IMA Journal of Numerical Analysis最新文献_第10页

Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator 带波算子的二维非线性Schrödinger方程的线性隐式保能积分因子方法及其收敛性分析

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-25 DOI: 10.1093/imanum/drad067

Xuelong Gu, Wenjun Cai, Yushun Wang, Chaolong Jiang

引用次数: 0

A general collocation analysis for weakly singular Volterra integral equations with variable exponent 变指数弱奇异Volterra积分方程的一般配位分析

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-19 DOI: 10.1093/imanum/drad072

Hui Liang, Martin Stynes

{"title":"A general collocation analysis for weakly singular Volterra integral equations with variable exponent","authors":"Hui Liang, Martin Stynes","doi":"10.1093/imanum/drad072","DOIUrl":"https://doi.org/10.1093/imanum/drad072","url":null,"abstract":"Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-alpha }$ for some constant $alpha in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $alpha = alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135108645","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

Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems 椭圆型和抛物型问题扩散系数的混合神经网络有限元逼近

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-18 DOI: 10.1093/imanum/drad073

Siyu Cen, Bangti Jin, Qimeng Quan, Zhi Zhou

{"title":"Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems","authors":"Siyu Cen, Bangti Jin, Qimeng Quan, Zhi Zhou","doi":"10.1093/imanum/drad073","DOIUrl":"https://doi.org/10.1093/imanum/drad073","url":null,"abstract":"Abstract In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard $L^2(varOmega )$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135208123","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 posteriori error estimates for fully coupled McKean–Vlasov forward-backward SDEs 完全耦合McKean-Vlasov正向向后SDEs的后验误差估计

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-15 DOI: 10.1093/imanum/drad060

Christoph Reisinger, Wolfgang Stockinger, Yufei Zhang

{"title":"<i>A posteriori</i> error estimates for fully coupled McKean–Vlasov forward-backward SDEs","authors":"Christoph Reisinger, Wolfgang Stockinger, Yufei Zhang","doi":"10.1093/imanum/drad060","DOIUrl":"https://doi.org/10.1093/imanum/drad060","url":null,"abstract":"Abstract Fully coupled McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the $L^2$-approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135438201","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

Group sparse optimization for inpainting of random fields on the sphere 球上随机场绘制的群稀疏优化

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-15 DOI: 10.1093/imanum/drad071

Chao Li, Xiaojun Chen

{"title":"Group sparse optimization for inpainting of random fields on the sphere","authors":"Chao Li, Xiaojun Chen","doi":"10.1093/imanum/drad071","DOIUrl":"https://doi.org/10.1093/imanum/drad071","url":null,"abstract":"Abstract We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $ell _2$ norm and non-Lipschitz $ell _p (0&lt;p&lt;1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled Karush–Kuhn–Tucker (KKT) point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Cosmic Microwave Background (CMB) data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135396059","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

Maximum norm error bounds for the full discretization of nonautonomous wave equations 非自治波动方程完全离散化的最大范数误差界

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-09 DOI: 10.1093/imanum/drad065

Benjamin Dörich, Jan Leibold, Bernhard Maier

引用次数: 0

Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves 固定域上的局部吸收边界条件对高频波产生一阶误差

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-09 DOI: 10.1093/imanum/drad058

Jeffrey Galkowski, David Lafontaine, Euan A. Spence

引用次数: 3

Error estimates of high-order compact finite difference schemes for the nonlinear abcd Boussinesq systems 非线性abcd Boussinesq系统高阶紧致有限差分格式的误差估计

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-09-08 DOI: 10.1093/imanum/drad069

Su-Cheol Yi, Kai Fu, Shusen Xie

引用次数: 0

Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term 离散gagliado - nirenberg不等式及其在具有焦耳效应项的对流扩散方程有限体积近似中的应用

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-08-30 DOI: 10.1093/imanum/drad063

C. Calgaro, C. Cancès, E. Creusé

引用次数: 0

An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem 多边形网格上任意阶离散rot-rot复形及其在四次rot问题中的应用

2区数学

IMA Journal of Numerical Analysis Pub Date : 2023-08-28 DOI: 10.1093/imanum/drad045

Daniele Antonio Di Pietro

引用次数: 0