Article
Mathematics, Applied
Fei Xu, Yasai Guo, Qiumei Huang, Hongkun Ma
Summary: In this paper, an efficient multigrid method is studied for solving semilinear interface problems. With an optimal finite element error estimate and a novel correction step design, the proposed method can efficiently handle semilinear elliptic problems and achieves a similar efficiency as the multigrid method for linear interface problems.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Software Engineering
Thomas C. Clevenger, Timo Heister, Guido Kanschat, Martin Kronbichler
Summary: The paper introduces a geometric multigrid method for massively parallel computations on adaptively refined meshes, utilizing local smoothing and partitioning through a space filling curve. Efficiency of mesh hierarchy distribution is modeled and compared to runtime measurements. The algorithm is implemented in the DEAL.II finite-element library for public use.
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
(2021)
Article
Environmental Sciences
Xianyang Huang, Changchun Yin, Luyuan Wang, Yunhe Liu, Bo Zhang, Xiuyan Ren, Yang Su, Jun Li, Hui Chen
Summary: In order to improve the computational efficiency of 3D magnetotelluric (MT) forward modeling, a novel geometric multigrid algorithm for the finite element method is proposed. The algorithm discretizes Maxwell's equations in the frequency domain using vector finite element and applies Dirichlet boundary conditions to obtain complex linear equations for the solution of EM responses. A divergence correction is used to improve the convergence of the solution at low frequencies, and a V-cycle geometric multigrid algorithm is developed to solve the linear equations system. Numerical results show that the proposed algorithm outperforms commonly used Krylov subspace algorithms in terms of iteration number, solution time, and stability, making it more suitable for large-scale 3D MT forward modeling.
Article
Engineering, Mechanical
Zhibao Zheng, Hongzhe Dai, Michael Beer
Summary: This paper proposes a novel methodology for structural reliability analysis using the stochastic finite element method (SFEM). The method decouples the stochastic displacement into a combination of deterministic displacements with random variable coefficients. An iterative algorithm is provided to solve for the deterministic displacements and corresponding random variables. The proposed method can calculate the limit state function and multidimensional integral encountered in reliability analysis accurately and efficiently. It can also be applied to high-dimensional stochastic problems without modification, effectively addressing the curse of dimensionality.
PROBABILISTIC ENGINEERING MECHANICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Zhuogang Peng, Ryan G. McClarren
Summary: The dynamical low-rank (DLR) approximation is an efficient technique for approximating matrix differential equation solutions. This study extends the low-rank scheme to the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries. The low-rank solution requires less memory and computational time while maintaining accuracy compared to solving the full rank equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Engineering, Multidisciplinary
Zhibao Zheng, Udo Nackenhorst
Summary: This article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. The method utilizes a series of (pseudo) time steps to describe history-dependent stochastic behavior and solves the corresponding stochastic solutions. By transforming the nonlinear stochastic problem into a set of linearized stochastic finite element equations, the method approximates the stochastic solution at each time step using the products of random variables and deterministic vectors. The proposed method successfully avoids the curse of dimensionality and demonstrates good performance in numerical cases.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Hardik Kothari, Rolf Krause
Summary: The article presents an unfitted Finite Element method for internal interfaces within a domain and a multigrid algorithm for solving contact problems on these interfaces. By utilizing structured background meshes and the method of Lagrange multipliers for discretizing non-penetration conditions, and constructing a hierarchy of nested FE spaces using pseudo-L-2 projection-based transfer operators, the method ensures global convergence while enforcing linear constraints locally.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Patrick E. Farrell, Abdalaziz Hamdan, Scott P. MacLachlan
Summary: This paper presents a finite element formulation for solving fourth-order problems and addresses the issue of boundary conditions. By introducing an explicit variable and using Lagrange multipliers to constrain the gradient, essential boundary conditions are enforced weakly using Nitsche's method. The problem is transformed into a saddle-point system, allowing for the development of monolithic multigrid solvers. Numerical results in two and three-dimensional cases demonstrate the accuracy of the formulation and the efficiency of the proposed solvers.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Ming Li
Summary: A new algebraic prolongation operator is developed for the standard V-cycle multigrid method to accelerate the process based on a coarsening strategy of adjacency matrix. An efficient algebraic multigrid (EAMG) method is proposed for solving large-scale linear systems arising from finite element discretization of second-order elliptic boundary value problems. Numerical experiments on polygonal domains demonstrate that the EAMG computation is more efficient than the standard method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Hong Zhu, Michael K. Ng, Guang-Jing Song
Summary: A new approximate method for solving nonnegative low-rank matrix approximation problem is developed in this study. It involves alternately projecting onto fixed-rank matrix manifold and nonnegative matrix manifold to ensure convergence, with numerical results demonstrating its performance.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Gwanghyun Jo, Do Y. Kwak, Young-Ju Lee
Summary: This paper introduces a locally conservative enriched immersed finite element method (EIFEM) for elliptic problems with interfaces, addressing the lack of conservative flux conservation in current methods under the IFEM framework. By introducing a local piecewise constant enrichment, locally conservative flux is provided, and an auxiliary space preconditioner based on algebraic multigrid method is constructed and analyzed for the resulting system. A new observation is made that imposing strong Dirichlet boundary conditions can remove zero eigen-modes of the EIFEM system while still weakly applying Dirichlet boundary conditions to the piecewise constant enrichment. Various issues related to the piecewise constant enrichment for mesh unfit to the interface are also discussed and clarified, with numerical tests confirming the theoretical developments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Alex Bespalov, David Silvester
Summary: This work recalls a multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data. The strategy extends the a posteriori error estimation framework to cover problems with a nonaffine parametric coefficient dependence. The results obtained using a potentially more efficient multilevel approximation strategy are discussed, showing that optimal convergence rates can be achieved for specific types of problems. The codes for generating the numerical results are available on GitHub.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Multidisciplinary
Zhibao Zheng, Hongzhe Dai
Summary: This paper introduces a new stochastic finite element method for computing structural stochastic responses, which decouples the stochastic response into deterministic responses with random variable coefficients through an iterative algorithm. The method is insensitive to stochastic dimensions and can be easily embedded into existing FEM structural analysis software, showing excellent performance in high-dimensional stochastic problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Yongle Hao, Siyu Liu, Lin Wang
Summary: This paper investigates the stochastic interface for diffraction grating by formulating the model as Helmholtz interface problems (HIPs) and using PML boundary for more accurate simulation. Shape-Taylor expansion for the solution of HIPs is developed, leading to approximate simulations of second and third order through perturbation method. Error estimation and efficient computation of solutions using low-rank approximation are provided, with the results illustrated through numerical simulations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Lukas Exl, Norbert J. Mauser, Sebastian Schaffer, Thomas Schrefl, Dieter Suess
Summary: A machine learning model is established for predicting the magnetization dynamics response to an external field. Through dimensionality reduction and kernel principal component analysis, accurate prediction of time steps is achieved. This approach reduces complexity in the learning process and can handle large training sample sets effectively.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Howard C. Elman, Minghao W. Rostami
IMA JOURNAL OF NUMERICAL ANALYSIS
(2016)
Article
Mathematics, Applied
Quan M. Bui, Howard C. Elman, J. David Moulton
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Kookjin Lee, Howard C. Elman
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Engineering, Multidisciplinary
Heyrim Cho, Howard Elman
INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION
(2018)
Article
Mathematics, Interdisciplinary Applications
Kookjin Lee, Kevin Carlberg, Howard C. Elman
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2018)
Article
Mathematics, Applied
Howard C. Elman, David J. Silvester
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2018)
Article
Computer Science, Interdisciplinary Applications
Quan M. Bui, Howard C. Elman
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Engineering, Multidisciplinary
Howard C. Elman, Tengfei Su
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2020)
Article
Computer Science, Interdisciplinary Applications
Howard C. Elman, Jiaxing Liang, Tonatiuh Sanchez-Vizuet
Summary: The equilibrium configuration of plasma in magnetic confinement fusion devices is determined by a balance of hydrostatic pressure, magnetic forces, and various uncertain physical parameters. Variations in current intensities through external coils are considered the dominant uncertainty source. A surrogate function and Monte Carlo strategy are used to explore the effects of parameter stochasticity on important plasma boundary features, reducing simulation time significantly.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Software Engineering
Kookjin Lee, Howard C. Elman, Catherine E. Powell, Dongeun Lee
Summary: This paper presents an efficient computational method for solving forward models of partial differential equations with uncertain coefficients. The method is based on an alternating energy minimization framework and improves the accuracy of the solution through enhanced procedures.
BIT NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Bedrich Sousedik, Howard C. Elman, Kookjin Lee, Randy Price
Summary: We study the linear stability of solutions to the Navier-Stokes equations with stochastic viscosity. Three surrogate models, namely generalized polynomial chaos, Gaussian process regression, and a shallow neural network, are used to analyze and compare the results with Monte Carlo simulation.
APPLICATIONS OF MATHEMATICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Kookjin Lee, Howard C. Elman, Bedrich Sousedik
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2019)
Article
Mathematics, Applied
Howard C. Elman, Tengfei Su
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2019)
Article
Engineering, Multidisciplinary
Howard C. Elman, Virginia Forstall
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2017)
