Numerical methods based on interval arithmetic are efficient means to reliably solve nonlinear systems of equations. Algorithm bc3revise is an interval method that tightens variables' domains by enforcing a property called box consistency. It has been successfully used on difficult problems whose solving eluded traditional numerical methods. We present a new algorithm to enforce box consistency that is simpler than bc3revise, faster, and easily data parallelizable. A parallel implementation with Intel SSE2 SIMD instructions shows that an increase in performance of up to an order of magnitude and more is achievable.
We propose and analyze a Vanka-type multigrid solver for solving a sequence of complex-shifted Laplacian systems arising in diagonalization-based parallel-in-time algorithms for evolutionary equations. Under suitable assumption, local Fourier analysis shows the proposed Vanka-type smoother achieves a uniform smoothing factor, which is verified by several numerical examples.
The DD-CPM software library provides a set of tools for the discretization and solution of problems arising from the closest point method (CPM) for partial differential equations on surfaces. The solvers are built on top of the well-known PETSc framework, and are supplemented by custom domain decomposition (DD) preconditioners specific to the CPM. These solvers are fully compatible with distributed memory parallelism through MPI. This library is particularly well suited to the solution of elliptic and parabolic equations, including many reaction-diffusion equations. The software is detailed he…
The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell equations and compare different preconditioners. We show numerically that standard preconditioners like incomplete LU and the additive Schwarz method lead to slow convergence for iterative solvers like generalized minimal residuals, especially for high wave numbers. Even more we show that also more specialized methods like the Schur complement method also yield…
A parallel solver for a preconditioned space-time boundary element method for the heat equation
Graz University of Technology（格拉茨工业大学）；Institute of Applied Mathematics, Graz University of Technology（格拉茨工业大学应用数学研究所）；VŠB – Technical University of Ostrava（俄斯特拉发理工大学）；IT4Innovations, VŠB – Technical University of Ostrava（俄斯特拉发理工大学IT4创新研究院）；Department of Applied Mathematics, VŠB – Technical University of Ostrava（俄斯特拉发理工大学应用数学系）
Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A fin…
This paper describes in detail the implementation of a finite element technique for solving the compressible Navier-Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is second-order accurate in time and space. Robustness here means that the method is proved to be invariant domain preserving under the hyperbolic CFL time step restriction, and the method delivers results that are reproducible. The proposed technique is shown to be accurate on challenging 2D and 3D realistic benchmarks.
In this paper, we study numerical methods for solving the coupled Allen-Cahn/Cahn-Hilliard system associated with a free energy functional of logarithmic type. To tackle the challenge posed by the special free energy functional, we propose a method to approximate the discrete variational derivatives in polynomial forms, such that the corresponding finite difference scheme is unconditionally energy stable and the energy dissipation law is maintained. To further improve the performance of the algorithm, a modified adaptive time stepping strategy is adopted such that the time step size can be fle…
Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, "predicting" the solution serially using a cheap (coarse) solver and "correcting" these values using an expensive (fine) solver that runs in parallel on a number of temporal subintervals. In this work, we propose a time-parallel algorithm (GParareal) that solves IVPs by modelling the correction term, i.e. the difference between fine and coarse solutions, using…
Numerical solutions to the Eikonal equation are computed using variants of the fast marching method, the fast sweeping method, and the fast iterative method. In this paper, we provide a unified view of these algorithms that highlights their similarities and suggests a wider class of Eikonal solvers. We then use this framework to justify applying concurrent priority scheduling techniques to Eikonal solvers. We demonstrate that doing so results in good parallel performance for a problem from seismology. We explain why existing Eikonal solvers may produce different results despite using the same …
In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners…
High-performance Parallel Solver for Integral Equations of Electromagnetics Based on Galerkin Method
SUNY New Paltz（纽约州立大学新帕尔茨分校）；Faculty of Computational Mathematics and Cybernetics, Lomonosov MSU（莫斯科国立大学计算数学与控制论系）；Institute of Geophysics, ETH Zurich（苏黎世联邦理工学院地球物理研究所）
A space-time parallel solver for the three-dimensional heat equation
Forschungszentrum Jülich（于利希研究中心）；Università della Svizzera italiana（瑞士意大利大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Bergische Universität Wuppertal（伍珀塔尔大学）
本文提出将时间并行的PFASST方法与空间并行的多网格方法结合，实现三维热方程的高效并行求解，通过在Cray XE6和IBM Blue Gene/Q上进行并行扩展测试，展示了空间和时间并行化的有效性。
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-to-co…
Optimally hybrid numerical solvers were constructed for massively parallel generalized eigenvalue problem (GEP).The strong scaling benchmark was carried out on the K computer and other supercomputers for electronic structure calculation problems in the matrix sizes of M = 10^4-10^6 with upto 105 cores. The procedure of GEP is decomposed into the two subprocedures of the reducer to the standard eigenvalue problem (SEP) and the solver of SEP. A hybrid solver is constructed, when a routine is chosen for each subprocedure from the three parallel solver libraries of ScaLAPACK, ELPA and EigenExa. Th…
Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We describe the specific choices made in the implementation of the explicit and implicit extrapolation methods which allow for generating low overhead static schedules to then exploit with optimized multi-threaded implementations. We demonstrate that wh…
