Numerische Mathematik最新文献_第2页

Moment-SoS methods for optimal transport problems 优化运输问题的矩-SoS 方法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-06-11 DOI: 10.1007/s00211-024-01422-x

Olga Mula, Anthony Nouy

{"title":"Moment-SoS methods for optimal transport problems","authors":"Olga Mula, Anthony Nouy","doi":"10.1007/s00211-024-01422-x","DOIUrl":"https://doi.org/10.1007/s00211-024-01422-x","url":null,"abstract":"Most common optimal transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. However, it is sometimes relevant to work only with moments of measures instead of the measure itself, and many common OT problems can be formulated as moment problems (the most relevant examples being (L^p)-Wasserstein distances, barycenters, and Gromov–Wasserstein discrepancies on Euclidean spaces). We leverage this fact to develop a generalized moment formulation that covers these classes of OT problems. The transport plan is represented through its moments on a given basis, and the marginal constraints are expressed in terms of moment constraints. A practical computation then consists in considering a truncation of the involved moment sequences up to a certain order, and using the polynomial sums-of-squares hierarchy for measures supported on semi-algebraic sets. We prove that the strategy converges to the solution of the OT problem as the order increases. We also show how to approximate linear quantities of interest, and how to estimate the support of the optimal transport map from the computed moments using Christoffel–Darboux kernels. Numerical experiments illustrate the good behavior of the approach.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503109","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}

引用次数: 0

Validated integration of semilinear parabolic PDEs 半线性抛物线 PDE 的验证整合

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-06-04 DOI: 10.1007/s00211-024-01415-w

Jan Bouwe van den Berg, Maxime Breden, Ray Sheombarsing

{"title":"Validated integration of semilinear parabolic PDEs","authors":"Jan Bouwe van den Berg, Maxime Breden, Ray Sheombarsing","doi":"10.1007/s00211-024-01415-w","DOIUrl":"https://doi.org/10.1007/s00211-024-01415-w","url":null,"abstract":"Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton–Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"59 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254147","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}

引用次数: 0

Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution 重叠网格上的时空 CutFEM II：简单不连续网格演化

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-27 DOI: 10.1007/s00211-024-01413-y

Mats G. Larson, Carl Lundholm

{"title":"Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution","authors":"Mats G. Larson, Carl Lundholm","doi":"10.1007/s00211-024-01413-y","DOIUrl":"https://doi.org/10.1007/s00211-024-01413-y","url":null,"abstract":"We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141167102","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}

引用次数: 0

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion 重叠网格上的时空 CutFEM I：简单连续网格运动

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-25 DOI: 10.1007/s00211-024-01417-8

Mats G. Larson, Anders Logg, Carl Lundholm

{"title":"Space-time CutFEM on overlapping meshes I: simple continuous mesh motion","authors":"Mats G. Larson, Anders Logg, Carl Lundholm","doi":"10.1007/s00211-024-01417-8","DOIUrl":"https://doi.org/10.1007/s00211-024-01417-8","url":null,"abstract":"We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.\u0000","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148070","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}

引用次数: 0

Drift approximation by the modified Boris algorithm of charged-particle dynamics in toroidal geometry 环状几何中带电粒子动力学的改进鲍里斯算法漂移近似

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-24 DOI: 10.1007/s00211-024-01416-9

Yanyan Shi

引用次数: 0

Finite Elements with Switch Detection for direct optimal control of nonsmooth systems 有限元与开关检测，用于非平滑系统的直接优化控制

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-24 DOI: 10.1007/s00211-024-01412-z

Armin Nurkanović, Mario Sperl, Sebastian Albrecht, Moritz Diehl

{"title":"Finite Elements with Switch Detection for direct optimal control of nonsmooth systems","authors":"Armin Nurkanović, Mario Sperl, Sebastian Albrecht, Moritz Diehl","doi":"10.1007/s00211-024-01412-z","DOIUrl":"https://doi.org/10.1007/s00211-024-01412-z","url":null,"abstract":"This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"28 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148206","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}

引用次数: 0

Super-localization of spatial network models 空间网络模型的超定位

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-21 DOI: 10.1007/s00211-024-01410-1

Moritz Hauck, Axel Målqvist

引用次数: 0

Multiresolution kernel matrix algebra 多分辨率核矩阵代数

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-09 DOI: 10.1007/s00211-024-01409-8

H. Harbrecht, M. Multerer, O. Schenk, Ch. Schwab

{"title":"Multiresolution kernel matrix algebra","authors":"H. Harbrecht, M. Multerer, O. Schenk, Ch. Schwab","doi":"10.1007/s00211-024-01409-8","DOIUrl":"https://doi.org/10.1007/s00211-024-01409-8","url":null,"abstract":"We propose a sparse algebra for samplet compressed kernel matrices to enable efficient scattered data analysis. We show that the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. The compression can be performed in cost and memory that scale essentially linearly with the number of data points for kernels of finite differentiability. The same holds true for the addition and multiplication of S-formatted matrices. We prove that the inverse of a kernel matrix, given that it exists, is compressible in the S-format as well. The use of selected inversion allows to directly compute the entries in the corresponding sparsity pattern. Moreover, S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as ({varvec{A}}^alpha ) or (exp ({varvec{A}})) of a matrix ({varvec{A}}). The matrix algebra is justified mathematically by pseudo differential calculus. As an application, we consider Gaussian process learning algorithms for implicit surfaces. Numerical results are presented to illustrate and quantify our findings.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"2672 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931362","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}

引用次数: 0

Interior estimates for the virtual element method 虚拟元素法的内部估计

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-09 DOI: 10.1007/s00211-024-01408-9

Silvia Bertoluzza, Micol Pennacchio, Daniele Prada

引用次数: 0

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds 保证特征值下限的自适应混合高阶方法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-05-06 DOI: 10.1007/s00211-024-01407-w

Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran

{"title":"Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds","authors":"Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran","doi":"10.1007/s00211-024-01407-w","DOIUrl":"https://doi.org/10.1007/s00211-024-01407-w","url":null,"abstract":"The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter (C_{text {st},1}) that is found not robust as the polynomial degree p increases. This is related to the (H^1) stability bound of the (L^{2}) projection onto polynomials of degree at most p and its growth (C_{textrm{st, 1}}propto (p+1)^{1/2}) as (p rightarrow infty ). A similar estimate for the Galerkin projection holds with a p-robust constant (C_{text {st},2}) and (C_{text {st},2} le 2) for right-isosceles triangles. This paper utilizes the new inequality with the constant (C_{text {st},2}) to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved (L^{2}) error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"111 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882670","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}

引用次数: 0