Algebraic Preconditioners Research Articles

Parallel Sparse Computation Toolkit

This paper presents a new software framework for solving large and sparse linear systems on current hybrid architectures, from small servers to high-end supercomputers, embedding multi-core CPUs and Nvidia GPUs at the node level. The framework has a modular structure and is composed of three main components, which separate basic functionalities for managing distributed sparse matrices and executing some sparse matrix computations involved in iterative Krylov projection methods, eventually exploiting multi-threading and CUDA-based programming models, from the functionalities for setup and application of different types of one-level and multi-level algebraic preconditioners.

Accelerating Certifiable Estimation with Preconditioned Eigensolvers

Convex (specifically semidefinite) relaxation provides a powerful approach to constructing robust machine perception systems, enabling the recovery of certifiably globally optimal solutions of challenging estimation problems in many practical settings. However, solving the large-scale semidefinite relaxations underpinning this approach remains a formidable computational challenge. A dominant cost in many state-of-the-art (Burer-Monteiro factorization-based) certifiable estimation methods is solution verification (testing the global optimality of a given candidate solution), which entails computing a minimum eigenpair of a certain symmetric certificate matrix. In this letter, we show how to significantly accelerate this verification step, and thereby the overall speed of certifiable estimation methods. First, we show that the certificate matrices arising in the Burer-Monteiro approach generically possess spectra that make the verification problem expensive to solve using standard iterative eigenvalue methods. We then show how to address this challenge using preconditioned eigensolvers; specifically, we design a specialized solution verification algorithm based upon the locally optimal block preconditioned conjugate gradient (LOBPCG) method together with a simple yet highly effective algebraic preconditioner. Experimental evaluation on a variety of simulated and real-world examples shows that our proposed verification scheme is very effective in practice, accelerating solution verification by up to 280x, and the overall Burer-Monteiro method by up to 16x, versus the standard Lanczos method when applied to relaxations derived from large-scale SLAM benchmarks.

Multigrid reduction preconditioning framework for coupled processes in porous and fractured media

Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Because of the wide range of processes and couplings that may be involved – e.g. formation and propagation of fractures, deformation of the solid porous medium, viscous flow of one or more fluids in the pores and fractures, complicated well sources and sinks, etc. – it is difficult to develop general-purpose but scalable linear solver frameworks. This challenge is further aggravated by the range of different discretization schemes that may be adopted, which have a direct impact on the linear system structure. To address this obstacle, we describe a flexible strategy based on multigrid reduction (MGR) that can produce purely algebraic preconditioners for a wide spectrum of relevant physics and discretizations. We demonstrate that MGR, guided by physics and theory in block preconditioning, can tackle several distinct and challenging problems, notably: a hybrid discretization of single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations. Extension to other systems can be handled quite naturally. We demonstrate the efficiency and scalability of the resulting solvers through numerical examples of difficult, field-scale problems.

Construction of a Preconditioner for General Elliptic Problems Using Riesz Map

The current work aspires to design and study the construction of an efficient preconditioner for linear symmetric systems in a Hilbert space setting. Compliantly to Josef Málek and Zdeněk Strakoš’s work [Preconditioning and the Conjugate Gradient Method in the Context of Solving[Formula: see text] PDEs, Vol. 1 (SIAM, USA).], we shed new light on the dependence of algebraic preconditioners with the resolution steps of partial differential equations (PDEs) and describe their impact on the final numerical solution. The numerical strength and efficiency of the proposed approach is demonstrated on a two-dimensional examples.

On parallel algebraic multilevel preconditioner with approximate inversion by basis matrix method

On parallel algebraic multilevel preconditioner with approximate inversion by basis matrix method

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix–vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges.

A Charge-Conservative Finite Element Method for Inductionless MHD Equations. Part II: A Robust Solver

In [L. Li, M. Ni, and W. Zheng, SIAM J. Sci. Comput., 41 (2019), pp. B796--B815] a charge-conservative finite element method is proposed for solving inductionless and incompressible magnetohydrodynamic (MHD) equations. The purpose of this paper is to propose a robust solver for the discrete problem. Using the framework of field-of-values-equivalence, we first study the preconditioned Krylov space method for the continuous problem in the setting of Hilbert spaces. The algebraic preconditioner for the discrete problem is then obtained by representing the preconditioner for the continuous problem in finite element spaces. By three numerical examples, the optimality of the solver to the number of unknowns is demonstrated for both stationary and time-dependent MHD problems.

A Scalable Preconditioner for a Primal Discontinuous Petrov--Galerkin Method

We show how a scalable preconditioner for the primal discontinuous Petrov--Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system arising from the DPG method. One of our intermediate results shows that a generic stable decomposition implies a stable decomposition for the Schur complement. This justifies the application of algebraic solvers directly to the interface degrees of freedom. Combining such results, we obtain the first massively scalable algebraic preconditioner for the DPG system.

Improved Algebraic Preconditioning for Multiscale Electromagnetic Problems

An improved scheme for the algebraic preconditioners is proposed in this letter to accelerate the iterative convergence rate of the multiscale electromagnetic problems, in which the overcrowded leaf boxes of the multilevel fast multipole algorithm are recursively bisected with the application of the multilevel accelerated Cartesian expansion algorithm to maintain the computational efficiency. Due to a lack of some strong couplings measured by the relative distance between the testing/basis functions, the algebraic preconditioners constructed by filtering the near-field interaction matrix of the fast algorithm are consequently ineffective. To alleviate the adverse effect of the multiscale-size boxes, the proposed scheme constructs the preconditioners on higher levels rather than the leaf level of the related fast algorithm, thus speeding up the iterative convergence rate significantly. Moreover, the rank-revealing adaptive cross approximation algorithm is first introduced to compute and store the enlarged computational domain of the preconditioner. As a result, there is only a modest increase in the computational cost. Numerical experiments have been performed to demonstrate the effectiveness and efficiency of the proposed method.

Parallel subdomain solver strategies for the algebraic additive Schwarz preconditioner

Hybrid domain-decomposition method with two levels of domain partitioning.Solution of the local problem by ShyLU or other parallel approximation techniques.Proposal of an inexact QR factorization to be used within ShyLU.Comparisons of parallel performances between UMFPACK, ShyLU, and inexact QR. Domain-decomposition (DD) methods are used in most, if not all, modern parallel implementations of finite element modeling software. In the solver stage, the algebraic additive Schwarz (AAS) domain-decomposition preconditioner represents a fundamental component and its performance and scalability are key to the overall performance of the solution process. The established approach to construct the preconditioner in a parallel MPI setting is with a 1-to-1 correspondence between the number of MPI processes and the number of AAS subdomains.In this paper, we describe our attempts to extend this paradigm with the possibility to assign more than one MPI process per AAS subdomain, with the goal of improving the overall performance of the AAS preconditioner on supercomputers with multicore nodes.We discuss the implementation of the new AAS preconditioner framework, based on two levels of MPI parallelism, and the performance of different subdomain solver strategies. Finally, we examine the behavior of our novel approach for a series of benchmark problems, performed with the LifeV parallel finite element library.

A Mixed Potential MLFMA for Higher Order Moment Methods With Application to the Generalized Method of Moments

The application of the multilevel fast multipole algorithm (MLFMA) to higher order moment method discretizations is a continuing open problem. Herein, we present a point-based mixed potential variant of the MLFMA algorithm that exhibits an MLFMA tree of arbitrary height with no restriction related to basis function support size, efficient nearfield precomputation, and that maintains favorable scaling for any mixture of low- and high-order bases. The flexibility of the algorithm is also leveraged to accelerate the precomputation of algebraic preconditioners. We demonstrate the method through application to the generalized method of moments (GMM), a recently introduced moment method discretization capable of combining both low- and high-order bases and geometries in the same simulation; however, the method may be used to accelerate other higher order moment methods as well.

Efficient solution of time‐domain boundary integral equations arising in sound‐hard scattering

SummaryWe consider the efficient numerical solution of the three‐dimensional wave equation with Neumann boundary conditions via time‐domain boundary integral equations. A space‐time Galerkin method with C∞‐smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time. Copyright © 2015 John Wiley & Sons, Ltd.

Hierarchical Bases Preconditioner to Enhance Convergence of CFIE With Multiscale Meshes

A hierarchical quasi-Helmholtz decomposition, originally developed to address the low-frequency and dense-discretization breakdowns for the EFIE, is applied together with an algebraic preconditioner to improve the convergence of the CFIE in multiscale problems. The effectiveness of the proposed method is studied first on some simple examples; next, test on real-life cases up to several hundreds wavelengths show its good performance.

A preconditioned dual–primal finite element tearing and interconnecting method for solving three-dimensional time-harmonic Maxwell's equations

A new preconditioned dual–primal nonoverlapping domain decomposition method is proposed for the finite element solution of three-dimensional large-scale electromagnetic problems. With the aid of two Lagrange multipliers, the new method converts the original volumetric problem to a surface problem by using a higher-order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation. Similar to the previous version, a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration, which further increases the rate of convergence. In addition, a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable. Perfectly matched layers (PMLs) are considered for the boundary truncation when solving open-region problems. The influence of the PML truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the PML. Numerical examples including wave propagation and antenna radiation problems truncated with PMLs are presented to demonstrate the validity and the capability of this method.

Improved Algebraic Preconditioning for MoM Solutions of Large-Scale Electromagnetic Problems

Algebraic preconditioners in method of moments (MoM) are often constructed from the filtered near-field interaction (NFI) matrix. However, the criterion for filtering NFI matrix is an art rather than science, and the resulting preconditioner suffers seriously from the lack of global interaction information. The proposed methodology is aimed to relax these limitations by the skeletonization. Numerical experiments have been performed to show efficiency and effectiveness of the proposed methodology.

Application of auxiliary space preconditioning in field-scale reservoir simulation

We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation. These methods are discussed in the framework of the auxiliary space preconditioning method for generality. Unlike in the case of classical algebraic preconditioning methods, we take several analytical and physical considerations into account. In addition, we choose appropriate auxiliary problems to design the robust solvers herein. More importantly, our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators. We test the efficiency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems. The numerical results show that our methods are both efficient and robust for large reservoir models.

Enhanced Block FSAI Preconditioning Using Domain Decomposition Techniques

Adaptive block factorized sparse approximate inverse (FSAI) (ABF) is a novel algebraic preconditioner for the cost-effective parallel solution of symmetric positive definite linear systems. However, a main drawback stems from its reduced scalability, as the iteration count to convergence tends to grow when the number of processors increases. A domain decomposition Schur complement approach can enhance both the ABF performance and scalability. It is demonstrated that the enhanced ABF preconditioner is superior to the native block FSAI, reducing at the same time the construction and communication computational burden. Numerical results from different large size applications show that the proposed algorithm can improve significantly the preconditioner, allowing for its efficient use in massively parallel simulations as well.

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper.

Improving parallel scalability for edge plasma transport simulations with neutral gas species

Simulating the transport of multi-species plasma and neutral species in the edge region of a tokamak magnetic fusion energy device is computationally intensive and difficult due to coupling among various components, strong nonlinearities and a broad range of temporal scales. In addition to providing boundary conditions for the core plasma, such models aid in the understanding and control of the associated plasma/material–wall interactions, a topic that is essential for the development of a viable fusion power plant. The governing partial differential equations are discretized to form a large nonlinear system that typically must be evolved in time to obtain steady-state solutions. Fully implicit techniques using preconditioned Jacobian-free Newton–Krylov methods with parallel domain-based preconditioners are shown to be robust and efficient for the plasma components. Inclusion of neutral gas components, however, increases the condition number of the system to the point where improved parallel preconditioning is needed. Standard algebraic preconditioners that provide sufficient coupling throughout the global domain to handle the neutrals are not generally scalable. We present a new preconditioner, termed FieldSplit, which exploits the character of the neutral equations to improve the scalability of the combined plasma/neutral system.

PyTrilinos: Recent Advances in the Python Interface to Trilinos

PyTrilinos is a set of Python interfaces to compiled Trilinos packages. This collection supports serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, algebraic and multilevel preconditioners, nonlinear solvers and continuation algorithms, eigensolvers and partitioning algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms and matrix generation. PyTrilinos vector objects are compatible with the popular NumPy Python package. As a Python front end to compiled libraries, PyTrilinos takes advantage of the flexibility and ease of use of Python, and the efficiency of the underlying C++, C and Fortran numerical kernels. This paper covers recent, previously unpublished advances in the PyTrilinos package.