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A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations 基于认证小波的物理信息神经网络，用于求解参数化偏微分方程
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-05 DOI: 10.1093/imanum/drae011
Lewin Ernst, Karsten Urban
{"title":"A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations","authors":"Lewin Ernst, Karsten Urban","doi":"10.1093/imanum/drae011","DOIUrl":"https://doi.org/10.1093/imanum/drae011","url":null,"abstract":"Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"125 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826237","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
Finite element methods for multicomponent convection-diffusion 多成分对流扩散有限元方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-04-28 DOI: 10.1093/imanum/drae001
Francis R A Aznaran, Patrick E Farrell, Charles W Monroe, Alexander J Van-Brunt
{"title":"Finite element methods for multicomponent convection-diffusion","authors":"Francis R A Aznaran, Patrick E Farrell, Charles W Monroe, Alexander J Van-Brunt","doi":"10.1093/imanum/drae001","DOIUrl":"https://doi.org/10.1093/imanum/drae001","url":null,"abstract":"We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140819116","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
An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization 用于双标准优化的显式光谱弗莱彻-里维斯共轭梯度法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-04-12 DOI: 10.1093/imanum/drae003
Y Elboulqe, M El Maghri
{"title":"An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization","authors":"Y Elboulqe, M El Maghri","doi":"10.1093/imanum/drae003","DOIUrl":"https://doi.org/10.1093/imanum/drae003","url":null,"abstract":"In this paper, we propose a spectral Fletcher–Reeves conjugate gradient-like method for solving unconstrained bi-criteria minimization problems without using any technique of scalarization. We suggest an explicit formulae for computing a descent direction common to both criteria. The latter further verifies a sufficient descent property that does not depend on the line search nor on any convexity assumption. After proving the existence of a bi-criteria Armijo-type stepsize, global convergence of the proposed algorithm is established. Finally, some numerical results and comparisons with other methods are reported.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"169 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140550430","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
On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation 论非局部卡恩-希利亚德方程的约西达近似的收敛速率
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-04-10 DOI: 10.1093/imanum/drae006
Piotr Gwiazda, Jakub Skrzeczkowski, Lara Trussardi
{"title":"On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation","authors":"Piotr Gwiazda, Jakub Skrzeczkowski, Lara Trussardi","doi":"10.1093/imanum/drae006","DOIUrl":"https://doi.org/10.1093/imanum/drae006","url":null,"abstract":"It is well-known that one can construct solutions to the nonlocal Cahn–Hilliard equation with singular potentials via Yosida approximation with parameter $lambda to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $sqrt{lambda }$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert–Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $lambda $ could be linked to the discretization parameters, yielding appropriate error estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"79 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140544700","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
Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media 断裂多孔介质输运问题的整体和局部时间步进解耦算法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-04-03 DOI: 10.1093/imanum/drae005
Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh
{"title":"Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media","authors":"Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh","doi":"10.1093/imanum/drae005","DOIUrl":"https://doi.org/10.1093/imanum/drae005","url":null,"abstract":"The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349099","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
On the necessity of the inf-sup condition for a mixed finite element formulation 论混合有限元公式中 inf-sup 条件的必要性
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-02-28 DOI: 10.1093/imanum/drae002
Fleurianne Bertrand, Daniele Boffi
{"title":"On the necessity of the inf-sup condition for a mixed finite element formulation","authors":"Fleurianne Bertrand, Daniele Boffi","doi":"10.1093/imanum/drae002","DOIUrl":"https://doi.org/10.1093/imanum/drae002","url":null,"abstract":"We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $textbf{H}(operatorname{textrm{div}};varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140000943","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
Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations 不可压缩纳维-斯托克斯方程的连续内部惩罚无发散有限元方法的压力和对流稳健边界
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1093/imanum/drad108
Bosco García-Archilla, Julia Novo
{"title":"Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations","authors":"Bosco García-Archilla, Julia Novo","doi":"10.1093/imanum/drad108","DOIUrl":"https://doi.org/10.1093/imanum/drad108","url":null,"abstract":"In this paper, we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705063","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
An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain 演化域中线性化纳维-斯托克斯问题的欧拉有限元法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-30 DOI: 10.1093/imanum/drad105
Michael Neilan, Maxim Olshanskii
{"title":"An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain","authors":"Michael Neilan, Maxim Olshanskii","doi":"10.1093/imanum/drad105","DOIUrl":"https://doi.org/10.1093/imanum/drad105","url":null,"abstract":"The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"34 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644033","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
Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems 基于均衡通量重构的目标导向误差估计，用于涡流问题中谐波近似公式的计算
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-29 DOI: 10.1093/imanum/drad107
Emmanuel Creusé, Serge Nicaise, Zuqi Tang
{"title":"Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems","authors":"Emmanuel Creusé, Serge Nicaise, Zuqi Tang","doi":"10.1093/imanum/drad107","DOIUrl":"https://doi.org/10.1093/imanum/drad107","url":null,"abstract":"In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $textbf {A}$-$varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139577496","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
Cauchy data for Levin’s method 列文方法的柯西数据
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-25 DOI: 10.1093/imanum/drad106
Anthony Ashton
{"title":"Cauchy data for Levin’s method","authors":"Anthony Ashton","doi":"10.1093/imanum/drad106","DOIUrl":"https://doi.org/10.1093/imanum/drad106","url":null,"abstract":"In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $|Bx|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate and multivariate case, which minimizes the mean-square of the derivative over the domain. The Cauchy data that generates this solution is then obtained, and this can be used to supplement the Levin equation in the computation of highly oscillatory integrals in the presence of stationary points.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"20 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139568229","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
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