SIAM Journal on Scientific Computing最新文献

{"title":"Cut Finite Element Discretizations of Cell-by-Cell EMI Electrophysiology Models","authors":"Nanna Berre, Marie E. Rognes, André Massing","doi":"10.1137/23m1580632","DOIUrl":"https://doi.org/10.1137/23m1580632","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B527-B553, August 2024. <br/> Abstract. The EMI (extracellular-membrane-intracellular) model describes electrical activity in excitable tissue, where the extracellular and intracellular spaces and cellular membrane are explicitly represented. The model couples a system of partial differential equations (PDEs) in the intracellular and extracellular spaces with a system of ordinary differential equations (ODEs) on the membrane. A key challenge for the EMI model is the generation of high-quality meshes conforming to the complex geometries of brain cells. To overcome this challenge, we propose a novel cut finite element method (CutFEM) where the membrane geometry can be represented independently of a structured and easy-to-generate background mesh for the remaining computational domain. Starting from a Godunov splitting scheme, the EMI model is split into separate PDE and ODE parts. The resulting PDE part is a nonstandard elliptic interface problem, for which we devise two different CutFEM formulations: one single-dimensional formulation with the intra/extracellular electrical potentials as unknowns, and a multi-dimensional formulation that also introduces the electrical current over the membrane as an additional unknown leading to a penalized saddle point problem. Both formulations are augmented by suitably designed ghost penalties to ensure stability and convergence properties that are insensitive to how the membrane surface mesh cuts the background mesh. For the ODE part, we introduce a new unfitted discretization to solve the membrane bound ODEs on a membrane interface that is not aligned with the background mesh. Finally, we perform extensive numerical experiments to demonstrate that CutFEM is a promising approach to efficiently simulate electrical activity in geometrically resolved brain cells. 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://zenodo.org/record/8068506.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"10 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942985","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}

{"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":"11 1","pages":""},"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}

{"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":"80 1","pages":""},"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}

{"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":"4 1","pages":""},"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}

{"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":"34 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"46 1","pages":""},"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}

{"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":"43 1","pages":""},"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}

{"title":"An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices","authors":"Shifan Zhao, Tianshi Xu, Hua Huang, Edmond Chow, Yuanzhe Xi","doi":"10.1137/23m1565139","DOIUrl":"https://doi.org/10.1137/23m1565139","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2351-A2376, August 2024. <br/> Abstract. The spectrum of a kernel matrix significantly depends on the parameter values of the kernel function used to define the kernel matrix. This makes it challenging to design a preconditioner for a regularized kernel matrix that is robust across different parameter values. This paper proposes the adaptive factorized Nyström (AFN) preconditioner. The preconditioner is designed for the case where the rank [math] of the Nyström approximation is large, i.e., for kernel function parameters that lead to kernel matrices with eigenvalues that decay slowly. AFN deliberately chooses a well-conditioned submatrix to solve with and corrects a Nyström approximation with a factorized sparse approximate matrix inverse. This makes AFN efficient for kernel matrices with large numerical ranks. AFN also adaptively chooses the size of this submatrix to balance accuracy and cost. 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/scalable-matrix/H2Pack/tree/AFN_precond and in the supplementary materials (H2Pack.zip [3.99MB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"42 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719236","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}

{"title":"An Efficient Frequency-Independent Numerical Method for Computing the Far-Field Pattern Induced by Polygonal Obstacles","authors":"Andrew Gibbs, Stephen Langdon","doi":"10.1137/23m1612160","DOIUrl":"https://doi.org/10.1137/23m1612160","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2324-A2350, August 2024. <br/> Abstract. For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that (i) they are highly sensitive to numerical errors in practice, and (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we address challenges (i) and (ii), supporting our theory with numerical experiments. Challenge (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Challenge (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the set of valid coefficient vectors. The coefficient vector can then be selected using either a least squares approach or column subset selection.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"2 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719237","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}