Advanced solution methods in structural optimization based on evolution strategies

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.

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.

A robust and efficient methodology is presented for treating large-scale reliability-based structural optimization problems. The optimization is performed with evolution strategies, while the reliability analysis is carried out with the Monte Carlo simulation method incorporating the importance sampling technique to reduce the sample size. Efficient hybrid methods are implemented to solve the reanalysis-type problems that arise in the optimization phase with evolution strategies and in the reliability analysis with Monte Carlo simulations. These hybrid solution methods are based on the preconditioned conjugate gradient algorithm using efficient preconditioning schemes. The numerical tests presented demonstrate the computational advantages of the proposed methods, which become more pronounced for large-scale optimization problems.

The effects of initial solution and order of observations on the solving of mixed model equations (MME) by traditional iterative algorithms-Gauss-Seidel, successive over-relaxation (SOR), and second-order Jacobi-and by preconditioned conjugate gradient algorithms-scaled conjugate gradient (SCG) and incomplete Cholesky conjugate gradient (ICCG) simulation were investigated by using data in a computer. With SOR, the use of adjusted initial solutions based on phenotype and heritability of the trait and order of the animal within the trait in the MME was effective for fast convergence. Adjustment of initial solutions was not an advantage in preconditioned conjugate gradient algorithms. Of the five algorithms, ICCG required the lowest number of iterations until convergence. However, ICCG needed the largest central processing unit (CPU) time to calculate one round of iteration. The SCG algorithm was an attractive alternative for solving MME because it required the second fewest rounds of iteration and had the shortest CPU time per round of iteration until convergence.

Parallel preconditioned conjugate gradient algorithm on GPU

It is well known that the preconditioned conjugate gradient algorithms (PCG) work very well (for both symmetric and nonsymmetric problems) if the preconditioned is “good enough”. But, in many cases, “good enough” means that for solving (during the application of (PCG)) the systems in which the preconditioning matrix appears too much computational work is necessary. In this paper we present, for systems arising from finite differences or finite element discretizations, both a method of preconditioning and some possibilities to approximate the exact solutions of the systems in which the preconditioner appears. We present a theoretical study of the influence of these approxiamtions on the convergence behaviour of the (PCG) algorithm. In the last section we present some test on the one-dimensional steady state heat convective transfer problem.

Phase Shifting Full-Field Interferometric Methods for Determination of In-Plane Tensorial Stress

A novel optimization method based on a differential evolution (DE) strategy for designing low-weight truss structures is presented for both continuous and discrete variables. Applications of this technique on the optimization of a benchmark-type truss structure with continuous and discrete variables are given to evaluate its effectiveness. Results are compared with various classical and evolutionary optimization methods which show that the proposed procedure based on DE outperforms other methods and can effectively be applied to the optimization problems of truss structures with both continuous and discrete design variables.

A multi-GPU parallel optimization model for the preconditioned conjugate gradient algorithm

Bundle block adjustment of large-scale remote sensing data with Block-based Sparse Matrix Compression combined with Preconditioned Conjugate Gradient

Most structural optimization methods are based on mathematical algorithms that require substantial gradient information. The selection of the starting values is also important to ensure that the algorithm converges to the global optimum. This paper describes a new structural configuration optimization method based on the harmony search (HS) meta-heuristic algorithm. The HS algorithm does not require initial values and uses a random search instead of a gradient search, so derivative information is unnecessary. A benchmark truss example is presented to demonstrate the effectiveness and robustness of the proposed approach compared to other optimization methods. Results reveal that the proposed approach is capable of solving configuration optimization problems, and may yield better solutions than those obtained using earlier methods.

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.

Two accelerated optimization algorithms are presented for computing approximate Tucker tensor decompositions, formulated using orthonormal factor matrices, by minimizing error as measured by the Frobenius norm. The first is a nonlinearly preconditioned conjugate gradient (NPCG) algorithm, wherein a nonlinear preconditioner is used to generate a direction which replaces the gradient in the nonlinear conjugate gradient iteration. The second is a nonlinear GMRES (N-GMRES) algorithm, in which a linear combination of past iterates and a tentative new iterate, generated by a nonlinear preconditioner, is minimized to produce an improved search direction. The Euclidean versions of these methods are extended to the manifold setting, where optimization on Grassmann manifolds is used to handle orthonormality constraints and to allow isolated minimizers. Several modifications are required for use on manifolds: logarithmic maps are used to determine required tangent vectors, retraction mappings are used in the line se...

Parallel PCG Algorithm On Distributed Network By PVM

Crystallization is an important part of many chemical industries. Efforts are being invested to improve the performance of the crystallization process by designing novel crystallizers. An important aspect in the development of new crystallizers is the ability to describe the behavior of such units in terms of rigorous dynamic mathematical models and solving the resulting models efficiently. The current bottleneck in modeling crystallization systems is the complexities associated with the crystal birth, growth, and death processes using population balance equations. In this article, various crystal birth, death, and growth models are introduced and reviewed. Population balance models as well as solution methods (e.g., analytical, moment methods, discretization (classes/sectional) methods and Monte Carlo methods) are also reviewed, and new advances in solution methods are described. Population balance equations are used in other fields, and developments from other fields that can be extended to crystal population balance equations are included in this review.