This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of the lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.
We present a geometric multigrid solver for the M1 model of radiative transfer without source terms. In radiative hydrodynamics applications, the radiative transfer needs to be solved implicitly because of the fast propagation speed of photons relative to the fluid velocity. The M1 model is hyperbolic and can be discretized with an HLL solver, whose time implicit integration can be done using a nonlinear Jacobi method. One can show that this iterative method always preserves the admissible states, such as positive radiative energy and reduced flux less than 1. To decrease the number of iterati…
The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation storage multigrid method. Starting from a second-order accurate centered finite difference approximation, we present a nonlinear Gauss-Seidel iterative method which has a mechanism for selecting the convex soluti…
It is well known that the choice of the iterative method is crucial in determining the speed of the converged solution. This article presents a detailed comparison between several iterative techniques for solving incmopressible Navier-Stokes equations. The numerical approaches implemented in the solver include Jacobi, Gauss-Siedel, Successive Over Relaxation, Alternating Direct Implicit and Multigrid methods. The results reveal that multigrid method is the most powerful iterative method among all other methods investigated in terms of the computational time and the number of iterations.
Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow solve and a pure advective transport solve with different right-hand sides. The new solver was compared with traditional preconditioned iterative methods and direct sparse solvers on several two- and three-dimensional benchmark problems spanning homogeneous and heterogeneous formations. For the groundwate…
In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmen…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our met…
A Nonlinear Multigrid Steady-State Solver for Microflow
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）
Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. The computationally most expensive components of the solver are the repeated applications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dram…
This article presents a systematic quantitative performance analysis for large finite element computations on extreme scale computing systems. Three parallel iterative solvers for the Stokes system, discretized by low order tetrahedral elements, are compared with respect to their numerical efficiency and their scalability running on up to $786\,432$ parallel threads. A genuine multigrid method for the saddle point system using an Uzawa-type smoother provides the best overall performance with respect to memory consumption and time-to-solution. The largest system solved on a Blue Gene/Q system h…
We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a ``fine-grain'' parallel algorithm for finding a maximal independent set from a graph of strong ne…
We present a monolithic geometric multigrid preconditioner for solving fluid-solid interaction problems in Stokes limit. The problems are discretized by a spatially adaptive high-order meshless method, the generalized moving least squares (GMLS) with adaptive $h$-refinement. In Stokes limit, solid kinematics can be dominated by the singularities governing the lubrication effects. Resolving those singularities with adaptive $h$-refinement can lead to an ill-conditioned linear system of equations. For constructing the interpolation and restriction operators - the key ingredients of the multigrid…
A Multigrid-like Algorithm for Probabilistic Domain Decomposition
Center for Mathematics and its Applications, Department of Mathematics, Instituto Superior Técnico（高等技术学院数学系数学及其应用中心）；ISCTE - Instituto Universitário de Lisboa Departamento de Ciências e Tecnologias de Informação（里斯本大学ISCTE信息与科技科学系）
Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find sol…
In this paper we present a new highly efficient calculation method for the far field amplitude pattern that arises from scattering problems governed by the d-dimensional Helmholtz equation and, by extension, Schrödinger's equation. The new technique is based upon a reformulation of the classical real-valued Green's function integral for the far field amplitude to an equivalent integral over a complex domain. It is shown that the scattered wave, which is essential for the calculation of the far field integral, can be computed very efficiently along this complex contour (or manifold, in multiple…
Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs), but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-…
This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations of the discretization and the algebraic error estimators, that are based on the respective duality error estimator and the difference of two consecutive iterates. Iterations are terminated when the algebraic estimator is of the same magnitude as the discretization estimator. Numerical results for multigrid $V(1,1)$-cycle and symmetric Gauss-Seidel iterative…
As CPU clock speeds have stagnated and high performance computers continue to have ever higher core counts, increased parallelism is needed to take advantage of these new architectures. Traditional serial time-marching schemes can be a significant bottleneck, as many types of simulations require large numbers of time-steps which must be computed sequentially. Parallel in Time schemes, such as the Multigrid Reduction in Time (MGRIT) method, remedy this by parallelizing across time-steps, and have shown promising results for parabolic problems. However, chaotic problems have proved more difficul…
Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University（湘潭大学湖南省科学与工程计算重点实验室）
Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid-transfer operators is crucial to the design of optimally efficient algorithms. It is well--known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes …
Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Run…
Combined Preconditioning with Applications in Reservoir Simulation
Kunming University of Science and Technology（昆明理工大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；PetroChina Company Limited（中国石油天然气股份有限公司）；ExxonMobil Upstream Research Company（埃克森美孚上游研究公司）；Academy of Mathematics and System Sciences（中国科学院数学与系统科学研究院）；Sichuan University（四川大学）
Kernel-learning parameter prediction and evaluation in algebraic multigrid method for several PDEs
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH（于利希超级计算中心，于利希研究中心有限公司）；Institute of Computational Science, Università della Svizzera italiana（意大利瑞士大学计算科学研究所）；Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室计算科学与工程中心）
This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the …
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
School of Mathematics, Jilin University（吉林大学数学学院）；Ocean University of China（中国海洋大学）；Shenzhen Loop Area Institute（深圳河套学院）；King Abdullah University of Science and Technology（卡布斯科学技术大学）
