SIAM Journal on Scientific Computing最新文献

{"title":"A Multiscale Hybrid Method","authors":"Gabriel R. Barrenechea, Antonio Tadeu A. Gomes, Diego Paredes","doi":"10.1137/22m1542556","DOIUrl":"https://doi.org/10.1137/22m1542556","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1628-A1657, June 2024. <br/> Abstract. In this work we propose, analyze, and test a new multiscale finite element method called Multiscale Hybrid (MH) method. The method is built as a close relative to the Multiscale Hybrid Mixed (MHM) method, but with the fundamental difference that a novel definition of the Lagrange multiplier is introduced. The practical implication of this is that both the local problems to compute the basis functions, as well as the global problem, are elliptic, as opposed to the MHM method (and also other previous methods) where a mixed global problem is solved and constrained local problems are solved to compute the local basis functions. The error analysis of the method is based on a hybrid formulation, and a static condensation process is done at the discrete level, so the final global system only involves the Lagrange multipliers. We tested the performance of the method by means of numerical experiments for problems with multiscale coefficients, and we carried out comparisons with the MHM method in terms of performance, accuracy, and memory requirements.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"153 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938967","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":"Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels","authors":"Nick Alger, Tucker Hartland, Noemi Petra, Omar Ghattas","doi":"10.1137/23m1584745","DOIUrl":"https://doi.org/10.1137/23m1584745","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1658-A1689, June 2024. <br/> Abstract. We present an efficient matrix-free point spread function (PSF) method for approximating operators that have locally supported nonnegative integral kernels. The PSF-based method computes impulse responses of the operator at scattered points and interpolates these impulse responses to approximate entries of the integral kernel. To compute impulse responses efficiently, we apply the operator to Dirac combs associated with batches of point sources, which are chosen by solving an ellipsoid packing problem. The ability to rapidly evaluate kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are then performed with fast H-matrix methods. This end-to-end procedure is illustrated on a blur problem. We demonstrate the PSF-based method’s effectiveness by using it to build preconditioners for the Hessian operator arising in two inverse problems governed by PDEs: inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low-rank structure, and hence a low-rank approximation is suitable, for many problems of practical interest, the numerical rank of the Hessian is still large. The Hessian impulse responses, on the other hand, typically become more local as the numerical rank increases, which benefits the PSF-based method. Numerical results reveal that the preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly [math]–[math] reductions in the required number of PDE solves, as compared to classical regularization-based preconditioning and no preconditioning. We also present a comprehensive numerical study for the influence of various parameters (that control the shape of the impulse responses and the rank of the Hessian) on the effectiveness of the advection-diffusion Hessian approximation. The results show that the PSF-based method is able to form good approximations of high-rank Hessians using only a small number of operator applications.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"32 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939197","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":"Matrix-Free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems","authors":"Daniel Jodlbauer, Ulrich Langer, Thomas Wick, Walter Zulehner","doi":"10.1137/22m1504184","DOIUrl":"https://doi.org/10.1137/22m1504184","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1599-A1627, June 2024. <br/> Abstract. We consider the widely used continuous [math]-[math] quadrilateral or hexahedral Taylor–Hood elements for the finite element discretization of the Stokes and generalized Stokes systems in two and three spatial dimensions. For the fast solution of the corresponding symmetric, but indefinite system of finite element equations, we propose and analyze matrix-free monolithic geometric multigrid solvers that are based on appropriately scaled Chebyshev–Jacobi smoothers. The analysis is based on results by Schöberl and Zulehner (2003). We present and discuss several numerical results for typical benchmark problems.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"21 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938884","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":"Observability and Effective Region of Partial Differential Equations with Application to Data Assimilation","authors":"Wei Kang, Liang Xu, Hong Zhou","doi":"10.1137/23m1586690","DOIUrl":"https://doi.org/10.1137/23m1586690","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page C249-C271, June 2024. <br/> Abstract. In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"18 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938910","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":"Modified Lawson Methods for Vlasov Equations","authors":"Benjamin Boutin, Anaïs Crestetto, Nicolas Crouseilles, Josselin Massot","doi":"10.1137/22m154301x","DOIUrl":"https://doi.org/10.1137/22m154301x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1574-A1598, June 2024. <br/>Abstract. In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov–Maxwell examples are presented to illustrate the good behavior of the approach.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"67 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938893","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":"Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree","authors":"Pablo D. Brubeck, Patrick E. Farrell","doi":"10.1137/22m1537370","DOIUrl":"https://doi.org/10.1137/22m1537370","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1549-A1573, June 2024. <br/> Abstract. The Riesz maps of the [math] de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work, we present multigrid solvers for high-order finite-element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e., with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the [math]- and [math]-inner products ([math] on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions in the separable case. In the nonseparable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal, but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree. 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.7358044 and in the supplementary materials (pmg_de_rham.zip [61.2KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"112 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883383","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":"Adaptive Space-Time Domain Decomposition for Multiphase Flow in Porous Media with Bound Constraints","authors":"Tianpei Cheng, Haijian Yang, Jizu Huang, Chao Yang","doi":"10.1137/23m1578139","DOIUrl":"https://doi.org/10.1137/23m1578139","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B306-B330, June 2024. <br/> Abstract. This paper proposes an adaptive space-time algorithm based on domain decomposition for the large-scale simulation of a recently developed thermodynamically consistent reservoir problem. In the approach, the bound constraints are represented by means of a minimum-type complementarity function to enforce the positivity of the reservoir model, and a space-time mixed finite element method is applied for the parallel-in-time monolithic discretization. In particular, we propose a time-adaptive strategy using the improved backward differencing formula of second order, to take full advantage of the high degree of space-time parallelism. Moreover, the complicated dynamics with higher nonlinearity of space-time discretization require some innovative nonlinear and linear solution strategies. Therefore, we present a class of modified semismooth Newton algorithms to enhance the convergence rate of nonlinear iterations. Multilevel space-time restricted additive Schwarz algorithms, whose subdomains cover both space and time variables, are also studied for domain decomposition-based preconditioning. Numerical experiments demonstrate the robustness and parallel scalability of the proposed adaptive space-time algorithm on a supercomputer with tens of thousands of processor cores.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"20 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883469","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 Framework for Implementing General Virtual Element Spaces","authors":"Andreas Dedner, Alice Hodson","doi":"10.1137/23m1573653","DOIUrl":"https://doi.org/10.1137/23m1573653","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B229-B253, June 2024. <br/> Abstract. In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a virtual element method (VEM) tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss [math]-conforming spaces for [math] as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package, and we demonstrate this within the open source software package Dune. 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://gitlab.dune-project.org/dune-fem/dune-vem-paper and in the supplementary materials (128492_2_supp_546442_s3hsrj.zip [22KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"82 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829929","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":"Finite Time Horizon Mixed Control of Vibrational Systems","authors":"Ivica Nakić, Marinela Pilj Vidaković, Zoran Tomljanović","doi":"10.1137/22m1488648","DOIUrl":"https://doi.org/10.1137/22m1488648","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B280-B305, June 2024. <br/> Abstract. We consider a vibrational system control problem over a finite time horizon. The performance measure of the system is taken to be a [math]-mixed [math] norm which generalizes the standard [math] norm. We present an algorithm for efficient calculation of this norm in the case when the system is parameter dependent and the number of inputs or outputs of the system is significantly smaller than the order of the system. Our approach is based on a novel procedure which is not based on solving Lyapunov equations and which takes into account the structure of the system. We use a characterization of the [math] norm given in terms of integrals which we solve using adaptive quadrature rules. This enables us to use recycling strategies as well as parallelization. The efficiency of the new algorithm allows for an analysis of the influence of various system parameters and different finite time horizons on the value of the [math]-mixed [math] norm. We illustrate our approach by numerical examples concerning an [math]-mass oscillator with one damper.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"11 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829835","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 Bound-Preserving and Positivity-Preserving High-Order Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Compressible Multi-Medium Flows","authors":"Fan Zhang, Jian Cheng","doi":"10.1137/23m1588810","DOIUrl":"https://doi.org/10.1137/23m1588810","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B254-B279, June 2024. <br/>Abstract. This work presents a novel bound-preserving and positivity-preserving direct arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) method for compressible multimedium flows by solving the five-equation transport model. The proposed method satisfies the discrete geometric conservation law (D-GCL) which indicates that uniform flow is precisely preserved during the simulation. More importantly, based on the D-GCL condition, we present a theoretical analysis on designing an efficient bound-preserving and positivity-preserving limiting strategy, which is able to maintain the boundedness of the volume fraction and the positivity of the partial density and internal energy, with the aim of avoiding the occurrence of inadmissible solutions and meanwhile improving the computational robustness. The accuracy and robustness of the proposed method are demonstrated by various one- and two-dimensional benchmark test cases. The numerical results verify the well capacity of the proposed high-order ALE-DG method for compressible multimedium flows with both the ideal and stiffened gas equation of state.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"17 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829924","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}