Multigrid and Krylov Solvers for Large Scale Finite Element Groundwater Flow Simulations on Distributed Memory Parallel Platforms

Efficient parallel multigrid based solvers for large scale groundwater flow simulations

The focus of this study is on the efficient solution for three-dimensional (3D) time-harmonic eddy current problems discretised by the finite-element method (FEM) with thin elements. The systems of equations arising from the finite-element formulations are solved by the geometric multigrid preconditioned conjugate gradient (MGCG) method. Numerical examples show that the MGCG method has stable convergence, which does not deteriorate for elements with high aspect ratios and its computation time is much less than that of the conjugate gradient method with incomplete Cholesky factorisation as the preconditioner (ICCG).

An effective wavelet based multigrid preconditioned conjugate gradient method is developed to solve electromagnetic large matrix problem for millimeter wave scattering application. By using wavelet transformation we restrict the large matrix equation to a relative smaller matrix and which can be solved rapidly. The solution is prolonged as the new improvement for the conjugate gradient (CG) method. Numerical results show that our developed wavelet based multigrid preconditioned CG method can reach large improvement of computational complexity. Due to the automaticity of wavelet transformation, this method is potential to be a block box solver without physical background.

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

Hiding global synchronization latency in the preconditioned Conjugate Gradient algorithm

An effective wavelet based multigrid preconditioned conjugate gradient method is developed to solve electromagnetic large matrix problem for millimeter wave scattering application. By using wavelet transformation we restrict the large matrix equation to a relative smaller matrix and which can be solved rapidly. The solution is prolonged as the new improvement for the conjugate gradient (CG) method. Numerical result shows that our developed wavelet based multigrid preconditioned CG method can reach large improvement of computational complexity. Due to the automaticity of wavelet transformation, this method is potential to be a block box solver without physical background.

The focus of this paper is on the efficient solution of 3-D time-harmonic eddy current problems discretized by the finite element method with bad quality meshes. The sys tems of equations arising from the finite element formula tions are solved by a geometric multigrid preconditioned conjugate gradient method. The efficiency of the proposed method will be analyzed by numerical examples.

As the increasing size of power grids, IR drop analysis has become more computationally challenging both in runtime and memory consumption. In this paper, we propose a linear complexity simulator named PowerRush, which consists of an efficient SPICE Parser, a robust circuit Builder and a linear solver. The proposed solver is a pure algebraic method which can provide an optimal convergence without geometric information. It is implemented by Algebraic Multigrid Preconditioned Conjugate Gradient method, in which an aggregation based algebraic multigrid with K-Cycle acceleration is adopted as a preconditioner to improve the robustness of conjugate gradient iterative method. In multigrid scheme, double pairwise aggregation technique is applied to the matrix graph in coarsening procedure to ensure low setup cost and memory requirement. Further, a K-Cycle multigrid scheme is adopted to provide Krylov subspace acceleration at each level to guarantee optimal or near optimal convergence. Experimental results on real power grids have shown that PowerRush has a linear complexity in runtime cost and memory consumption. The DC analysis of a 60 Million nodes power grid can be solved by PowerRush for 0.01mV accuracy in 170 seconds with 21.89GB memory used.

In this study, we analyze the real-time solution of a series of augmented least squares problems, which are generated by adding information to an original least squares model repetitively. Instead of solving the least squares problems directly, we transform them into a batch of saddle point linear systems and subsequently solve the linear systems using restrictively preconditioned conjugate gradient (RPCG) methods. Approximation of the new Schur complement is generated effectively based on a previously approximated Schur complement. Owing to the variations of the preconditioned conjugate gradient method, the proposed methods generate convergence results similar to the conjugate gradient method and achieve a very fast convergent iterative sequence when the coefficient matrix is well preconditioned. Numerical tests show that the new methods are more effective than some standard Krylov subspace methods. Updated RPCG methods meet the requirement of real-time computing successfully for multifactor models.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

Sparsified Randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation

The Newton-based global gradient algorithm (GGA) (also known as the Todini and Pilati method) is a widely used method for computing the steady-state solution of the hydraulic variables within a water distribution system (WDS). The Newton-based computation involves solving a linear system of equations arising from the Jacobian of the WDS equations. This step is the most computationally expensive process within the GGA, particularly for large networks involving up to O(105) variables. An increasingly popular solver for large linear systems of the M-matrix class is the algebraic multigrid (AMG) method, a hierarchical-based method that uses a sequence of smaller dimensional systems to approximate the original system. This paper studies the application of AMG to the steady-state solution of WDSs through its incorporation as the linear solver within the GGA. The form of the Jacobian within the GGA is proved to be an M-matrix (under specific criteria on the pipe resistance functions), and thus able to be...

Multigrid preconditioned conjugate gradient method is proposed for the algebraic systems resulting from two 8-node hexahedron combined hybrid elements with high performance for the linear elasticity problem. Numerical computations are presented, which demonstrate the convergence and effectiveness of the method.

Simulating Spatially Explicit Problems on High Performance Architectures

Conjugate Gradient (CG) method is an iterative linear solver which is used by many scientific and engineering applications to solve a linear system of algebraic equations. CG generates a heavy load of computation and therefore it slows the performance of the applications using it. Parallelizing CG is considered as a way to increase its performance. However, CG suffers from communication dependencies among its divisible loads. Most of the studies to parallelize CG concentrate on parallelizing its matrix-vector multiplication. In this paper, we answer the following questions: 1) what are the divisible loads in the CG, and 2) where is communication involved in the parallel CG. To answer 1), we highlight the different divisible data blocks in CG. To answer 2), we introduce a dependency graph among the different data blocks. We conduct experiments on a parallel CG implementation and evaluate communication cost.