Deflated Conjugate Gradient Research Articles

An adapted deflated conjugate gradient solver for robust extended/generalised finite element solutions of large scale, 3D crack propagation problems

An adapted deflation preconditioner is employed to accelerate the solution of linear systems resulting from the discretisation of fracture mechanics problems with well-conditioned extended/generalised finite elements. The deflation space typically used for linear elasticity problems is enriched with additional vectors, accounting for the enrichment functions used, thus effectively removing low frequency components of the error. To further improve performance, deflation is combined, in a multiplicative way, with a block-Jacobi preconditioner, which removes high frequency components of the error as well as near-linear dependencies introduced by enrichment. The resulting scheme is tested on a series of non-planar crack propagation problems and compared to alternative linear solvers in terms of performance.

Fast deflated conjugate gradient method with proper orthogonal decomposition for topology optimization

Fast deflated conjugate gradient method with proper orthogonal decomposition for topology optimization

Comparison of selected FETI coarse space projector implementation strategies

This paper deals with scalability improvements of the FETI (Finite Element Tearing and Interconnecting) domain decomposition method solving elliptic PDEs. The main bottleneck of FETI is the solution of a coarse problem that is part of the projector onto the natural coarse space. This paper introduces and compares two strategies for the FETI coarse problem solution. The first one is a classical solution with either direct (factorization + forward/backward substitutions) or iterative solvers (conjugate gradient and deflated conjugate gradient methods). The second one is the assembly of an explicit inverse using a direct solver with the coarse problem solution realised by dense matrix-vector products. MPI subcommunicators are employed to increase arithmetic intensity and, crucially, to decrease the communication cost. PERMON library for quadratic programming implementing the Total FETI variant of FETI was used for the numerical experiments.

Large-scale buckling-constrained topology optimization based on assembly-free finite element analysis

In this article, we propose a fast method to solve large-scale three-dimensional topology optimization problems subject to buckling constraints. Buckling analysis entails the solution of a generalized eigenvalue problem. For problems with large degrees of freedom, the current numerical methods tend to be memory-hungry, leading to high computational costs. First, a low-memory assembly-free linear buckling analysis method is proposed. Specifically, this method is based on the voxelization model, an assembly-free version of the deflated conjugate gradient is used to accelerate the iteration solution of linear systems of equations, where neither the stiffness matrix nor the deflation matrix is assembled, and the parallelization of matrix–vector multiplication is achieved by the congruency of voxels. Due to the particularity of the stress stiffness matrix in the buckling analysis, the inverse iteration is used to solve the general eigenvalue problem, which can reduce the operations of stress stiffness matrix considerably. Based on the efficient buckling analysis method, we extend the level-set method for buckling constraints in a semi-analytical manner. Several numerical experiments demonstrate that the proposed method can solve large-scale three-dimensional buckling analysis and topology optimization against buckling constraints effectively.

Rigid body modes deflation of the Preconditioned Conjugate Gradient in the solution of discretized structural problems

In the numerical simulation of structural problems, a crucial aspect concern the solution of the linear system arising from the discretization of the governing equations. In fact, ill-conditioned system, related to an unfavorable eigenspectrum, are quite common in several engineering applications. In these cases the Preconditioned Conjugate Gradient enhanced with the deflation technique seems to be a very promising approach in particular because an effective deflation space is already at hand. In fact, it is possible to utilize rigid body motions of the system, that can be calculated easily and cheaply, and only the knowledge of the geometry of problem is required. This paper investigates the advantages of using a Rigid Body Modes Deflated Conjugate Gradient in the solution of challenging systems arising from structural problems. Two different situations are analyzed: the ill-conditioning caused by low constraining is addressed deflating the total rigid body modes, while the one concerning the heterogeneity of the problem by using the rigid body modes of separate components. Moreover, the implemented method is highly parallel and therefore suitable for High Performance Computing. Numerical results show how both approaches performed successfully in reducing the overall system solution time cost and iterations required for convergence.

The deflated conjugate gradient method: Convergence, perturbation and accuracy

Deflation techniques for Krylov subspace methods have seen a lot of attention in recent years. They provide means to improve the convergence speed of these methods by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors, but also more general subspaces are applicable.In this paper we discuss two results concerning the accuracy requirements within the deflated CG method. First we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace. Second, we discuss the accuracy required in calculating the deflating projection. This is crucial concerning the overall convergence of the method, and also allows to save some computational work.To show these results, we use the fact that as a projection approach deflation has many similarities to multigrid methods. In particular, recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods. In the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property.

A Deflated Assembly Free Approach to Large-Scale Implicit Structural Dynamics

The primary computational bottle-neck in implicit structural dynamics is the repeated inversion of the underlying stiffness matrix. In this paper, a fast inversion technique is proposed by merging four distinct but complementary concepts: (1) voxelization with adaptive local refinement, (2) assembly-free (a.k.a. matrix-free or element-by-element) finite element analysis (FEA), (3) assembly-free deflated conjugate gradient (AF-DCG), and (4) multicore parallelization. In particular, we apply these concepts to the well-known Newmark-beta method, and the resulting AF-DCG is well-suited for large-scale problems. It can be easily ported to many-core central processing unit (CPU) and multicore graphics-programmable unit (GPU) architectures, as demonstrated through numerical experiments.

Large Scale Finite Element Analysis Via Assembly-Free Deflated Conjugate Gradient

Large-scale finite element analysis (FEA) with millions of degrees of freedom (DOF) is becoming commonplace in solid mechanics. The primary computational bottleneck in such problems is the solution of large linear systems of equations. In this paper, we propose an assembly-free version of the deflated conjugate gradient (DCG) for solving such equations, where neither the stiffness matrix nor the deflation matrix is assembled. While assembly-free FEA is a well-known concept, the novelty pursued in this paper is the use of assembly-free deflation. The resulting implementation is particularly well suited for large-scale problems and can be easily ported to multicore central processing unit (CPU) and graphics-programmable unit (GPU) architectures. For demonstration, we show that one can solve a 50 × 106 degree of freedom system on a single GPU card, equipped with 3 GB of memory. The second contribution is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration.” The latter exploits classic plate and beam theories for efficient deflation of highly ill-conditioned problems arising from thin structures.

A deflated conjugate gradient method for multiple right hand sides and multiple shifts

We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right-hand sides. In the past, Krylov subspace methods have been developed which exploit either the need for solutions to multiple right-hand sides (e.g. deflation type methods and block methods) or multiple shifts (e.g. shifted CG) with some success. In this paper we present a block Krylov subspace method which, based on a block Lanczos process, exploits both features--shifts and multiple right-hand sides--at once. Such situations arise, for example, in lattice quantum chromodynamics (QCD) simulations within the Rational Hybrid Monte Carlo (RHMC) algorithm. We present numerical evidence that our method is superior compared to applying other iterative methods to each of the systems individually as well as, in typical situations, to shifted or block Krylov subspace methods.

Optimization of the deflated Conjugate Gradient algorithm for the solving of elliptic equations on massively parallel machines

The discretization of Partial Differential Equations often leads to the need of solving large symmetric linear systems. In the case of the Navier–Stokes equations for incompressible flows, solving the elliptic pressure Poisson equation can represent the most important part of the computational time required for the massively parallel simulation of the flow. The need for efficiency that this issue induces is completed with a need for stability, in particular when dealing with unstructured meshes. Here, a stable and efficient variant of the Deflated Preconditioned Conjugate Gradient (DPCG) solver is first presented. This two-level method uses an arbitrary coarse grid to reduce the computational cost of the solving. However, in the massively parallel implementation of this technique for very large linear systems, the coarse grids generated can count up to millions of cells, which makes direct solvings on the coarse level impossible. The solving on the coarse grid, performed with a Preconditioned Conjugate Gradient (PCG) solver for this reason, may involve a large number of communications, which reduces dramatically the performances on massively parallel machines. To this effect, two methods developed in order to reduce the number of iterations on the coarse level are introduced, that is the creation of improved initial guesses and the adaptation of the convergence criterion. The design of these methods make them easy to implement in any already existing DPCG solver. The structural requirements for an efficient massively parallel unstructured solver and the implementation of this solver are described. The novel DPCG method is assessed for applications involving turbulence, heat transfers and two-phase flows, with grids up to 17.8billion elements. Numerical results show a two- to 12-fold reduction of the number of iterations on the coarse level, which implies a reduction of the computational time of the Poisson solver up to 71% and a global reduction of the proportion of communication times up to 53%. As a result, the weak scaling of the LES solver is shown to be clearly improved for massively parallel uses.