A Parallel Solver for Graph Laplacians

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm

Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations

On Block Triangular Preconditioners for the Interior Point Solution of PDE-Constrained Optimization Problems

We consider the numerical solution of saddle point systems of equations resulting from the discretization of PDE-constrained optimization problems, with additional bound constraints on the state and control variables, using an interior point method. In particular, we derive a Bramble-Pasciak Conjugate Gradient method and a tailored block triangular preconditioner which may be applied within it. Crucial to the usage of the preconditioner are carefully chosen approximations of the (1,1)-block and Schur complement of the saddle point system. To apply the inverse of the Schur complement approximation, which is computationally the most expensive part of the preconditioner, one may then utilize methods such as multigrid or domain decomposition to handle individual sub-blocks of the matrix system

A Hybrid a-ϕ Cell Method for Solving Eddy-Current Problems in 3-D Multiply-Connected Domains

A hybrid a-Cell Method formulation for solving eddy-current problems in 3-D multiply-connected regions is presented. By using the magnetic scalar potential the number of degrees of freedom in the exterior domain with respect to the A,V-A formulation, typically implemented in commercial software for electromagnetic design, can be almost halved. On the other hand, the use of the magnetic vector potential in the interior domain improves the flexibility with respect to T-Omega formulation, since both conductive and magnetic parts can be easily modeled. By using a Cell Method variant, based on an augmented dual grid for discretization, electric and magnetic variables can be consistently coupled at the interface between interior and exterior domain. Global basis functions needed for representing the magnetic field in the insulating region are obtained by using for the first time iterative solvers relying on auxiliary space preconditioner and aggregation-based algebraic multigrid, with linear optimal complexity. These represent highly-efficient alternatives to traditional computational topology algorithms based on the concept of thick cut. As a result, an indefinite symmetric matrix system, amenable to fast iterative solution, is obtained. Numerical tests show high accuracy and fast convergence of the a method on test cases with complex topology. Computational cost for both matrix assembly and linear system solution is limited even for large problems. Comparisons show that the a-method provides better performance than existing methods such as A,V-A and h

A Hybrid a\u2013\u3c6 Cell Method for Solving Eddy\u2013Current Problems in 3\u2013D Multiply\u2013Connected Domains

A hybrid a\u2014\u3c6 Cell Method formulation for solving eddy\u2013current problems in 3\u2013D multiply\u2013connected regions is presented. By using the magnetic scalar potential the number of degrees of freedom in the exterior domain with respect to the A, V\u2014A formulation, typically implemented in commercial software for electromagnetic design, can be almost halved. On the other hand, the use of the magnetic vector potential in the interior domain improves the flexibility with respect to T\u2013\u3a9 formulation, since both conductive and magnetic parts can be easily modeled. By using a Cell Method variant, based on an augmented dual grid for discretization, electric and magnetic variables can be consistently coupled at the interface between interior and exterior domain. Global basis functions needed for representing the magnetic field in the insulating region are obtained by using for the first time iterative solvers relying on auxiliary space preconditioner and aggregation\u2013based algebraic multigrid, with linear optimal complexity. These represent highly\u2013efficient alternatives to traditional computational topology algorithms based on the concept of thick cut. As a result, an indefinite symmetric matrix system, amenable to fast iterative solution, is obtained. Numerical tests show high accuracy and fast convergence of the a\u2014\u3c6 method on test cases with complex topology. Computational cost for both matrix assembly and linear system solution is limited even for large problems. Comparisons show that the a\u2014\u3c6 method provides better performance than existing methods such as A, V\u2014A and h\u2014\u3c6

Algebraic multigrid for moderate order finite elements

We investigate the use of algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is required by standard analyses of classical AMG and aggregation-based AMG methods. A unified approach is presented that allows us to extend these analyses. It uses an intermediate M-matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix. This constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations, provided that jumps are located at elements' boundaries. For two-dimensional problems, it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound. This analysis validates the application of the AMG methods to the considered problems. On the other hand, because the intermediate M-matrix can be computed automatically, an alternative strategy is to define the AMG preconditioners from this matrix, instead of defining them from the original matrix. Numerical experiments are presented that assess both strategies using publicly available state-of-theart implementations of classical AMG and aggregation-based AMG methods.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Fast Solution of 3-D Eddy-Current Problems in Multiply Connected Domains by a, v-phi and t-phi Formulations With Multigrid-Based Algorithm for Cohomology Generation

The fast solution of three-dimensional eddy current problems is still an open problem, especially when real-size finite element models with millions of degrees of freedom are considered. In order to lower the number of degrees of freedom a magnetic scalar potential can be used in the insulating parts of the model. This may become difficult when the model geometry presents some conductive parts which are multiply connected. In this work a multigrid-based algoritm is proposed that allows for a calculation in linear-time of cohomology, which is needed to introduce the scalar potential without cuts. This algorithm relies on an algebraic multigrid solver for curl-curl field problems, which ensures optimal computational complexity. Numerical results show that the novel algorithm outperforms state-of-the-art methods for cohomology generation based on homological algebra. In addition, based on this algoritm, novel a, v-phi and t-phi formulations to analyze three-dimensional eddy current problems in multiply connected domains are proposed. Both formulations, after discretization by the cell method, lead to a complex symmetric system of linear equations amenable to fast iterative solution by Krylov-subspace solvers. These formulations are able to provide very accurate numerical results, with a minimum amount of degrees of freedoms to represent the eddy current model. In this way the computational performance is improved compared to the classical A, V-A formulation typically implemented in finite element software for electromagnetic design

Smoothing factor, order of prolongation and actual multigrid convergence

We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V-cycle multigrid. We derive a two-sided bound that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow and away from 1 when both the smoothing factor and an additional parameter are small enough. Besides the smoothing factor, the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix associated with small eigenvalues. Nice V-cycle convergence is guaranteed if the tangent of this angle has an upper bound proportional to the eigenvalue, whereas nice two-grid convergence requires a bound proportional to the square root of the eigenvalue. We also discuss the well-known rule which relates the order of the prolongation to that of the differential operator associated to the problem. We first define a frequency based order which in most cases amounts to the so-called high frequency order as defined in Hemker (J Comput Appl Math 32:423-429, 1990). We give a firmer basis to the related order rule by showing that, together with the requirement of having the smoothing factor away from one, it provides necessary and sufficient conditions for having the two-grid convergence rate away from 1. A stronger condition is further shown to be sufficient for optimal convergence with the V-cycle. The presented results apply to rigorous Fourier analysis for regular discrete PDEs, and also to local Fourier analysis via the discussion of semi-positive systems as may arise from the discretization of PDEs with periodic boundary conditions. © 2011 Springer-Verlag.SCOPUS: ar.jinfo:eu-repo/semantics/publishe