Calcolo最新文献_第5页

Optimal approximation of infinite-dimensional holomorphic functions 无限维全貌函数的最优逼近

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-29 DOI: 10.1007/s10092-023-00565-x

Ben Adcock, Nick Dexter, Sebastian Moraga

{"title":"Optimal approximation of infinite-dimensional holomorphic functions","authors":"Ben Adcock, Nick Dexter, Sebastian Moraga","doi":"10.1007/s10092-023-00565-x","DOIUrl":"https://doi.org/10.1007/s10092-023-00565-x","url":null,"abstract":"Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called ((varvec{b},varepsilon ))-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139649608","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

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon 多边形中的半线性椭圆偏微分方程的解析正则性和解法近似

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-11 DOI: 10.1007/s10092-023-00562-0

Yanchen He, Christoph Schwab

引用次数: 0

The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem 四阶椭圆问题的带内部惩罚的赫米特型虚拟元素法

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-03 DOI: 10.1007/s10092-023-00555-z

Jikun Zhao, Teng Chen, Bei Zhang, Xiaojing Dong

引用次数: 0

Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa–Holm system with small energy 小能量旋转两分量卡马萨-霍尔姆系统的保不变差分方案的误差估计

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00558-w

Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang

{"title":"Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa–Holm system with small energy","authors":"Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang","doi":"10.1007/s10092-023-00558-w","DOIUrl":"https://doi.org/10.1007/s10092-023-00558-w","url":null,"abstract":"A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are (O(tau ^2 + h^2)) for the velocity in the (L^infty )-norm and the surface elevation in the (L^2)-norm, where (tau ) denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"43 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079807","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 algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors 弱本质不可还原非负张量谱半径的算法

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00561-1

引用次数: 0

$$L^2(I;H^1(Omega )^d)$$ and $$L^2(I;L^2(Omega )^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems 瞬态斯托克斯问题 Galerkin 解决方案的 $$L^2(I;H^1(Omega )^d)$$ 和 $$L^2(I;L^2(Omega )^d)$$ 最佳近似型误差估计值

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-29 DOI: 10.1007/s10092-023-00560-2

Dmitriy Leykekhman, Boris Vexler

引用次数: 0

A convection–diffusion problem with a large shift on Durán meshes 杜兰网格上的大位移对流扩散问题

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-28 DOI: 10.1007/s10092-023-00559-9

引用次数: 0

On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions 关于三维非结构化有限元离散的离散格林函数的实在性

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-22 DOI: 10.1007/s10092-023-00556-y

Andrew P. Miller

引用次数: 0

A class of two stage multistep methods in solutions of time dependent parabolic PDEs 求解随时间变化的抛物型 PDE 的一类两阶段多步骤方法

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-21 DOI: 10.1007/s10092-023-00557-x

Moosa Ebadi, Mohammad Shahriari

引用次数: 0

High-order schemes based on extrapolation for semilinear fractional differential equation 基于外推法的半线性分数微分方程高阶方案

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-11 DOI: 10.1007/s10092-023-00553-1

Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan

{"title":"High-order schemes based on extrapolation for semilinear fractional differential equation","authors":"Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan","doi":"10.1007/s10092-023-00553-1","DOIUrl":"https://doi.org/10.1007/s10092-023-00553-1","url":null,"abstract":"By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order (alpha in (1,2).) The error has the asymptotic expansion ( big ( d_{3} tau ^{3- alpha } + d_{4} tau ^{4-alpha } + d_{5} tau ^{5-alpha } + cdots big ) + big ( d_{2}^{*} tau ^{4} + d_{3}^{*} tau ^{6} + d_{4}^{*} tau ^{8} + cdots big ) ) at any fixed time (t_{N}= T, N in {mathbb {Z}}^{+}), where (d_{i}, i=3, 4,ldots ) and (d_{i}^{*}, i=2, 3,ldots ) denote some suitable constants and (tau = T/N) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order (alpha in (1,2)) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566643","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