Preconditioned Conjugate Gradient Iteration Research Articles

A new aggregation algorithm based on coordinates partitioning recursively for algebraic multigrid method

Aggregation based algebraic multigrid is widely used to solve sparse linear systems, due to its potential to achieve asymptotic optimal convergence and cheap cost to set up. In this kind of method, it is vital to construct coarser grids based on aggregation. In this paper, we provide a new aggregation method based on coordinates partitioning recursively. The adjacent graph of the original coefficient matrix is partitioned into sub-graphs and each sub-graph is recursively partitioned until the minimal number of nodes over the sub-graphs on some level is small enough. In this way, a hierarchy of grids can be constructed from top to bottom, which is completely different from the classical schemes. The results from the solution of model partial differential equations with the preconditioned conjugate gradient iteration show that the new algorithm has better performance and is more robust than the widely used classical algorithms in most cases.

Massively Parallel Implementation of Divide-and-Conquer Jacobi Iterations Using Particle-Mesh Ewald for Force Field Polarization.

To accelerate the evaluation of the self-consistent polarization in large condensed-phase systems with polarizable force fields, the new divide-and-conquer Jacobi iterations (DC-JI) solver is adapted for periodic boundary conditions with particle-mesh Ewald and implemented in a massively parallel fashion within the Tinker-HP software package. DC-JI captures the mutual polarization of close-range interactions within subclusters of atoms using Cholesky decomposition and couples in the polarization effects between these clusters iteratively. Iterative convergence is accelerated with direct inversion of the iterative subspace (DIIS) extrapolation. Compared to widely used preconditioned conjugate gradient (PCG) or conventional Jacobi iterations (JI/DIIS) algorithms, DC-JI/DIIS solves the polarization equations ∼20-30% faster in protein systems ranging from ∼10,000-175,000 atoms run on hundreds of processor cores. This translates to ∼10-15% speed-ups in the number of nanoseconds of simulation time that can be achieved per day. Not only is DC-JI/DIIS faster than PCG, but it also gives more energetically robust solutions for a given convergence threshold. These improvements make numerically robust polarizable force field simulations more computationally tractable for chemical systems of interest.

Parallel Implementation of the Coordinates-partitioning Based Aggregation-type Algebraic Multigrid Preconditioners

The coordinates-partitioning based aggregation-type algebraic multigrid preconditioners have been proven to be very efficient in the solution of sparse linear systems with conjugate gradient iterations. In this paper, a parallel algorithm for the setup is provided and the parallelization of the preconditioning process is also considered. The parallel algorithm is based on a good property of the coordinates-partitioning based aggregation, that is, the aggregation process is performed in a fashion from the coarsest level to the finest step by step. Thus, the original adjacent graph is partitioned into a number of sub-graphs, where each sub-graph is related to a node in the coarsest level and is assigned to a processor. The aggregation process can then proceed forward on each processor independently from the assigned sub-graph. When this kind of multigrid preconditioner is applied in Krylov subspace iterations, only the computation on the coarsest level and the matrix-vector multiplications related to the smoothing on each level require communication for V- and W-cycle versions, and require only some extra communication related to dot products for the K-cycle version. The size of the derived linear system on the coarsest level can be controlled by the adjustable arguments and this system can be solved again in parallel with some preconditioned Krylov subspace iterations. The structure information related to the matrix-vector multiplications is invariable to the iterations and can be derived in the setup and be stored, and is used in the latter iterations. Finally, the parallel algorithm is validated for the Gauss-Seidel smoother and some kinds of popular multigrid cycles in solving sparse linear systems from some two-dimensional model partial equations with preconditioned conjugate gradient iterations. The results show that the parallel efficiencies of both the setup and the iteration processes are satisfied.

A GPU based iteration approach to efficiently evaluate radiation symmetry for laser driven inertial confinement fusion

• Efficient computing approach presented for laser driven inertial confinement fusion • Guaranteed symmetry, strictly diagonally dominant and positive definite properties. • GPU based Preconditioned Conjugate gradient iteration approach. • The radiation models are efficiently solved and validated with two examples. Radiation computation is very important for high energy density experiments design in the laser-driven Inertial Confinement Fusion. The view-factor based models are often used to calculate the radiation on the capsule inside a hohlraum. However, it usually takes much time to solve them when the number of equations is very large. In this paper, an efficient iteration approach GPU is presented. The core idea is: (1) guaranteed symmetry, strictly diagonally dominant, and positive definite properties underlying the models are described, (2) a preconditioned conjugate gradient iteration approach is presented to compute the radiation based on such guaranteed properties, and (3) such approach is then parallelized and implemented for GPU so that the large scale models, especially for the non-linear model, can be efficiently solved in reasonable time. Finally, two experimental targets for Shenguang laser facilities built in China are demonstrated and compared to validate the efficiency of the presented approach. The results show that, the models’ computation (1) can be speeded up with successive over-relax iteration method by eight times as compared with Cholesky factorization based direct approach, (2) can be accelerated more with the preconditioned conjugate gradient iteration approach by almost eight times, and (3) can be further accelerated about 2 to 4 times as it parallelized and run on the GPU, which enables the large scale models, can be efficiently solved in reasonable time on the usual desktop computers.

Spectral preconditioners for the efficient numerical solution of a continuous branched transport model

We consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the Monge–Kantorovich equations of optimal transport. Galerkin Finite Element discretization combined with explicit Euler time stepping yield a linear system to be solved at each time step, characterized by a large sparse very ill conditioned symmetric positive definite (SPD) matrix A . Extreme cases even prevent the convergence of Preconditioned Conjugate Gradient (PCG) with standard preconditioners such as an Incomplete Cholesky (IC) factorization of A , which cannot always be computed. We investigate several preconditioning strategies that incorporate partial approximated spectral information. We present numerical evidence that the proposed techniques are efficient in reducing the condition number of the preconditioned systems, thus decreasing the number of PCG iterations and the overall CPU time.

Explicit time integration of transient eddy current problems

SummaryFor time integration of transient eddy current problems, commonly implicit time integration methods are used, where in every time, step 1 or several nonlinear systems of equations have to be linearized with the Newton‐Raphson method because of ferromagnetic materials involved. In this paper, a generalized Schur complement is applied to the magnetic vector potential formulation, which converts a differential‐algebraic equation system of index 1 into a system of ordinary differential equations with reduced stiffness. For the time integration of this ordinary differential equations system of equations, the explicit Euler method is applied. The Courant‐Friedrich‐Levy stability criterion of explicit time integration methods may result in small time steps. Applying a pseudoinverse of the discrete curl‐curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudoinverse, the preconditioned conjugate gradient method is used. The cascaded subspace extrapolation method is presented to produce suitable start vectors for these preconditioned conjugate gradient iterations. The resulting scheme is validated using the nonlinear TEAM 10 benchmark problem.

Technical note: Avoiding the direct inversion of the numerator relationship matrix for genotyped animals in single-step genomic best linear unbiased prediction solved with the preconditioned conjugate gradient.

This paper evaluates an efficient implementation to multiply the inverse of a numerator relationship matrix for genotyped animals () by a vector (). The computation is required for solving mixed model equations in single-step genomic BLUP (ssGBLUP) with the preconditioned conjugate gradient (PCG). The inverse can be decomposed into sparse matrices that are blocks of the sparse inverse of a numerator relationship matrix () including genotyped animals and their ancestors. The elements of were rapidly calculated with the Henderson's rule and stored as sparse matrices in memory. Implementation of was by a series of sparse matrix-vector multiplications. Diagonal elements of , which were required as preconditioners in PCG, were approximated with a Monte Carlo method using 1,000 samples. The efficient implementation of was compared with explicit inversion of with 3 data sets including about 15,000, 81,000, and 570,000 genotyped animals selected from populations with 213,000, 8.2 million, and 10.7 million pedigree animals, respectively. The explicit inversion required 1.8 GB, 49 GB, and 2,415 GB (estimated) of memory, respectively, and 42 s, 56 min, and 13.5 d (estimated), respectively, for the computations. The efficient implementation required <1 MB, 2.9 GB, and 2.3 GB of memory, respectively, and <1 sec, 3 min, and 5 min, respectively, for setting up. Only <1 sec was required for the multiplication in each PCG iteration for any data sets. When the equations in ssGBLUP are solved with the PCG algorithm, is no longer a limiting factor in the computations.

Numerical modeling and validation for 3D coupled-nonlinear thermo-hydro-mechanical problems in masonry dams

We introduce in this article a fully coupled thermo-hydro-mechanical (THM) model describing the physical phenomenon present in masonry materials. A series of 3D numerical simulations are carried out for the THM analysis of a dam during its impounding process. The results of the coupled THM analyzes are validated with measurements recorded during that process in terms of transient displacement, pore-water pressure and temperature values. An agreement between numerical simulations and measured data proves that the coupled THM model can well reproduce the multi-physical behavior of masonry dams. Furthermore, we introduce herein a solution for a large system of balance equations by combining parallel computation, storage in Compressed Sparse Row format, and iterative pre-conditioned Conjugate Gradient Squared method in order to improve the computational cost.

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

SummaryThis paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley &amp; Sons Ltd.

An Adaptive MultiPreconditioned Conjugate Gradient Algorithm

This article introduces and analyzes a new adaptive algorithm for solving symmetric positive definite linear systems in cases where several preconditioners are available or the usual preconditioner is a sum of contributions. A new theoretical result allows us to select, at each iteration, whether a classical preconditioned conjugate gradient (CG) iteration is sufficient (i.e., the error decreases by a factor of at least some chosen ratio) or whether convergence needs to be accelerated by performing an iteration of multipreconditioned CG [4]. This is first presented in an abstract framework with the one strong assumption being that a bound for the smallest eigenvalue of the preconditioned operator is available. Then, the algorithm is applied to the balancing domain decomposition method and its behavior is illustrated numerically. In particular, it is observed to be optimal in terms of local solves, for both well-conditioned and ill-conditioned test cases, which makes it a good candidate to be a default par...

Improving the Reconstruction of Vector Fields Using Mixed Finite Element Methods and Optimal Preconditioning

In this article, we study numerically a diagnostic model, based on mass conservation, to recover solenoidal vector fields from experimental data. Based on a reformulation of the mathematical model as a saddle‐point problem, we introduce an iterative preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To obtain a stable algorithm, we use a second‐order mixed finite element approximation for discretization. We show, using synthetic vector fields, that this new approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1137–1154, 2016

A PRECONDITIONED ITERATIVE SOLUTION SCHEME FOR NONLINEAR PARABOLIC SYSTEMS ARISING IN AIR POLLUTION MODELING

A preconditioned iterative solution method is presented for nonlinear parabolic transport systems. The ingredients are implicit Euler discretization in time and finite element discretization in space, then an outer-inner (outer damped inexact Newton method with inner preconditioned conjugate gradient) iteration, further, as a main part, preconditioning via an l-tuple of independent elliptic operators. Numerical results show that the suggested method works properly for a test problem in air pollution modeling.

Adaptive and Stochastic Algorithms for Electrical Impedance Tomography and DC Resistivity Problems with Piecewise Constant Solutions and Many Measurements

This article develops fast numerical methods for the practical solution of the famous electrical impedance tomography and DC resistivity problems in the presence of discontinuities and potentially many experiments or data. Based on a Gauss–Newton (GN) approach coupled with preconditioned conjugate gradient (PCG) iterations, we propose two algorithms. One determines adaptively the number of inner PCG iterations required to stably and effectively carry out each GN iteration. The other algorithm, useful especially in the presence of many experiments, employs a randomly chosen subset of experiments at each GN iteration that is controlled using a cross validation approach. Numerical examples demonstrate the efficacy of our algorithms.

Decomposition and regularization of nonlinear anisotropic curl‐curl DAEs

PurposeThe space discretization of eddy‐current problems in the magnetic vector potential formulation leads to a system of differential‐algebraic equations. They are typically time discretized by an implicit method. This requires the solution of large linear systems in the Newton iterations. The authors seek to speed up this procedure. In most relevant applications, several materials are non‐conducting and behave linearly, e.g. air and insulation materials. The corresponding matrix system parts remain constant but are repeatedly solved during Newton iterations and time‐stepping routines. The paper aims to exploit invariant matrix parts to accelerate the system solution.Design/methodology/approachFollowing the principle “reduce, reuse, recycle”, the paper proposes a Schur complement method to precompute a factorization of the linear parts. In 3D models this decomposition requires a regularization in non‐conductive regions. Therefore, the grad‐div regularization is revisited and tailored such that it takes anisotropies into account.FindingsThe reduced problem exhibits a decreased effective condition number. Thus, fewer preconditioned conjugate gradient iterations are necessary. Numerical examples show a decrease of the overall simulation time, if the step size is small enough. 3D simulations with large time step sizes might not benefit from this approach, because the better condition does not compensate for the computational costs of the direct solvers used for the Schur complement. The combination of the Schur approach with other more sophisticated preconditioners or multigrid solvers is subject to current research.Originality/valueThe Schur complement method is adapted for the eddy‐current problem. Therefore, a new partitioning approach into linear/non‐linear and static/dynamic domains is proposed. Furthermore, a new variant of the grad‐div gauging is introduced that allows for anisotropies and enables the Schur complement method in 3D.

Toward large-scale Hybrid Monte Carlo simulations of the Hubbard model on graphics processing units

One of the most efficient non-perturbative methods for the calculation of thermal properties of quantum systems is the Hybrid Monte Carlo algorithm, as evidenced by its use in large-scale lattice quantum chromodynamics calculations. The performance of this algorithm is determined by the speed at which the fermion operator is applied to a given vector, as it is the central operation in the preconditioned conjugate gradient iteration. We study a simple implementation of these operations for the fermion matrix of the Hubbard model in d+1 spacetime dimensions, and report a performance comparison between a 2.66 GHz Intel Xeon E5430 CPU and an NVIDIA Tesla C1060 GPU using double-precision arithmetic. We find speedup factors ranging between 30 and 350 for d=1, and in excess of 40 for d=3. We argue that such speedups are of considerable impact for large-scale simulational studies of quantum many-body systems.

Reduced numerical solution times for combined boundary‐initial value problems using parallel computing

Purpose — The purpose of this paper is to seek improved solution techniques for combined boundary‐initial value problems (IVPs) associated with the time‐dependent creep deformation and rupture of engineering structures at high temperatures and hence to reconfigure a parallel iterative preconditioned conjugate gradient (PCG) solver and the DAMAGE XXX software, for 3‐D finite element creep continuum damage mechanics (CDM) analysis.Design/methodology/approach — The potential to speed up the computer numerical solution of the combined BV‐IVPs is addressed using parallel computers. Since the computational bottleneck is associated with the matrix solver, the parallelisation of a direct and an iterative solver has been studied. The creep deformation and rupture of a tension bar has been computed for a range of the number of degrees of freedom (ndf), and the performance of the two solvers is compared and assessed.Findings — The results show the superior scalability of the iterative solver compared to the direct solver, with larger speed‐ups gained by the PCG solver for higher degrees of freedom. Also, a new algorithm for the first trial solution of the PCG solver provides additional speed‐ups.Research limitations/implications — The results show that the ideal parallel speed‐up of the iterative solver of 16, relative to two processors, is achieved when using 32 processors for a mesh of ndf = 153,238. Originality/value — Techniques have been established in this paper for the parallelisation of CDM creep analysis software using an iterative equation solver. The significant computational speed‐ups achieved will enable the analysis of failures in weldments of industrial significance.

Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints

Optimality systems and their linearizations arising in optimal control of partial differential equations with pointwise control and (regularized) state constraints are considered. The preconditioned conjugate gradient (PCG) method in a nonstandard inner product is employed for their efficient solution. Preconditioned condition numbers are estimated for problems with pointwise control constraints, mixed control-state constraints, and of Moreau–Yosida penalty type. Numerical results for elliptic problems demonstrate the performance of the PCG iteration. Regularized state-constrained problems in three dimensions with more than 750,000 variables are solved.

Incomplete factorization preconditioners for the iterative solution of Stochastic Finite Element equations

This work is focused on enhancing the computational efficiency in Monte Carlo simulation-based Stochastic Finite Element (SFE) analysis of large-scale structural models. Such analyses require the solution of successive systems of equations derived during simulations, which can be efficiently treated using customized versions of the iterative Preconditioned Conjugate Gradient (PCG) solution method. PCG-customization is localized at the preconditioning matrix employed to accelerate convergence. Thus, specialized preconditioners following the rationale of incomplete factorization are presented, which retain only essential numerical information during factorization. The resulting PCG-based solution schemes allow for computationally efficient SFE analyses with low storage demands in computer memory.

Unconditionally stable integration of Maxwell’s equations

Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit – finite difference time domain scheme. In this paper, we discuss unconditionally stable integration for a general semi-discrete Maxwell system allowing non-Cartesian space grids as encountered in finite-element discretizations. Such grids exclude the alternating direction implicit approach. Particular attention is given to the second-order trapezoidal rule implemented with preconditioned conjugate gradient iteration and to second-order exponential integration using Krylov subspace iteration for evaluating the arising φ -functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second-order implicit–explicit integrator.

Reuse, Recycle, Reduce (3R) – strategies for the calculation of transient magnetic fields

The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge–Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies (“ reuse, recycle, reduce”). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge–Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements.