SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2528-A2556, August 2024. Abstract. We present a structure-preserving Eulerian algorithm for solving [math]-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system’s free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.
{"title":"Energetic Variational Neural Network Discretizations of Gradient Flows","authors":"Ziqing Hu, Chun Liu, Yiwei Wang, Zhiliang Xu","doi":"10.1137/22m1529427","DOIUrl":"https://doi.org/10.1137/22m1529427","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2528-A2556, August 2024. <br/> Abstract. We present a structure-preserving Eulerian algorithm for solving [math]-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system’s free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942986","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C421-C447, August 2024. Abstract. In a recent work [Q. Li and S. Evje, Netw. Heterog. Media, 18 (2023), pp. 48–79], it was explored how to identify the unknown flux function in a one-dimensional scalar conservation law. Key ingredients are symbolic neural networks to represent the candidate flux functions, entropy-satisfying numerical schemes, and a proper combination of initial data. The purpose of this work is to extend this methodology to a two-dimensional scalar conservation law ([math]) [math]. Straightforward extension of the method from the 1D to the 2D problem results in poor identification of the unknown [math] and [math]. Relying on ideas from joint and alternating equations training, a learning strategy is designed that enables accurate identification of the flux functions, even when 2D observations are sparse. It involves an alternating flux training approach where a first set of candidate flux functions obtained from joint training is improved through an alternating direction-dependent training strategy. Numerical investigations demonstrate that the method can effectively identify the true underlying flux functions [math] and [math] in the general case when they are nonconvex and unequal.
SIAM 科学计算期刊》，第 46 卷第 4 期，第 C421-C447 页，2024 年 8 月。 摘要最近的一项研究 [Q. Li and S. Evje, Netw. Heterog. Media, 18 (2023), pp.其中的关键要素是表示候选通量函数的符号神经网络、满足熵的数值方案以及初始数据的适当组合。这项工作的目的是将这一方法扩展到二维标量守恒定律（[math]）[math]。将该方法从一维问题直接扩展到二维问题会导致对未知数[math]和[math]的识别不清。根据联合方程和交替方程训练的思想，我们设计了一种学习策略，即使在二维观测数据稀少的情况下，也能准确识别通量函数。它涉及一种交替通量训练方法，即通过交替方向相关训练策略改进从联合训练中获得的第一组候选通量函数。数值研究表明，在通量函数[math]和[math]非凸且不相等的一般情况下，该方法可以有效地识别真正的基本通量函数[math]和[math]。
{"title":"An Alternating Flux Learning Method for Multidimensional Nonlinear Conservation Laws","authors":"Qing Li, Steinar Evje","doi":"10.1137/23m1556605","DOIUrl":"https://doi.org/10.1137/23m1556605","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C421-C447, August 2024. <br/> Abstract. In a recent work [Q. Li and S. Evje, Netw. Heterog. Media, 18 (2023), pp. 48–79], it was explored how to identify the unknown flux function in a one-dimensional scalar conservation law. Key ingredients are symbolic neural networks to represent the candidate flux functions, entropy-satisfying numerical schemes, and a proper combination of initial data. The purpose of this work is to extend this methodology to a two-dimensional scalar conservation law ([math]) [math]. Straightforward extension of the method from the 1D to the 2D problem results in poor identification of the unknown [math] and [math]. Relying on ideas from joint and alternating equations training, a learning strategy is designed that enables accurate identification of the flux functions, even when 2D observations are sparse. It involves an alternating flux training approach where a first set of candidate flux functions obtained from joint training is improved through an alternating direction-dependent training strategy. Numerical investigations demonstrate that the method can effectively identify the true underlying flux functions [math] and [math] in the general case when they are nonconvex and unequal.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882638","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C399-C420, August 2024. Abstract. The integration of graph neural networks (GNNs) and neural ordinary and partial differential equations has been extensively studied in recent years. GNN architectures powered by neural differential equations allow us to reason about their behavior, and develop GNNs with desired properties such as controlled smoothing or energy conservation. In this paper we take inspiration from Turing instabilities in a reaction diffusion (RD) system of partial differential equations, and propose a novel family of GNNs based on neural RD systems, called RDGNN. We show that our RDGNN is powerful for the modeling of various data types, from homophilic, to heterophilic, and spatiotemporal datasets. We discuss the theoretical properties of our RDGNN, its implementation, and show that it improves or offers competitive performance to state-of-the-art methods.
{"title":"Graph Neural Reaction Diffusion Models","authors":"Moshe Eliasof, Eldad Haber, Eran Treister","doi":"10.1137/23m1576700","DOIUrl":"https://doi.org/10.1137/23m1576700","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C399-C420, August 2024. <br/> Abstract. The integration of graph neural networks (GNNs) and neural ordinary and partial differential equations has been extensively studied in recent years. GNN architectures powered by neural differential equations allow us to reason about their behavior, and develop GNNs with desired properties such as controlled smoothing or energy conservation. In this paper we take inspiration from Turing instabilities in a reaction diffusion (RD) system of partial differential equations, and propose a novel family of GNNs based on neural RD systems, called RDGNN. We show that our RDGNN is powerful for the modeling of various data types, from homophilic, to heterophilic, and spatiotemporal datasets. We discuss the theoretical properties of our RDGNN, its implementation, and show that it improves or offers competitive performance to state-of-the-art methods.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882641","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}
Celia Caballero-Cárdenas, Manuel Jesús Castro, Christophe Chalons, Tomás Morales de Luna, María Luz Muñoz-Ruiz
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2503-A2527, August 2024. Abstract. This article focuses on the design of semi-implicit schemes that are fully well-balanced for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady states of the system and not just water-at-rest equilibria. The proposed methods outperform standard explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity, eliminating the need for a large number of iterations on large time intervals. In this work, splitting and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the preservation of all the steady states while avoiding the complexities associated with Lagrangian formalism.
{"title":"A Semi-Implicit Fully Exactly Well-Balanced Relaxation Scheme for the Shallow Water System","authors":"Celia Caballero-Cárdenas, Manuel Jesús Castro, Christophe Chalons, Tomás Morales de Luna, María Luz Muñoz-Ruiz","doi":"10.1137/23m1621289","DOIUrl":"https://doi.org/10.1137/23m1621289","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2503-A2527, August 2024. <br/> Abstract. This article focuses on the design of semi-implicit schemes that are fully well-balanced for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady states of the system and not just water-at-rest equilibria. The proposed methods outperform standard explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity, eliminating the need for a large number of iterations on large time intervals. In this work, splitting and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the preservation of all the steady states while avoiding the complexities associated with Lagrangian formalism.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882640","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C369-C398, August 2024. Abstract. Physics-informed neural networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN, by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition, and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/hhjc-web/SEPINN.git and in the supplementary materials (M160119_SuppMat.pdf [399KB]).
{"title":"Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks","authors":"Tianhao Hu, Bangti Jin, Zhi Zhou","doi":"10.1137/23m1601195","DOIUrl":"https://doi.org/10.1137/23m1601195","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C369-C398, August 2024. <br/> Abstract. Physics-informed neural networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN, by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition, and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/hhjc-web/SEPINN.git and in the supplementary materials (M160119_SuppMat.pdf [399KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871331","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2475-A2502, August 2024. Abstract. In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes. In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in Lemaire, Thieullen, and Thomas [Adv. Appl. Probab., 52 (2020), pp. 138–172], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameters of the algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.
{"title":"Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes","authors":"Ajay Jasra, Kengo Kamatani, Mohamed Maama","doi":"10.1137/23m1600505","DOIUrl":"https://doi.org/10.1137/23m1600505","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2475-A2502, August 2024. <br/> Abstract. In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes. In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in Lemaire, Thieullen, and Thomas [Adv. Appl. Probab., 52 (2020), pp. 138–172], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameters of the algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785251","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2445-A2474, August 2024. Abstract. With the aid of hardware and software developments, there has been a surge of interest in solving PDEs by deep learning techniques, and the integration with domain decomposition strategies has recently attracted considerable attention due to its enhanced representation and parallelization capacity of the network solution. While there are already several works that substitute the numerical solver of overlapping Schwarz methods with the deep learning approach, the nonoverlapping counterpart has not been thoroughly studied yet because of the inevitable interface overfitting problem that would propagate the errors to neighboring subdomains and eventually hamper the convergence of outer iteration. In this work, a novel learning approach, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable the flux transmission across subregion interfaces with guaranteed accuracy, thereby allowing us to construct effective learning algorithms for realizing the more general nonoverlapping domain decomposition methods in the presence of overfitted interface conditions. Numerical experiments on a series of elliptic boundary value problems, including the regular and irregular interfaces, low and high dimensions, and smooth and high-contrast coefficients on multidomains, are carried out to validate the effectiveness of our proposed domain decomposition learning algorithms. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available" as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/AI4SC-TJU or in the supplementary materials.
{"title":"Domain Decomposition Learning Methods for Solving Elliptic Problems","authors":"Qi Sun, Xuejun Xu, Haotian Yi","doi":"10.1137/22m1515392","DOIUrl":"https://doi.org/10.1137/22m1515392","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2445-A2474, August 2024. <br/> Abstract. With the aid of hardware and software developments, there has been a surge of interest in solving PDEs by deep learning techniques, and the integration with domain decomposition strategies has recently attracted considerable attention due to its enhanced representation and parallelization capacity of the network solution. While there are already several works that substitute the numerical solver of overlapping Schwarz methods with the deep learning approach, the nonoverlapping counterpart has not been thoroughly studied yet because of the inevitable interface overfitting problem that would propagate the errors to neighboring subdomains and eventually hamper the convergence of outer iteration. In this work, a novel learning approach, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable the flux transmission across subregion interfaces with guaranteed accuracy, thereby allowing us to construct effective learning algorithms for realizing the more general nonoverlapping domain decomposition methods in the presence of overfitted interface conditions. Numerical experiments on a series of elliptic boundary value problems, including the regular and irregular interfaces, low and high dimensions, and smooth and high-contrast coefficients on multidomains, are carried out to validate the effectiveness of our proposed domain decomposition learning algorithms. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available\" as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/AI4SC-TJU or in the supplementary materials.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772438","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2421-A2444, August 2024. Abstract. We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a joint extension of both the full approximation scheme multigrid technique for nonlinear partial differential equations, due to A. Brandt, and the constraint decomposition (CD) method introduced by X.-C. Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates. The full multigrid cycle version is an optimal solver. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://bitbucket.org/pefarrell/fascd/, where the software used to produce the results in section 8 is archived at tag v1.0, and at https://doi.org/10.5281/zenodo.10476845 or in the supplementary materials (pefarrell-fascd-6407e9f547d6.zip [225KB]). The authors used Firedrake master revision c5e939dde.
SIAM 科学计算期刊》，第 46 卷第 4 期，第 A2421-A2444 页，2024 年 8 月。 摘要我们提出了求解变分不等式（VIs）的全近似方案约束分解（FASCD）多层次方法。FASCD 是 A. Brandt 提出的非线性偏微分方程全近似方案多网格技术和 X.-C. Tai 提出的约束分解 (CD) 方法的联合扩展。Tai 针对优化中出现的 VIs 提出的约束分解（CD）方法。我们利用多级网格分层产生的某些子集分解的伸缩性，扩展了 CD 的思想。当应用缩减空间（活动集）牛顿方法作为平滑器时，其功与给定网格层次上的未知数数量成正比，FASCD V 循环表现出几乎与网格无关的收敛速度。全多网格循环版本是一种最佳求解器。示例问题包括单边和双边 VI 问题中的对称线性、非对称线性和非线性微分算子。计算结果的可重复性。本文被授予 "SIAM 可重复性徽章：可用代码和数据"，以表彰作者遵循了 SISC 和科学计算界重视的可重复性原则。读者可通过以下网址获取代码和数据以重现本文结果：https://bitbucket.org/pefarrell/fascd/，其中用于生成第8节结果的软件以标签v1.0存档；https://doi.org/10.5281/zenodo.10476845，或在补充材料（pefarrell-fascd-6407e9f547d6.zip [225KB]）中获取。作者使用的是 Firedrake 主修订版 c5e939dde。
{"title":"A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities","authors":"Ed Bueler, Patrick E. Farrell","doi":"10.1137/23m1594200","DOIUrl":"https://doi.org/10.1137/23m1594200","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2421-A2444, August 2024. <br/> Abstract. We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a joint extension of both the full approximation scheme multigrid technique for nonlinear partial differential equations, due to A. Brandt, and the constraint decomposition (CD) method introduced by X.-C. Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates. The full multigrid cycle version is an optimal solver. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://bitbucket.org/pefarrell/fascd/, where the software used to produce the results in section 8 is archived at tag v1.0, and at https://doi.org/10.5281/zenodo.10476845 or in the supplementary materials (pefarrell-fascd-6407e9f547d6.zip [225KB]). The authors used Firedrake master revision c5e939dde.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785252","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2398-A2420, August 2024. Abstract. We develop a method to compute the [math]-conforming finite element approximation to planar fourth order elliptic problems without having to implement [math] elements. The algorithm consists of replacing the original [math]-conforming scheme with preprocessing and postprocessing steps that require only an [math]-conforming Poisson type solve and an inner Stokes-like problem that again only requires at most [math]-conformity. We then demonstrate the method applied to the Morgan–Scott elements with three numerical examples. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.5281/zenodo.10070565.
{"title":"Computing [math]-Conforming Finite Element Approximations Without Having to Implement [math]-Elements","authors":"Mark Ainsworth, Charles Parker","doi":"10.1137/23m1615486","DOIUrl":"https://doi.org/10.1137/23m1615486","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2398-A2420, August 2024. <br/> Abstract. We develop a method to compute the [math]-conforming finite element approximation to planar fourth order elliptic problems without having to implement [math] elements. The algorithm consists of replacing the original [math]-conforming scheme with preprocessing and postprocessing steps that require only an [math]-conforming Poisson type solve and an inner Stokes-like problem that again only requires at most [math]-conformity. We then demonstrate the method applied to the Morgan–Scott elements with three numerical examples. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.5281/zenodo.10070565.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745580","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}
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2377-A2397, August 2024. Abstract. We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber [math]. On a coarse mesh of width [math], the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width [math] and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter [math] is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition [math] with [math] denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.
{"title":"Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems","authors":"Philip Freese, Moritz Hauck, Daniel Peterseim","doi":"10.1137/21m1465950","DOIUrl":"https://doi.org/10.1137/21m1465950","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2377-A2397, August 2024. <br/> Abstract. We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber [math]. On a coarse mesh of width [math], the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width [math] and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter [math] is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition [math] with [math] denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737205","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}
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