Parallel Preconditioned Conjugate Gradient Solver Research Articles

An octree pattern-based massively parallel PCG solver for elasto-static and dynamic problems

In this paper, an octree pattern-based massively parallel preconditioned conjugate gradient (PCG) solver is developed for large-scale elastostatic and implicit elastodynamic applications. The problem domain is discretised using octree cells to enable a fully automatic mesh generation. The compatibility between neighbouring octree cells of different sizes is satisfied by employing polyhedral elements. Matrix operations within the solver are carried out by adopting an octree pattern-based pre-computation approach. Here, a limited number of master cells in a balanced octree mesh is exploited to reduce memory requirements and perform highly efficient matrix product computations. The parallelism is achieved by employing a mesh-partitioning strategy and message-passing-interface (MPI) directives. The results show that the developed PCG solver attains a significant parallel speed-up and efficiency with an increasing number of cores on distributed memory systems. The practical applications of the proposed framework are demonstrated using large-scale examples for both CAD and image-based analyses.

A GPU implementation of the PCG method for large-scale image-based finite element analysis in heterogeneous periodic media

Image-based physics simulations in heterogeneous media at a microscopic scale are a growing trend in various fields within and around Scientific Computing. We consider the scope of Numerical Homogenization, where micro-computed tomography is used to obtain digital models of physical samples, naturally leading to pixel and voxel-based solutions. In this context, the Finite Element Method (FEM) is commonly employed to solve the governing differential equations, via a system of algebraic equations. As image dimensions increase, the memory allocation due to the matrix associated with the FEM quickly becomes unfeasible, even in sparse format. Assembly-free strategies are adopted to reduce memory usage, with the caveat of increased computational cost. The Preconditioned Conjugate Gradient (PCG) method is widely employed to solve this sort of large-scale sparse linear systems , and is fitting to be adapted for assembly-free implementations. This work focuses on a massively parallel PCG solver applied to finite element analyses of heat conduction and linear elasticity on image-based models. Memory-efficiency is one of our main concerns, in an attempt to make feasible the employment of personal-use GPUs for large-scale simulations. The resulting solver is validated with an analytical benchmark, and by comparing the obtained results for a microtomographic model of a cast iron sample against experimental values found in the literature. Time and memory metrics are presented and discussed. It is shown that the developed program allows for homogenization studies of nearly 500 million degrees-of-freedom to be conducted in personal computers equipped with CUDA-enabled devices of 8 GB RAM, taking seconds or a few minutes per system solution with the PCG method. Up to 400× speed-up was observed in comparison to an analogous solver running in a 16-thread CPU. Our GPU implementation makes it possible to conduct, in a matter of minutes, homogenization studies that would take hours, or even days, in personal CPUs.

Efficient Parallel Preconditioned Conjugate Gradient Solver on GPU for FE Modeling of Electromagnetic Fields in Highly Dissipative Media

We present a performance analysis of a parallel implementation of preconditioned conjugate gradient solvers using graphic processing units with compute unified device architecture programming model. The solvers were optimized for the solution of sparse systems of equations arising from finite-element analysis of electromagnetic phenomena involved in the diffusion of underground currents in both steady state and under time-harmonic current excitation. We used both shifted incomplete Cholesky factorization and incomplete LU factorization as preconditioners. The results show a significant speedup using the graphics processing unit compared with a serial CPU implementation.

Preconditioning in parallel for fractional step Navier--Stokes solvers

Preconditioning for the Pressure Poisson Equation, used with the fractional step Navier--Stokes solvers, is studied. The Pressure Poisson Equation results from the segregated calculation of the velocity and pressure in the momentum equations, with the divergence of the velocity as the source term. The coefficient matrix of the Pressure Poisson Equation is dependent only upon grid-size, and thus preconditioners need to be constructed only once initially, and used for all subsequent time steps. Several preconditioning techniques are studied, including Jacobi, incomplete matrix decomposition variants, and sparse approximate inverses. The test case is a three dimensional turbulent channel flow, with the domain discretised using a structured nonstaggered grid. Parallel computing is performed on a cluster of processors by message passing, with domain partitioning to avoid the use of global gather and scatter operations and to minimise the effect of communication bandwidth. The preconditioners are all constructed based on the local grid partition. For the sparse approximate inverse preconditioners, cell dependencies are bounded to limit communication across grid partition boundaries. The effect of a defined sparsity pattern is also investigated. The optimum sparsity pattern and the dependency on the neighbouring cells are found to be influenced by the grid ratios in each axis direction. References M. Benzi. Preconditioning techniques for large linear systems: A survey. J. of Comp. Physics , 182(2):418--477, 2002. doi:10.1006/jcph.2002.7176 Y. Saad. Iterative Methods for Sparse Linear Systems . SIAM, Philadelphia, PA, USA, 2nd edition, 2003. V. S. Djanali, S. W. Armfield, and M. P. Kirkpatrick. Comparison of approximate inverse preconditioners for the fractional step Navier--Stokes equations. ANZIAM J. , 52:C581--C595, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3890 M. Ament, G. Knittel, D. Weiskopf, and W. Strasser. A parallel preconditioned conjugate gradient solver for the Poisson problem on a multi-gpu platform. Parallel, Distributed and Network-Based Processing (PDP), 2010 18th Euromicro International Conference , 583--592, 2010. doi:10.1109/PDP.2010.51 S.M. Xu, H.X. Lin, K. Wang, and W. Xue. Utilizing cuda for preconditioned gmres solvers. Proc. 8th Int‚Aol Symp. on Distributed Computing and Applications to Business, Engineering and Sciences (DCABES 2009) . S. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional‚aistep methods for the Navier--Stokes equations on staggered grid. Int. J. Numer. Methods Fluids , 38:255--282, 2002. doi:10.1002/fld.217 H. L. Stone. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. , 5(3):530--558, 1968. doi:10.1137/0705044 G. E. Schneider and M. Zedan. A Modified strongly implicit procedure for the numerical solution of field problems. Numer. Heat Transfer , 4(1):1--19, 1981. doi:10.1080/01495728108961775 M. J. Grote and T. Huckle. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. , 18(3):838--853, May 1997. doi:10.1137/S1064827594276552 M. Benzi and M. Tuma. A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. , 19(3):968--994, 1998. doi:10.1137/S1064827595294691 R. D. Moser, J. Kim, and N. N. Mansour. {Direct numerical simulation of turbulent channel flow up to Re$_\tau $ = 590}. Phys. Fluids , 11(4):943--945, 1999. doi:10.1063/1.869966

A Parallel PCG Solver for MODFLOW

In order to simulate large-scale ground water flow problems more efficiently with MODFLOW, the OpenMP programming paradigm was used to parallelize the preconditioned conjugate-gradient (PCG) solver with in this study. Incremental parallelization, the significant advantage supported by OpenMP on a shared-memory computer, made the solver transit to a parallel program smoothly one block of code at a time. The parallel PCG solver, suitable for both MODFLOW-2000 and MODFLOW-2005, is verified using an 8-processor computer. Both the impact of compilers and different model domain sizes were considered in the numerical experiments. Based on the timing results, execution times using the parallel PCG solver are typically about 1.40 to 5.31 times faster than those using the serial one. In addition, the simulation results are the exact same as the original PCG solver, because the majority of serial codes were not changed. It is worth noting that this parallelizing approach reduces cost in terms of software maintenance because only a single source PCG solver code needs to be maintained in the MODFLOW source tree.

Elastoplastic finite element analysis on a mimd parallel‐computer

This paper presents the development of a parallel finite element algorithm for a MIMD parallel computer. The elements are distributed, onto the processors, in such a way that neighbouring elements are placed onto neighbouring processors. This guarantees a good load‐balancing even in physically non‐linear computations. The distribution of the columns of the global system of equations is done by an election algorithm. For solving the global system of equations we use a parallel preconditioned conjugate gradient solver. Tests were done with an elastoplastic material model for proving the efficiency of assembling and solving the system of equations.