Alexandros Markopoulos

Cholesky decomposition of a positive semidefinite matrix with known kernel

The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments.

A scalable TFETI algorithm for two-dimensional multibody contact problems with friction

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid of rigid body motions is adapted to the solution of two-dimensional multibody contact problems of elasticity with the Coulomb friction and proved to be scalable for the Tresca friction. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on analysis of the yielding clamp connection with the Coulomb friction.

On the Moore–Penrose inverse in solving saddle-point systems with singular diagonal blocks‡

SUMMARY This paper deals with the role of the generalized inverses in solving saddle-point systems arising naturally in the solution of many scientific and engineering problems when finite-element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems. Copyright

Massively Parallel Hybrid Total FETI (HTFETI) Solver

This paper describes the Hybrid Total FETI (HTFETI) method and its parallel implementation in the ESPRESO library. HTFETI is a variant of the FETI type domain decomposition method in which a small number of neighboring subdomains is aggregated into clusters. This can be also viewed as a multilevel decomposition approach which results into a smaller coarse problem - the main scalability bottleneck of the FETI and FETI-DP methods. The efficiency of our implementation which employs hybrid parallelization in the form of MPI and Cilk++ is evaluated using both weak and strong scalability tests. The weak scalability of the solver is shown on the 3 dimensional linear elasticity problem of a size up to 30 billion of Degrees Of Freedom (DOF) executed on 4096 compute nodes. The strong scalability is evaluated on the problem of size 2.6 billion DOF scaled from 1000 to 4913 compute nodes. The results show the super-linear scaling of the single iteration time and linear scalability of the solver runtime. The latter combines both numerical and parallel scalability and shows overall HTFETI solver performance. The large scale tests use our own parallel synthetics benchmark generator that is also described in the paper. The last set of results shows that HTFETI is very efficient for problems of size up 1.7 billion DOF and provide better time to solution when compared to TFETI method.

Massively parallel solution of elastoplasticity problems with tens of millions of unknowns using PermonCube and FLLOP packages

In this paper we are presenting our PermonCube and FLLOP packages, and their use for massively parallel solution of elastoplasticity problems. PermonCube provides simple cubical meshes, partitioned in a non-overlapping manner. By means of finite element method it assembles all linear algebra objects required for solution of the physical problem. Two chosen nonlinear material models are presented, and a solving strategy based on the Newtons method is briefly discussed. PermonCube uses our FLLOP library as a linear system solver. FLLOP is able to solve problems decomposed in a non-overlapping manner using domain decomposition methods of the FETI type. It extends PETSc (Portable, Extensible Toolkit for Scientific Computation). In the last section, large-scale numerical experiments with problem size up to 60 million of degrees of freedom are presented.

Intel Xeon Phi acceleration of Hybrid Total FETI solver

Abstract This paper describes an approach for acceleration of the Hybrid Total FETI (HTFETI) domain decomposition method using the Intel Xeon Phi coprocessors. The HTFETI method is a memory bound algorithm which uses sparse linear BLAS operations with irregular memory access pattern. The presented local Schur complement (LSC) method has regular memory access pattern, that allows the solver to fully utilize the Intel Xeon Phi fast memory bandwidth. This translates to speedup over 10.9 of the HTFETI iterative solver when solving 3 billion unknown heat transfer problem (3D Laplace equation) on almost 400 compute nodes. The comparison is between the CPU computation using sparse data structures (PARDISO sparse direct solver) and the LSC computation on Xeon Phi. In the case of the structural mechanics problem (3D linear elasticity) of size 1 billion DOFs the respective speedup is 3.4. The presented speedups are asymptotic and they are reached for problems requiring high number of iterations (e.g., ill-conditioned problems, transient problems, contact problems). For problems which can be solved with under hundred iterations the local Schur complement method is not optimal. For these cases we have implemented sparse matrix processing using PARDISO also for the Xeon Phi accelerators.

Implementation of the efficient communication layer for the highly parallel total FETI and hybrid total FETI solvers

Implementation, performance, and scalability results of communication layer for Total FETI and Hybrid Total FETI solver.In HTFETI several neighboring subdomains aggregated into clusters. This reduces the size of coarse problem and improves scalability.Optimization of nearest neighbor communication - global gluing matrix.Implementation of communication hiding and avoiding techniques inside the communication layerBenchmarks - elastic 3D cube up to 1.6 billion DOF and realistic car engine benchmark.Large test executed on Total FETI to see the real potential of communication layer on smaller clusters. This paper describes the implementation, performance, and scalability of our communication layer developed for Total FETI (TFETI) and Hybrid Total FETI (HTFETI) solvers. HTFETI is based on our variant of the Finite Element Tearing and Interconnecting (FETI) type domain decomposition method. In this approach a small number of neighboring subdomains is aggregated into clusters, which results in a smaller coarse problem. To solve the original problem TFETI method is applied twice: to the clusters and then to the subdomains in each cluster.The current implementation of the solver is focused on the performance optimization of the main CG iteration loop, including: implementation of communication hiding and avoiding techniques for global communications; optimization of the nearest neighbor communication - multiplication with a global gluing matrix; and optimization of the parallel CG algorithm to iterate over local Lagrange multipliers only.The performance is demonstrated on a linear elasticity 3D cube and real world benchmarks.

The R-linear convergence rate of an algorithm arising from the semi-smooth Newton method applied to 2D contact problems with friction

The goal is to analyze the semi-smooth Newton method applied to the solution of contact problems with friction in two space dimensions. The primal-dual algorithm for problems with the Tresca friction law is reformulated by eliminating primal variables. The resulting dual algorithm uses the conjugate gradient method for inexact solving of inner linear systems. The globally convergent algorithm based on computing a monotonously decreasing sequence is proposed and its R-linear convergence rate is proved. Numerical experiments illustrate the performance of different implementations including the Coulomb friction law.

Treatment of Singular Matrices in the Hybrid Total FETI Method

The goal is to show how to treat singular matrices via kernel detection so that any robust direct sparse solver for non-singular matrices may be used. This technique applied in the Hybrid total FETI method enables us to process large coarse problems, for which the classical FETI method fails. Efficiency of the algorithms is illustrated in our ESPRESO library for problems with billions of unknowns.

Hybrid parallelization of the total FETI solver

Hybrid parallelization of the Finite Element Tearing and Interconnecting method.Performance comparison of the hybrid parallelization to MPI-only parallelization.TFETI implementation for better utilization of the multi-core computer cluster. This paper describes our new hybrid parallelization of the Finite Element Tearing and Interconnecting (FETI) method for the multi-socket and multi-core computer cluster. This is an essential step in our development of the Hybrid FETI solver were small number of neighboring subdomains is aggregated into clusters and each cluster is processed by a single compute node.In our previous work we have implemented FETI solver using MPI parallelization into our ESPRESO solver. The proposed hybrid implementation provides better utilization of resources of modern HPC machines using advanced shared memory runtime systems such as Cilk++ runtime. Cilk++ is an alternative to OpenMP which is used by ESPRESO for shared memory parallelization.We have compared the performance of the hybrid parallelization to MPI-only parallelization. The results show that we have reduced both solver runtime and memory utilization. This allows a solver to use a larger number of smaller sub-domains and in order to solve larger problems using a limited number of compute nodes. This feature is essential for users with smaller computer clusters.In addition, we have evaluated this approach with large-scale benchmarks of size up to 1.3 billion of unknowns to show that the hybrid parallelization also reduces runtime of the FETI solver for these types of problems.