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Unfitted finite element method for the quad-curl interface problem 四旋度界面问题的非拟合有限元法
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-27 DOI: 10.1007/s10444-024-10213-9
Hailong Guo, Mingyan Zhang, Qian Zhang, Zhimin Zhang

In this paper, we introduce a novel unfitted finite element method to solve the quad-curl interface problem. We adapt Nitsche’s method for ({operatorname {curl}}{operatorname {curl}})-conforming elements and double the degrees of freedom on interface elements. To ensure stability, we incorporate ghost penalty terms and a discrete divergence-free term. We establish the well-posedness of our method and demonstrate an optimal error bound in the discrete energy norm. We also analyze the stiffness matrix’s condition number. Our numerical tests back up our theory on convergence rates and condition numbers.

本文提出了一种求解四旋度界面问题的非拟合有限元方法。我们采用Nitsche的方法求解({operatorname {curl}}{operatorname {curl}}) -一致性单元,并将界面单元的自由度加倍。为了保证稳定性,我们加入了鬼罚项和一个离散的无发散项。建立了该方法的适定性,并给出了离散能量范数下的最优误差界。分析了刚度矩阵的条件数。我们的数值测试支持了我们关于收敛速率和条件数的理论。
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引用次数: 0
A nonsingular-kernel Dirichlet-to-Dirichlet mapping method for the exterior Stokes problem 外部Stokes问题的非奇核Dirichlet-to-Dirichlet映射方法
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-18 DOI: 10.1007/s10444-024-10216-6
Xiaojuan Liu, Maojun Li, Tao Yin, Shangyou Zhang

This paper studies the finite element method for solving the exterior Stokes problem in two dimensions. A nonlocal boundary condition is defined using a nonsingular-kernel Dirichlet-to-Dirichlet (DtD) mapping, which maps the Dirichlet data on an interior circle to the Dirichlet data on another circular artificial boundary based on the Poisson integral formula of the Stokes problem. The truncated problem is then solved using the MINI-element method and a simple DtD iteration strategy, resulting into a sequence of unique and geometrically (h- independent) convergent solutions. The quasi-optimal error estimate is proved for the iterative solution at the end of the iteration process. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

本文研究了求解二维外斯托克斯问题的有限元方法。基于Stokes问题的泊松积分公式,利用非奇异核Dirichlet-to-Dirichlet (DtD)映射定义了一个非局部边界条件,该映射将内圆上的Dirichlet数据映射到另一个圆形人工边界上的Dirichlet数据。然后使用MINI-element方法和简单的DtD迭代策略求解截断问题,得到一系列唯一且几何上(h-无关)收敛的解。在迭代过程结束时,证明了迭代解的拟最优误差估计。数值实验验证了该方法的准确性和有效性。
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引用次数: 0
Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation 利用列导数公式实现奥尔德罗伊德-B 粘弹性流体流动的离散化
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-17 DOI: 10.1007/s10444-024-10211-x
Ben S. Ashby, Tristan Pryer

In this article, we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with a detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.

在这篇文章中,我们介绍了奥尔德罗伊德-B 流体斯托克斯流的数值方法。粘弹性应力根据上对流时间导数制定的构成定律演变。采用有限差分法沿流体轨迹离散,以简单、廉价和内聚的方式逼近上对流导数的平流和变形项,并确保离散构象张量为正定。本文介绍了与流体流动耦合的完整实施方案,并详细讨论了此类方案中出现的问题。我们在盖子驱动的空腔设置中进行了详细的数值实验,证明了该方法的性能。数值结果与已公布的数据进行了比对,结果表明该方法在这种具有挑战性的情况下表现良好。
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引用次数: 0
A posteriori error control for a discontinuous Galerkin approximation of a Keller-Segel model Keller-Segel模型的不连续Galerkin近似的后验误差控制
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-13 DOI: 10.1007/s10444-024-10212-w
Jan Giesselmann, Kiwoong Kwon

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional in the sense that an a posteriori computable quantity needs to be small enough—which can be ensured by mesh refinement—and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove the existence of a weak solution up to a certain time based on numerical results.

给出了二维或三维抛物-椭圆型Keller-Segel系统的不连续Galerkin格式的后测误差估计。估计是有条件的,因为后验可计算量需要足够小,这可以通过网格细化来保证;估计是最优的,因为误差估计器的衰减顺序与网格细化下的误差相同。我们的误差估计器的一个特点是,它可以用来证明一个弱解的存在到一定时间的数值结果。
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引用次数: 0
Efficient iterative methods for hyperparameter estimation in large-scale linear inverse problems 大规模线性反问题超参数估计的有效迭代方法
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-09 DOI: 10.1007/s10444-024-10208-6
Khalil A. Hall-Hooper, Arvind K. Saibaba, Julianne Chung, Scot M. Miller

We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for small problems but are not computationally feasible for problems with a very large number of unknown inverse parameters. In this work, we describe an empirical Bayes (EB) method to estimate hyperparameters that maximize the marginal posterior, i.e., the probability density of the hyperparameters conditioned on the data, and then we use the estimated hyperparameters to compute the posterior of the unknown inverse parameters. For problems where the computation of the square root and inverse of prior covariance matrices are not feasible, we describe an approach based on the generalized Golub-Kahan bidiagonalization to approximate the marginal posterior and seek hyperparameters that minimize the approximate marginal posterior. Numerical results from seismic and atmospheric tomography demonstrate the accuracy, robustness, and potential benefits of the proposed approach.

我们研究了大规模线性逆问题的贝叶斯方法,重点研究了超参数估计这一具有挑战性的任务。遵循马尔可夫链蒙特卡罗方法的典型层次贝叶斯公式对于小问题是可能的,但对于具有大量未知逆参数的问题在计算上是不可行的。在这项工作中,我们描述了一种经验贝叶斯(EB)方法来估计最大化边际后验的超参数,即数据条件下超参数的概率密度,然后我们使用估计的超参数来计算未知逆参数的后验。对于无法计算先验协方差矩阵的平方根和逆的问题,我们描述了一种基于广义Golub-Kahan双对角化的方法来近似边际后验,并寻求使近似边际后验最小的超参数。地震和大气层析成像的数值结果证明了该方法的准确性、鲁棒性和潜在的优势。
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引用次数: 0
Analysis of a time filtered finite element method for the unsteady inductionless MHD equations 非定常无感应MHD方程的时间滤波有限元分析
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-09 DOI: 10.1007/s10444-024-10215-7
Xiaodi Zhang, Jialin Xie, Xianzhu Li

This paper studies a time filtered finite element method for the unsteady inductionless magnetohydrodynamic (MHD) equations. The method uses the semi-implicit backward Euler scheme with a time filter in time and adopts the standard inf-sup stable fluid pairs to discretize the velocity and pressure, and the inf-sup stable face-volume elements for solving the current density and electric potential in space. Since the time filter for the velocity is added as a separate post-processing step, the scheme can be easily incorporated into the existing backward Euler code and improves the time accuracy from first order to second order. The unique solvability, unconditional energy stability, and charge conservativeness of the scheme are also proven. In terms of the energy arguments, we establish the error estimates for the velocity, current density, and electric potential. Numerical experiments are performed to verify the theoretical analysis.

研究了求解非定常无感应磁流体动力学方程的时间滤波有限元方法。该方法在时间上采用带时间滤波器的半隐式后向欧拉格式,在空间上采用标准的中流稳定流体对离散速度和压力,在空间上采用中流稳定面体积元求解电流密度和电势。由于速度的时间滤波器是作为一个单独的后处理步骤添加的,因此该方案可以很容易地合并到现有的向后欧拉代码中,并将时间精度从一阶提高到二阶。证明了该方案的唯一可解性、无条件能量稳定性和电荷保守性。在能量参数方面,我们建立了速度、电流密度和电势的误差估计。通过数值实验验证了理论分析的正确性。
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引用次数: 0
On the recovery of initial status for linearized shallow-water wave equation by data assimilation with error analysis 用数据同化法恢复线性化浅水波动方程初始状态及误差分析
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-05 DOI: 10.1007/s10444-024-10210-y
Jun-Liang Fu, Jijun Liu

We recover the initial status of an evolution system governed by linearized shallow-water wave equations in a 2-dimensional bounded domain by data assimilation technique, with the aim of determining the initial wave height from the measurement of wave distribution in an interior domain. Since we specify only one component of the solution to the governed system and the observation is only measured in part of the interior domain, taking into consideration of the engineering restriction on the measurement process, this problem is ill-posed. Based on the known well-posedness result of the forward problem, this inverse problem is reformulated as an optimizing problem with data-fit term and the penalty term involving the background of the wave amplitude as a-prior information. We establish the Euler-Lagrange equation for the optimal solution in terms of its adjoint system. The unique solvability of this Euler-Lagrange equation is rigorously proven. Then the optimal approximation error of the regularizing solution to the exact solution is established in terms of the noise level of measurement data and the a-prior background distribution, based on the Lax-Milgram theorem. Finally, we propose an iterative algorithm to realize this process, with several numerical examples to validate the efficacy of our proposed method.

本文利用数据同化技术在二维有界域中恢复线性化浅水波方程控制的演化系统的初始状态,目的是通过测量内域中的波分布来确定初始波高。由于我们只指定了被控制系统的解的一个组成部分,并且观测仅在内部域的一部分进行测量,考虑到测量过程的工程限制,该问题是不适定的。在已知正问题的适定性结果的基础上,将反问题重新表述为以数据拟合项和以波幅背景为先验信息的惩罚项为先验信息的优化问题。建立了其伴随系统最优解的欧拉-拉格朗日方程。严格证明了该欧拉-拉格朗日方程的唯一可解性。然后,基于Lax-Milgram定理,根据测量数据的噪声级和a先验背景分布,建立了正则化解对精确解的最优逼近误差。最后,我们提出了一种迭代算法来实现这一过程,并通过几个数值算例验证了我们提出的方法的有效性。
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引用次数: 0
Inverting the fundamental diagram and forecasting boundary conditions: how machine learning can improve macroscopic models for traffic flow 反转基本图和预测边界条件:机器学习如何改善交通流的宏观模型
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-04 DOI: 10.1007/s10444-024-10206-8
Maya Briani, Emiliano Cristiani, Elia Onofri

In this paper, we develop new methods to join machine learning techniques and macroscopic differential models, aimed at estimate and forecast vehicular traffic. This is done to complement respective advantages of data-driven and model-driven approaches. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: (1) if congestion is appearing under the sensor, and (2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident/bottleneck happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better predict the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A.

在本文中,我们开发了新的方法来结合机器学习技术和宏观微分模型,旨在估计和预测车辆交通。这样做是为了补充数据驱动和模型驱动方法各自的优势。我们在这里考虑一个数据集,其中包含高速公路上行驶的车辆的流量和速度数据,由固定传感器收集,并按车道和车辆类别分类。通过基于LSTM递归神经网络的机器学习模型,我们推断出两个重要的信息:(1)传感器下是否出现拥堵,以及(2)下一个未来(30分钟)将通过传感器下的车辆总量。然后使用这些信息片段来提高基于lhr的一阶多类模型的准确性,该模型描述了传感器之间的交通流动态。利用第一段信息反演(凹)基本图,从而从通量数据中恢复车辆密度,然后直接将密度基准注入模型。这使得人们可以更好地近似传感器之间的动态,特别是如果事故/瓶颈发生在不受监控的路段。第二部分信息被用作交通模型基础方程的边界条件,以便更好地预测未来任何时间道路上的车辆总量。我们将讨论一些基于真实场景的例子。真实数据由意大利高速公路公司Autovie Venete S.p.A提供。
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引用次数: 0
Solving the quadratic eigenvalue problem expressed in non-monomial bases by the tropical scaling 用热带标度法求解非一元基表示的二次特征值问题
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-12-03 DOI: 10.1007/s10444-024-10214-8
Hongjia Chen, Teng Wang, Chun-Hua Zhang, Xiang Wang

In this paper, we consider the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange bases. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues for QEP solved by a class of block Kronecker linearizations. To improve the backward error and condition number of the QEP expressed in a non-monomial basis, we combine the tropical scaling with the block Kronecker linearization. We then establish upper bounds for the backward error of an approximate eigenpair of the QEP relative to the backward error of an approximate eigenpair of the block Kronecker linearization with and without tropical scaling. Moreover, we get bounds for the normwise condition number of an eigenvalue of the QEP relative to that of the block Kronecker linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings.

本文考虑了用泰勒基、牛顿基和拉格朗日基等几种常用基表示的二次特征值问题。我们提出研究用一类块Kronecker线性化方法求解QEP的计算特征对和特征值的条件数的后向误差。为了改善非一元基表示的QEP的后向误差和条件数,我们将热带标度与块Kronecker线性化相结合。然后,我们建立了QEP的近似特征对相对于具有和不具有热带尺度的块Kronecker线性化的近似特征对的向后误差的上界。此外,我们得到了相对于块Kronecker线性化的QEP特征值的正态条件数的界。我们的研究伴随着足够的数值实验来证明我们的理论发现。
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引用次数: 0
Discontinuous Galerkin schemes for Stokes flow with Tresca boundary condition: iterative a posteriori error analysis 具有 Tresca 边界条件的斯托克斯流的非连续 Galerkin 方案:迭代后验误差分析
IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-11-25 DOI: 10.1007/s10444-024-10207-7
J.K. Djoko, T. Sayah

In two dimensions, we propose and analyse an iterative a posteriori error indicator for the discontinuous Galerkin finite element approximations of the Stokes equations under boundary conditions of friction type. Two sources of error are identified here, namely; the discretisation error and the linearization error. Under a smallness assumption on data, we prove that the devised error estimator is reliable. Balancing these two errors is crucial to design an adaptive strategy for mesh refinement. We illustrate the theory with some representative numerical examples.

在两个维度上,我们提出并分析了摩擦型边界条件下斯托克斯方程的非连续 Galerkin 有限元近似的迭代后验误差指标。这里确定了两个误差来源,即离散化误差和线性化误差。在数据较小的假设条件下,我们证明了所设计的误差估算器是可靠的。平衡这两个误差对于设计网格细化的自适应策略至关重要。我们用一些有代表性的数值示例来说明这一理论。
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引用次数: 0
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Advances in Computational Mathematics
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