Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equation

Classical iterative methods when applied to the partial solution of the generalized eigenvalue problem Ax =λBx, may yield very poor convergence rates particularly when ill‐conditioned problems are considered. In this paper the preconditioned conjugate gradient (CG) method via the minimization of the Rayleigh quotient and the reverse power method is employed for the partial eigenproblem. The triangular splitting preconditioners employed are obtained from an incomplete Choleski factorization and a partial Evans preconditioner. This approach can dramatically improve the convergence rate of the basic CG method and is applicable to any symmetric eigenproblem in which one of the matrices A, B is positive definite. Because of the renewed interest in CG techniques for FE work on microprocessors and parallel computers, it is believed that this improved approach to the generalized eigenvalue problem is likely to be very promising.

We consider the problem of vectorizing the recursive calculations found in modified incomplete factorizations and SSOR preconditioners for the conjugate gradient method. We examine matrix problems derived from partial differential equations which are discretized on regular 2-D and 3-D grids, where the grid nodes are ordered in the natural ordering. By performing data dependency analyses of the computations, we show that there is concurrency in both the factorization and the forward and backsolves. The computations may be performed with an average vector length of $O(n)$ on an $n^2 $ or $n^3 $ grid in two and three dimensions. Numerical studies on four model problems show that the conjugate gradient method using these vectorized implementations of the modified incomplete factorizations and SSOR preconditioners achieves overall speeds approaching 100 megaflops on a Cray X-MP/24 vector computer. Furthermore, these methods require considerably less overall execution time than diagonal scaling and no-fill red-black incomplete factorization preconditioners, both of which allow full vectorization but are not as convergent.

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.

A partial preconditioned conjugate gradient method for large eigenproblems

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).

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

A preconditioned Chebyshev basis communication-avoiding conjugate gradient method (P-CBCG) is applied to the pressure Poisson equation in a multiphase thermal-hydraulic CFD code JUPITER, and its computational performance and convergence properties are compared against a preconditioned conjugate gradient (P-CG) method and a preconditioned communication-avoiding conjugate gradient (P-CACG) method on the Oakforest-PACS, which consists of 8,208 KNLs. The P-CBCG method reduces the number of collective communications with keeping the robustness of convergence properties. Compared with the P-CACG method, an order of magnitude larger communication-avoiding steps are enabled by the improved robustness. It is shown that the P-CBCG method is 1.38 $$\times $$ and 1.17 $$\times $$ faster than the P-CG and P-CACG methods at 2,000 processors, respectively.

A parallel preconditioned conjugate gradient package for solving sparse linear systems on a Cray Y-MP

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.

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.

Groundwater flow simulation has become one of the top international issues in new generation of environmental applications. When managing large-scale groundwater flow problems, the intensive computational ability and large amounts of memory space required for modeling are the main bottlenecks for researchers. In order to solve three-dimensional large-scale groundwater flow problems more rapidly, the Open MP was adopted to parallelize the preconditioned conjugate gradient (PCG) algorithm in this paper. And this paper carried out a numerical experiment of the three-dimensional groundwater flow model on a computer with four cores. Based on the numerical experiment, it is found that the execution time of the original serial PCG program is about 1.74 to 2.86 times of the parallel PCG program executed with different number of threads. The experimental results also demonstrate that the PCG solver based on Open MP is an effective way for solving large-scale three-dimensional groundwater flow problem.

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of \(M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}\) can be computed using \({\mathcal {O}}(T^{2d} N^d)\) operations, and the number of nonzeros in the factorization factors is of \({\mathcal {O}}(T^{d} N^d)\), \(d=1,2,3\). A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of \({\mathcal {O}}(N^d)\). To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of \({\mathcal {O}}(N^d (\log _2 N)^{2})\). In view that T is independent of N in one dimension and is set to be of order \({\mathcal {O}}(\log _2 N)\) in two and three dimensions, the PCG method has a \({\mathcal {O}}(N^d (\log _2 N)^{2d})\) quasi-optimal complexity for a d dimensional domain with \((N-1)^d\) unknows, \(d=1,2,3\). In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of \({\mathcal {O}}(N^{3} (\log _2 N)^{2})\) and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.

Iterative 3D PET reconstruction represents a very computational challenge due to the large number of lines of response (LOR) collected for each data set. This iterative 3D reconstruction also needs a lot of iterations to achieve an acceptable PET reconstructed image. A preconditioned conjugate gradient (PCG) method was previously shown to have faster convergence rate than expectation maximization (EM) type algorithms. For the microPET, images suffer from crystal penetration blurring due to small scanner radius. An exact 2D blur model is needed to achieve high resolution image. A Preconditioned Conjugate Gradient (PCG) method is described for reconstruction of high-resolution 3D images from the microPET Inveon small-animal scanner from Siemens. The projector pair is used as part of a factored system matrix that takes into account detector-pair response by using shift-variant sinogram blur kernels, attenuation correction, and detector efficiency corrections. The system matrix for geometric projection is based on depth dependent solid angle calculation in combination with a spatially variant detector response model. The mircoPET PCG is combined with OSEM to accelerate convergence. This reconstruction model achieves a high resolution animal image; however, it took an hour to reconstruct a frame. Therefore, we describe an ultrafast forward and back projector pair based on Symmetry and SIMD projector (SSP). The proposed method produces similar quality images when compared to those obtained with the software package from Siemens and requires order of magnitude less computation.

The preconditioned conjugate gradient (PCG) algorithm is a well-known iterative method for solving sparse linear systems in scientific computations. GPU-accelerated PCG algorithms for large-sized problems have attracted considerable attention recently. However, on a specific multi-GPU platform, producing a highly parallel PCG implementation for any large-sized problem requires significant time because several manual steps are involved in adjusting the related parameters and selecting an appropriate storage format for the matrix block that is assigned to each GPU. This motivates us to propose adaptive optimization modeling of PCG on multi-GPUs, which mainly involves the following parts: (1) an optimization multi-GPU parallel framework of PCG and (2) the profile-based optimization modeling for each one of the main components of the PCG algorithm, including vector operation, inner product, and sparse matrix-vector multiplication (SpMV). Our model does not construct a new storage format or kernel but automatically and rapidly generates an optimal parallel PCG algorithm for any problem on a specific multi-GPU platform by integrating existing storage formats and kernels. We take a vector operation kernel, an inner-product kernel, and five popular SpMV kernels for an example to present the idea of constructing the model. Given that our model is general, independent of the problems, and dependent on the resources of devices, this model is constructed only once for each type of GPU. The experiments validate the high efficiency of our proposed model.

In order to solve Toeplitz linear systems An(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed An(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=An−1(g) An(f). In this paper we prove that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2−1] and takes the maximum value 2−1 when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.