve a high-precision detection of the points of topology change, followed by a high-order approximation of the boundaries of the graph of F. We further discuss the case of set-valued functions whose graphs have ‘holes’ with Hölder-type singularities at the points of change of topology. To treat this case we apply some special approximation ideas near the singular points of the holes. We analyze the approximation order of the algorithm, including the error in approximating the points of change of topology, and show by several numerical examples the capability of obtaining high-order approximation of the holes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141452781","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

hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations 非线性二阶初值问题的 hp 版本 C1 连续 Petrov-Galerkin 方法在波方程中的应用

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-06-21 DOI: 10.1093/imanum/drae036

Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi

引用次数: 0