Narrow Results

This study concerns the solution of a class of nonlinear second order partial differential equations on the Connection Machine CM-2 massively parallel computer. To solve the nonlinear problem, an “outer” successive approximation technique is first applied in order to iteratively linearize the problem. This results in a sequence of systems of linear equations to be solved for successive nonlinear iterates. A preconditioned conjugate gradient (CG) method is used to solve each “inner” linear system. Either Jacobi preconditioning or a fast direct solver is employed as preconditioner for the conjugate gradient method. The implementation of these methods on the Connection Machine is discussed, and the results of numerical experiments with nested iteration are presented.

Several deterministic and stochastic multi-variable global optimization algorithms (Conjugate Gradient, Nelder–Mead, Quasi-Newton and global) are investigated in conjunction with energy minimization principle to resolve the pressure and volumetric flow rate fields in single ducts and networks of interconnected ducts. The algorithms are tested with seven types of fluid: Newtonian, power law, Bingham, Herschel–Bulkley, Ellis, Ree–Eyring and Casson. The results obtained from all those algorithms for all these types of fluid agree very well with the analytically derived solutions as obtained from the traditional methods which are based on the conservation principles and fluid constitutive relations. The results confirm and generalize the findings of our previous investigations that the energy minimization principle is at the heart of the flow dynamics systems. The investigation also enriches the methods of computational fluid dynamics for solving the flow fields in tubes and networks for various types of Newtonian and non-Newtonian fluids.

Not only in the field of high-performance computing (HPC), field programmable gate arrays (FPGAs) are a soaringly popular accelerator technology. However, they use a completely different programming paradigm and tool set compared to central processing units (CPUs) or even graphics processing units (GPUs), adding extra development steps and requiring special knowledge, hindering widespread use in scientific computing. To bridge this programmability gap, domain-specific languages (DSLs) are a popular choice to generate low-level implementations from an abstract algorithm description. In this work, we demonstrate our approach for the generation of numerical solver implementations based on the multigrid method for FPGAs from the same code base that is also used to generate code for CPUs using a hybrid parallelization of MPI and OpenMP. Our approach yields in a hardware design that can compute up to 11 V-cycles per second with an input grid size of 4096$\times$4096 and solution on the coarsest using the conjugate gradient (CG) method on a mid-range FPGA, beating vectorized, multi-threaded execution on an Intel Xeon processor.

The shortest-residual family of conjugate gradient methods was first proposed by Hestenes and was studied by Pytlak, and Dai and Yuan. Recently, a no-line-search scheme in conjugate gradient methods was given by Sun and Zhang, and Chen and Sun. In this paper, we show the global convergence of two shortest-residual conjugate gradient methods (FRSR and PRPSR) without line search. In addition, computational results are presented to show that the methods with line search have similar numerical behavior to the methods without line search.

Under the Bayesian Ying–Yang (BYY) harmony learning theory, a harmony function has been developed on a BI-directional architecture of the BYY system for Gaussian mixture with an important feature that, via its maximization through a general gradient rule, a model selection can be made automatically during parameter learning on a set of sample data from a Gaussian mixture. This paper further proposes the conjugate and natural gradient rules to efficiently implement the maximization of the harmony function, i.e. the BYY harmony learning, on Gaussian mixture. It is demonstrated by simulation experiments that these two new gradient rules not only work well, but also converge more quickly than the general gradient ones.

We study an optimization-based, non-overlapping, domain decomposition method for a bonded structure theoretically and numerically. The problem is restated as a constrained minimization problem for which the objective functional controls the normal component of stresses across the interface and the constraints are equlibrium equations in subdomains represented by adherents. We show that the minimization problem has a unique solution. We propose a domain decomposition method, based on the conjugate gradient minimization algorithm, which converges theoretically, in a finite number of steps.

We consider the problem of recovering surface vibrations from acoustic pressure measurements taken in the interior or the exterior of a region. We give two formulations of the problem. One is based on a representation of the pressure as layer potentials and the other is based on approximation by a class of specific solutions to the Helmholtz equation. Boundary element methods are developed to approximate the integral operators. A conjugate gradient algorithm based on Lanczos bidiagonalization is applied to compute regularized solutions to the ill-conditioned equations in the presence of measurement errors. Numerical examples which compare the two formulations are presented.

We present a novel parameter choice strategy for the conjugate gradient regularization algorithm which does not assume a priori information about the magnitude of the measurement error. Our approach is to regularize within the Krylov subspaces associated with the normal equations. We implement conjugate gradient via the Lanczos bidiagonalization process with reorthogonalization, and then we construct regularized solutions using the SVD of a bidiagonal projection constructed by the Lanczos process. We compare our method with the one proposed by Hanke and Raus and illustrate its performance with numerical experiments, including detection of acoustic boundary vibrations.

We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient (CG) algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called “fast convergence rates” depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the ℒ² (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.

This paper describes a wavelet-based preconditioning technique for conjugate gradient method for linear systems derived from the Poisson equation. The linear systems solved with a conventional iterative matrix solver resulted in a marked increase in computing time with respect to an increase in grid points. Use of our wavelet-based technique leads to a matrix with a bounded condition number so that computing time is reduced significantly. In this study, one of the simplest wavelets, the Haar wavelet, is used for the purpose of developing a simple but efficient preconditioning algorithm. Simple wavelets having low data communication property such as the Haar wavelet are expected to be suitable for the purpose of improving computing performance. In this study, we also pay attention to the basic characteristics of the Haar-wavelet-based preconditioning method for a Poisson equation solver.

In this paper, we present a new hybrid method to solve a nonlinear unconstrained optimization problem by using conjugate gradient, which is a convex combination of Liu–Storey (LS) conjugate gradient method and Hager–Zhang (HZ) conjugate gradient method. This method possesses the sufficient descent property with Strong Wolfe line search and the global convergence with the strong Wolfe line search. In the end of this paper, we illustrate our method by giving some numerical examples.

In the conjugate gradient iteration-multilevel fast multipole algorithm (CGMLFMA), the matrix-vector multiplication can be implemented by the interactions from nearby region and non-nearby region. Based on CG-MLFMA, an efficient partly approximate iteration technique is developed for solving the scattering from the object with large electrical sizes in this paper. When the iterative error is low than the critical iteration error (CIE), determined by the contributions of nearby region and the accuracy required, the matrix-vector multiplication can be computed approximately by the interactions from nearby region. Compared to the interactions from non-nearby region, the interactions from nearby region only possess a very small portion of total complexity. So the total CPU time can be reduced greatly, although the total number of iteration needed in the present method is nearly same to the one in CG-MLFMA. Numerical results are given to demonstrate the validity and efficiency of the present method.

The aim of this paper is to use the theory of biorthogonal polynomials to derive many algorithms for solving a system of linear equations in a unified framework. These algorithms include extrapolation and projection algorithms, the most important ones being conjugate gradient-type algorithms. New algorithms are also obtained.