Article
Engineering, Multidisciplinary
Bedrich Sousedik
Summary: In this paper, we propose a framework to approximately solve the coarse problem in the FETI-DP method by utilizing the multilevel BDDC method as the main tool. We demonstrate that the spectra of the multilevel FETI-DP and BDDC preconditioned operators are essentially the same and provide numerical experiments to support our theory.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2022)
Article
Mathematics, Applied
Ngoc Mai Monica Huynh, Luca F. Pavarino, Simone Scacchi
Summary: This paper constructs two novel parallel solvers and optimizes them using deluxe scaling. Through numerical experiments and theoretical analysis, it is proved that these two solvers are scalable and quasi-optimal.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Multidisciplinary
Serafeim Bakalakos, Manolis Georgioudakis, Manolis Papadrakakis
Summary: The extended finite element method (XFEM) has been successfully applied to solve crack propagation problems without remeshing. However, the enrichment in XFEM leads to ill-conditioned algebraic systems and slow convergence of iterative solvers. In this paper, two efficient domain decomposition solvers, FETI-DP and P-FETI-DP, are proposed for large-scale 3D XFEM crack propagation analysis, offering parallelization and reduced computational time.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
O. B. Widlund, S. Zampini, S. Scacchi, L. F. Pavarino
Summary: A block FETI-DP/BDDC preconditioner is constructed and analyzed for mixed formulations of almost incompressible elasticity. The strategy combines mixed isogeometric analysis with continuous pressure fields and deluxe scaling algorithms. Parallel numerical experiments validate the theory and demonstrate robustness in various scenarios.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Alexander Heinlein, Axel Klawonn, Martin Lanser, Janine Weber
Summary: The hybrid ML-FETI-DP algorithm combines adaptive coarse spaces and neural networks to improve solver robustness. Extending to three dimensions requires complex data preprocessing and representative training data. Numerical experiments show significant savings in the number of eigenvalue problems.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Alexander Heinlein, Axel Klawonn, Martin Lanser
Summary: This article explores different nonlinear domain decomposition methods for solving nonlinear problems with highly heterogeneous coefficient functions with jumps. The use of adaptive coarse spaces is employed to achieve robust solvers for both nonlinear and linear convergence. The results show that combining the nonlinear domain decomposition methods with adaptive coarse spaces leads to the best linear and nonlinear convergence, as compared to classical coarse spaces and Newton-Krylov methods with adaptive coarse spaces.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Xuemin Tu, Jinjin Zhang
Summary: A preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with HDG discretization. The use of BDDC as a preconditioner showed promising results, with the number of iterations being independent of subdomain quantity for large viscosity. However, convergence deteriorates with decreasing viscosity, similar to standard finite element discretizations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Engineering, Multidisciplinary
Xiao Xu, Christian Glusa, Marta D'Elia, John T. Foster
Summary: A domain decomposition method is proposed for efficient simulation of nonlocal problems, utilizing a multi-domain formulation with nonlocal interfaces. A distributed projected gradient algorithm is used to solve the Lagrange multiplier system, demonstrating high scalability and outperforming standard approaches.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Engineering, Multidisciplinary
Zouhair Samir, Ludovic Chamoin, Mickael Abbas
Summary: The modified constitutive relation error is a commonly used functional in structural dynamics for model updating. It involves a two-step iterative process to minimize the functional by localizing regions with high modeling errors and updating their associated parameters. The proposed domain decomposition formulation makes the method more flexible, efficient, and suitable for industrial models.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Jie Peng, Shi Shu, Junxian Wang, Liuqiang Zhong
Summary: An adaptive BDDC preconditioner is proposed for advection-diffusion problems, which extends the method to solve nonsymmetric and positive definite bilinear forms. By decomposing the original form and designing local generalized eigenvalue problems, adaptive coarse components are formed for improved performance. Published by Elsevier B.V. with all rights reserved.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Y. El Gharbi, A. Parret-Freaud, C. Bovet, P. Gosselet
Summary: This paper introduces a new parallel mesh generation method aimed at creating subdomains well-suited for Schur based domain decomposition methods, with a focus on limiting pathological situations to improve convergence. The method distributes and parallelizes the mesh generation step in the early phases, and has been evaluated on various test cases.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2021)
Article
Engineering, Multidisciplinary
Shourong Hao, Yongxing Shen
Summary: In this article, an efficient parallel explicit-implicit solution scheme for the phase field approach to dynamic fracture is proposed. The scheme updates the displacement field using an explicit algorithm and solves the phase field using an implicit algorithm. Lagrange multipliers are introduced to ensure interface continuity. The scheme has advantages in terms of computational cost and size, and is flexible in terms of code extensibility.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Jinjin Zhang, Xuemin Tu
Summary: This study applies the balancing domain decomposition by constraints (BDDC) methods to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. It enforces edge/face average constraints across the subdomain interface to ensure that the BDDC preconditioned conjugate gradient iterations stay in a special subspace. Numerical experiments confirm the theory.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Engineering, Multidisciplinary
Gopika Ajith, Debraj Ghosh
Summary: The numerical solution of stochastic partial differential equations often faces the challenge of high dimensionality. Recently, domain decomposition methods have successfully reduced computational complexity and achieved parallelization. To make this approach more amenable to parallel implementation, a non-intrusive formulation based on stochastic collocation method is proposed, which uses collocation method at both subdomain and interface levels. The method shows significant improvement in speed-up and scalable parallel performance.
INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION
(2022)
Article
Mathematics, Applied
Monica Montardini, Giancarlo Sangalli, Rainer Schneckenleitner, Stefan Takacs, Mattia Tani
Summary: In this paper, we develop solvers for an isogeometric multi-patch discretization using discontinuous Galerkin approach. We solve the resulting linear system using the Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We approximate the patch-local problems using the Fast Diagonalization method and introduce an orthogonal splitting of the local function spaces to obtain the tensor structure needed for the Fast Diagonalization method. The numerical experiments confirm the effectiveness of our approach and show good performance in both two-dimensional and three-dimensional problems.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Duk-Soon Oh, Olof B. Widlund, Stefano Zampini, Clark R. Dohrmann
MATHEMATICS OF COMPUTATION
(2018)
Article
Mathematics, Applied
Gustavo Chavez, George Turkiyyah, Stefano Zampini, David Keyes
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
L. F. Pavarino, S. Scacchi, O. B. Widlund, S. Zampini
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2018)
Article
Engineering, Chemical
Giuseppe Genduso, Eric Litwiller, Xiaohua Ma, Stefano Zampini, Ingo Pinnau
JOURNAL OF MEMBRANE SCIENCE
(2019)
Article
Computer Science, Interdisciplinary Applications
Lisandro Dalcin, Diego Rojas, Stefano Zampini, David C. Del Rey Fernandez, Mark H. Carpenter, Matteo Parsani
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Computer Science, Interdisciplinary Applications
Irving Reyna Nolasco, Lisandro Dalcin, David C. Del Rey Fernandez, Stefano Zampini, Matteo Parsani
COMPUTERS & FLUIDS
(2020)
Article
Mathematics, Applied
Robert Anderson, Julian Andrej, Andrew Barker, Jamie Bramwell, Jean-Sylvain Camier, Jakub Cerveny, Veselin Dobrev, Yohann Dudouit, Aaron Fisher, Tzanio Kolev, Will Pazner, Mark Stowell, Vladimir Tomov, Ido Akkerman, Johann Dahm, David Medina, Stefano Zampini
Summary: MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. Its goal is to provide access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to develop and test new algorithms in general, unstructured, high-order, parallel and GPU-accelerated settings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Matteo Parsani, Radouan Boukharfane, Irving Reyna Nolasco, David C. Del Rey Fernandez, Stefano Zampini, Bilel Hadri, Lisandro Dalcin
Summary: This study presents a fully-discrete hp-adaptive entropy stable discontinuous collocated Galerkin method for the compressible Navier-Stokes equations, utilizing the SSDC framework. The method demonstrates high-order numerical performance and systematic design, showcasing its potential as a base scheme for future unstructured computational fluid dynamics tools. Results indicate efficient scaling of the parallel SSDC solver over 100,000 processes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Pierre Jolivet, Jose E. Roman, Stefano Zampini
Summary: This paper explains the interfacing of PETSc and HPDDM libraries to provide robust preconditioners and advanced Krylov methods for solving linear systems of different structures. The flexibility of the implementation is showcased through minimalist examples covering various application domains.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
O. B. Widlund, S. Zampini, S. Scacchi, L. F. Pavarino
Summary: A block FETI-DP/BDDC preconditioner is constructed and analyzed for mixed formulations of almost incompressible elasticity. The strategy combines mixed isogeometric analysis with continuous pressure fields and deluxe scaling algorithms. Parallel numerical experiments validate the theory and demonstrate robustness in various scenarios.
MATHEMATICS OF COMPUTATION
(2021)
Article
Computer Science, Theory & Methods
Richard Tran Mills, Mark F. Adams, Satish Balay, Jed Brown, Alp Dener, Matthew Knepley, Scott E. Kruger, Hannah Morgan, Todd Munson, Karl Rupp, Barry F. Smith, Stefano Zampini, Hong Zhang, Junchao Zhang
Summary: The PETSc library offers scalable solvers for solving differential and algebraic equations as well as numerical optimization, addressing basic GPU accelerator challenges. The design of PETSc emphasizes performance portability, flexibility, and extensibility, allowing application developers to use their preferred programming model on different heterogeneous computing systems.
PARALLEL COMPUTING
(2021)
Article
Mathematics, Applied
Stefano Zampini, Wajih Boukaram, George Turkiyyah, Omar Knio, David Keyes
Summary: This paper presents high-performance, distributed-memory GPU-accelerated algorithms and implementations for matrix-vector multiplication and matrix recompression of hierarchical matrices in the H(2)format. The algorithms demonstrate near-ideal scalability up to 1024 NVIDIA V100 GPUs on Summit, with performance exceeding 2.3 Tflop/s/GPU for matrix-vector multiplication, and 670 Gflop/s/GPU for matrix compression. The flexibility and efficiency of the library are illustrated through solving a 2D variable diffusivity integral fractional diffusion problem with scalability up to 16M degrees of freedom problems on 64 GPUs.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Franco Dassi, Stefano Zampini, S. Scacchi
Summary: This paper introduces the Virtual Element Method (VEM) and its linear solver for solving three-dimensional elliptic equations. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner effectively controls the condition number of the system. Experimental results confirm the reliability and adaptability of the method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Computer Science, Theory & Methods
Junchao Zhang, Jed Brown, Satish Balay, Jacob Faibussowitsch, Matthew Knepley, Oana Marin, Richard Tran Mills, Todd Munson, Barry F. Smith, Stefano Zampini
Summary: PetscSF is the communication component of PETSc designed for exascale computers utilizing GPUs and other accelerators, providing a simple API for managing communication patterns in scientific computations. It supports various implementations based on MPI and NVSHMEM, essential for implementing large-scale applications.
IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
(2022)
Article
Mathematics, Applied
Ilona Ambartsumyan, Wajih Boukaram, Tan Bui-Thanh, Omar Ghattas, David Keyes, Georg Stadler, George Turkiyyah, Stefano Zampini
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
