Scaled Conjugate Gradient Method Research Articles

Iterative Linear Solvers in a 2D Radiation–Hydrodynamics Code: Methods and Performance

Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion (ICF) phenomena. A crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations. We present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation–hydrodynamics problems. The linear solvers used are diagonally scaled conjugate gradient, GMRES with incomplete LU preconditioning, conjugate gradient with incomplete Cholesky preconditioning, multigrid, and multigrid-preconditioned conjugate gradient. These problems involve shock propagation, opacities varying over 5–6 orders of magnitude, tabular equations of state, and dynamic ALE (Arbitrary Lagrangian Eulerian) meshes. We perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1000 to 100,000 zones. The fundamental question we address in this paper is: Is it more efficient to invert the matrix in many inexpensive steps (like diagonally scaled conjugate gradient) or in fewer expensive steps (like multigrid)? In addition, what is the answer to this question as a function of problem size and is the answer problem dependent? We find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size, increasing in both iteration count and overall CPU time with the size of the problem and also increasing for larger time steps. For all problems considered, the multigrid algorithms scale almost perfectly (i.e., the iteration count is approximately independent of problem size and problem time step). For pure radiation flow problems (i.e., no hydrodynamics), we see speedups in CPU time of factors of ≈15–30 for the largest problems, when comparing the multigrid solvers relative to diagonal scaled conjugate gradient. For the incomplete factorization preconditioners, we see a weak dependence of iteration count on problem size. The speedups observed for pure radiation flow are typically on the order of 10 relative to diagonal scaled conjugate gradient. For radiation–hydrodynamics problems, we again see multigrid scaling perfectly. However, for the problems considered, we see speedups relative to diagonal scaled conjugate gradient of no more than ≈10, with incomplete Cholesky in fact either equaling or outperforming multigrid. We trace these observations to the time step control and the feature of ALE to relax distorted zones.

Classification of normal and abnormal electrogastrograms using multilayer feedforward neural networks.

A neural network approach is proposed for the automated classification of the normal and abnormal EGG. Two learning algorithms, the quasi-Newton and the scaled conjugate gradient method for the multilayer feedforward neural networks (MFNN), are introduced and compared with the error backpropagation algorithm. The configurations of the MFNN are determined by experiment. The raw EGG data, its power spectral data, and its autoregressive moving average (ARMA) modelling parameters are used as the input to the MFNN and compared with each other. Three indexes (the percent correct, sum-squared error and complexity per iteration) are used to evaluate the performance of each learning algorithm. The results show that the scaled conjugate gradient algorithm performs best, in that it is robust and provides a super-linear convergence rate. The power spectral representation and the ARMA modelling parameters of the EGG are found to be better types of the input to the network for this specific application, both yielding a percent correctness of 95% on the test set. Although the results are focused on the classification of the EGG, this paper should provide useful information for the classification of other biomedical signals.

Fast Iterative Solution of Stabilised Stokes Systems. Part I: Using Simple Diagonal Preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. The use of stabilisation methods for convenient (but unstable) mixed elements introduces stabilisation parameters. The authors show how these can be chosen to obtain rapid iterative convergence. The authors propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) that is applicable to symmetric indefinite problems, describe the effects of stabilisation on the algebraic structure of the discrete Stokes operator, and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends. The simple case of diagonal preconditioning is discussed. The results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. It is demonstrated that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete Laplacian are approachable for the Stokes problem.

Preconditioning finite element subsurface flow solutions on distributed memory parallel computers

Due to their low cost and high expansion capabilities, distributed memory parallel computers, consisting of relatively large arrays of processor elements with local memory, offer great potential for solving large scale three-dimensional subsurface flow problems. We consider here two possible solution methods for such detailed modelling using the finite element method. One is based on the commonly used diagonally scaled conjugate gradient method. The other uses an element-by-element preconditioner in the conjugate gradient method which exploits the natural parallelism in the finite element method. Results using both methods are documented for saturated and variably saturated flow simulations carried out on a single Intel i860 processor and an array of Inmos transputer/Intel i860 processors at Lancaster.

Low-latency message communication support for the AP1000

Low-latency communication is the key to achieving a high-performance parallel computer. In using state-of-the-art processors, we must take cache memory into account. This paper presents an architecture for low-latency message comunication and implementation, and performance evaluation. We developed a message controller (MSC) to support low-latency message passing communication for the AP1000, to minimize message handling overhead. MSC sends messages directly from cache memory and automatically receives messages in the circular buffer. We designed communication functions between cells and evaluated communication performance by running benchmark programs such as the Pingpong benchmark, the LINPACK benchmark, the SLALOM benchmark, and a solver using the scaled conjugate gradient method.