arXiv - MATH - Numerical Analysis最新文献_第4页

A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications 频域毕奥孔弹性方程的稳定总压公式：数值分析与应用

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10465

Cristian Cárcamo, Alfonso Caiazzo, Felipe Galarce, Joaquín Mura

{"title":"A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications","authors":"Cristian Cárcamo, Alfonso Caiazzo, Felipe Galarce, Joaquín Mura","doi":"arxiv-2409.10465","DOIUrl":"https://doi.org/arxiv-2409.10465","url":null,"abstract":"This work focuses on the numerical solution of the dynamics of a poroelastic\u0000material in the frequency domain. We provide a detailed stability analysis\u0000based on the application of the Fredholm alternative in the continuous case,\u0000considering a total pressure formulation of the Biot's equations. In the\u0000discrete setting, we propose a stabilized equal order finite element method\u0000complemented by an additional pressure stabilization to enhance the robustness\u0000of the numerical scheme with respect to the fluid permeability. Utilizing the\u0000Fredholm alternative, we extend the well-posedness results to the discrete\u0000setting, obtaining theoretical optimal convergence for the case of linear\u0000finite elements. We present different numerical experiments to validate the\u0000proposed method. First, we consider model problems with known analytic\u0000solutions in two and three dimensions. As next, we show that the method is\u0000robust for a wide range of permeabilities, including the case of discontinuous\u0000coefficients. Lastly, we show the application for the simulation of brain\u0000elastography on a realistic brain geometry obtained from medical imaging.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems 针对参数化界面问题的物理信息定制有限点算子网络

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10284

Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang

{"title":"Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems","authors":"Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang","doi":"arxiv-2409.10284","DOIUrl":"https://doi.org/arxiv-2409.10284","url":null,"abstract":"Learning operators for parametric partial differential equations (PDEs) using\u0000neural networks has gained significant attention in recent years. However,\u0000standard approaches like Deep Operator Networks (DeepONets) require extensive\u0000labeled data, and physics-informed DeepONets encounter training challenges. In\u0000this paper, we introduce a novel physics-informed tailored finite point\u0000operator network (PI-TFPONet) method to solve parametric interface problems\u0000without the need for labeled data. Our method fully leverages the prior\u0000physical information of the problem, eliminating the need to include the PDE\u0000residual in the loss function, thereby avoiding training challenges. The\u0000PI-TFPONet is specifically designed to address certain properties of the\u0000problem, allowing us to naturally obtain an approximate solution that closely\u0000matches the exact solution. Our method is theoretically proven to converge if\u0000the local mesh size is sufficiently small and the training loss is minimized.\u0000Notably, our approach is uniformly convergent for singularly perturbed\u0000interface problems. Extensive numerical studies show that our unsupervised\u0000PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised\u0000deep operator networks in terms of accuracy and versatility.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Structure-preserving learning for multi-symplectic PDEs 多折射 PDE 的保结构学习

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10432

Süleyman Yıldız, Pawan Goyal, Peter Benner

{"title":"Structure-preserving learning for multi-symplectic PDEs","authors":"Süleyman Yıldız, Pawan Goyal, Peter Benner","doi":"arxiv-2409.10432","DOIUrl":"https://doi.org/arxiv-2409.10432","url":null,"abstract":"This paper presents an energy-preserving machine learning method for\u0000inferring reduced-order models (ROMs) by exploiting the multi-symplectic form\u0000of partial differential equations (PDEs). The vast majority of\u0000energy-preserving reduced-order methods use symplectic Galerkin projection to\u0000construct reduced-order Hamiltonian models by projecting the full models onto a\u0000symplectic subspace. However, symplectic projection requires the existence of\u0000fully discrete operators, and in many cases, such as black-box PDE solvers,\u0000these operators are inaccessible. In this work, we propose an energy-preserving\u0000machine learning method that can infer the dynamics of the given PDE using data\u0000only, so that the proposed framework does not depend on the fully discrete\u0000operators. In this context, the proposed method is non-intrusive. The proposed\u0000method is grey box in the sense that it requires only some basic knowledge of\u0000the multi-symplectic model at the partial differential equation level. We prove\u0000that the proposed method satisfies spatially discrete local energy conservation\u0000and preserves the multi-symplectic conservation laws. We test our method on the\u0000linear wave equation, the Korteweg-de Vries equation, and the\u0000Zakharov-Kuznetsov equation. We test the generalization of our learned models\u0000by testing them far outside the training time interval.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC 通过 McKean-Vlasov 动力学向前传播低差异：从 QMC 到 MLQMC

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09821

Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz

{"title":"Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC","authors":"Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz","doi":"arxiv-2409.09821","DOIUrl":"https://doi.org/arxiv-2409.09821","url":null,"abstract":"This work develops a particle system addressing the approximation of\u0000McKean-Vlasov stochastic differential equations (SDEs). The novelty of the\u0000approach lies in involving low discrepancy sequences nontrivially in the\u0000construction of a particle system with coupled noise and initial conditions.\u0000Weak convergence for SDEs with additive noise is proven. A numerical study\u0000demonstrates that the novel approach presented here doubles the respective\u0000convergence rates for weak and strong approximation of the mean-field limit,\u0000compared with the standard particle system. These rates are proven in the\u0000simplified setting of a mean-field ordinary differential equation in terms of\u0000appropriate bounds involving the star discrepancy for low discrepancy sequences\u0000with a group structure, such as Rank-1 lattice points. This construction\u0000nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An\u0000asymptotic error analysis reveals that the proposed approach outperforms\u0000methods based on the classic particle system with independent initial\u0000conditions and noise.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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 Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws 双曲守恒定律的高阶振荡消除赫米特 WENO 方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09632

Chuan Fan, Kailiang Wu

{"title":"High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws","authors":"Chuan Fan, Kailiang Wu","doi":"arxiv-2409.09632","DOIUrl":"https://doi.org/arxiv-2409.09632","url":null,"abstract":"This paper proposes high-order accurate, oscillation-eliminating Hermite\u0000weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for\u0000hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after\u0000each Runge--Kutta stage, dampening the first-order moments of the HWENO\u0000solution to suppress spurious oscillations without any problem-dependent\u0000parameters. This OE procedure acts as a filter, derived from the solution\u0000operator of a novel damping equation, solved exactly without discretization. As\u0000a result, the OE-HWENO method remains stable with a normal CFL number, even for\u0000strong shocks producing highly stiff damping terms. To ensure the method's\u0000non-oscillatory property across varying scales and wave speeds, we design a\u0000scale- and evolution-invariant damping equation and propose a dimensionless\u0000transformation for HWENO reconstruction. The OE-HWENO method offers several\u0000advantages over existing HWENO methods: the OE procedure is efficient and easy\u0000to implement, requiring only simple multiplication of first-order moments; it\u0000preserves high-order accuracy, local compactness, and spectral properties. The\u0000non-intrusive OE procedure can be integrated seamlessly into existing HWENO\u0000codes. Finally, we analyze the bound-preserving (BP) property using optimal\u0000cell average decomposition, relaxing the BP time step-size constraint and\u0000reducing decomposition points, improving efficiency. Extensive benchmarks\u0000validate the method's accuracy, efficiency, resolution, and robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition 非结构化三角形网格上的稳健 DG 方案：通过最优凸分解消除振荡和保持边界

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09620

Shengrong Ding, Shumo Cui, Kailiang Wu

{"title":"Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition","authors":"Shengrong Ding, Shumo Cui, Kailiang Wu","doi":"arxiv-2409.09620","DOIUrl":"https://doi.org/arxiv-2409.09620","url":null,"abstract":"Discontinuous Galerkin (DG) schemes on unstructured meshes offer the\u0000advantages of compactness and the ability to handle complex computational\u0000domains. However, their robustness and reliability in solving hyperbolic\u0000conservation laws depend on two critical abilities: suppressing spurious\u0000oscillations and preserving intrinsic bounds or constraints. This paper\u0000introduces two significant advancements in enhancing the robustness and\u0000efficiency of DG methods on unstructured meshes for general hyperbolic\u0000conservation laws, while maintaining their accuracy and compactness. First, we\u0000investigate the oscillation-eliminating (OE) DG methods on unstructured meshes.\u0000These methods not only maintain key features such as conservation, scale\u0000invariance, and evolution invariance but also achieve rotation invariance\u0000through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for\u0000the first time, the optimal convex decomposition for designing efficient\u0000bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal\u0000convex decomposition that maximizes the BP CFL number is an important yet\u0000challenging problem.While this challenge was addressed for rectangular meshes,\u0000it remains an open problem for triangular meshes. This paper successfully\u0000constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$\u0000spaces on triangular cells, significantly improving the efficiency of BP DG\u0000methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and\u0000280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our\u0000RIOE procedure and optimal decomposition technique can be integrated into\u0000existing DG codes with little and localized modifications. These techniques\u0000require only edge-neighboring cell information, thereby retaining the\u0000compactness and high parallel efficiency of original DG methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior 利用总变异先验对贝叶斯图像去模糊进行局部 MALA-within-Gibbs 处理

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09810

Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong

{"title":"Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior","authors":"Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong","doi":"arxiv-2409.09810","DOIUrl":"https://doi.org/arxiv-2409.09810","url":null,"abstract":"We consider Bayesian inference for image deblurring with total variation (TV)\u0000prior. Since the posterior is analytically intractable, we resort to Markov\u0000chain Monte Carlo (MCMC) methods. However, since most MCMC methods\u0000significantly deteriorate in high dimensions, they are not suitable to handle\u0000high resolution imaging problems. In this paper, we show how low-dimensional\u0000sampling can still be facilitated by exploiting the sparse conditional\u0000structure of the posterior. To this end, we make use of the local structures of\u0000the blurring operator and the TV prior by partitioning the image into\u0000rectangular blocks and employing a blocked Gibbs sampler with proposals\u0000stemming from the Metropolis-Hastings adjusted Langevin Algorithm (MALA). We\u0000prove that this MALA-within-Gibbs (MLwG) sampling algorithm has\u0000dimension-independent block acceptance rates and dimension-independent\u0000convergence rate. In order to apply the MALA proposals, we approximate the TV\u0000by a smoothed version, and show that the introduced approximation error is\u0000evenly distributed and dimension-independent. Since the posterior is a Gibbs\u0000density, we can use the Hammersley-Clifford Theorem to identify the posterior\u0000conditionals which only depend locally on the neighboring blocks. We outline\u0000computational strategies to evaluate the conditionals, which are the target\u0000densities in the Gibbs updates, locally and in parallel. In two numerical\u0000experiments, we validate the dimension-independent properties of the MLwG\u0000algorithm and demonstrate its superior performance over MALA.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics PROSE-FD：用于学习流体力学预测多算子的多模态 PDE 基础模型

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09811

Yuxuan Liu, Jingmin Sun, Xinjie He, Griffin Pinney, Zecheng Zhang, Hayden Schaeffer

引用次数: 0

Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms 带有四次弹性项的向列液晶 Landau-de Gennes 模型的有限元分析

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09837

Jacob Elafandi, Franziska Weber

引用次数: 0

Analysis of Centrifugal Clutches in Two-Speed Automatic Transmissions with Deep Learning-Based Engagement Prediction 利用基于深度学习的接合预测分析双速自动变速器中的离心离合器

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09755

Bo-Yi Lin, Kai Chun Lin

引用次数: 0