STENMIN: A software package for large, sparse unconstrained optimization using tensor methods

We describe a new package for minimizing an unconstrained nonlinear function where the Hessian is large and sparse. The software allows the user to select between a tensor method and a standard method based upon a quadratic model. The tensor method models the objective function by a fourth-order model, where the third- and fourth-order terms are chosen such that the extra cost of forming and solving the model is small. The new contribution of this package consists of the incorporation of an entirely new way of minimizing the tensor model that makes it suitable for solving large, sparse optimization problems efficiently. The test results indicate that, in general, the tensor method is significantly more efficient and more reliable than the standard Newton method for solving large, sparse unconstrained optimization problems. 12 refs., 1 tab

Tensor methods for large, sparse unconstrained optimization

Tensor methods for unconstrained optimization were first introduced by Schnabel and Chow [SIAM J. Optimization, 1 (1991), pp. 293-315], who describe these methods for small to moderate size problems. This paper extends these methods to large, sparse unconstrained optimization problems. This requires an entirely new way of solving the tensor model that makes the methods suitable for solving large, sparse optimization problems efficiently. We present test results for sets of problems where the Hessian at the minimizer is nonsingular and where it is singular. These results show that tensor methods are significantly more efficient and more reliable than standard methods based on Newton`s method

TENSOLVE: A software package for solving systems of nonlinear equations and nonlinear least squares problems using tensor methods

This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns, in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F({ital x}) by a quadratic model, where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two- dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations

Tensor methods for large sparse systems of nonlinear equations

This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time

A laboratory investigation on shear strength behavior of sandy soil: effect of glass fiber and clinker residue content

A study was undertaken to investigate the shear strength parameters of treated sands reinforced with randomly distributed glass fibers by carrying out direct shear test after seven days curing periods. Firstly, we studied the fiber content and fiber length effect on the peak shear strength on samples. The second part gives a parametric analysis on the effect of glass fiber and clinker residue content on the shear strength parameters for two types of uniform Algerian sands having different particle sizes (Chlef sand and Rass sand) with an average relative density Dr = 50%. Finally, the test results show that the combination of glass fiber and clinker residue content can effectively improve the shear strength parameters of soil in comparison with unreinforced soil. For instance, there is a significant gain for the cohesion and friction angle of reinforced sand of Chlef. Compared to unreinforced sand, the cohesion for sand reinforced with different ratios of clinker residue increased by 4.36 to 43.08 kPa for Chlef sand and by 3.1 to 28.64 kPa for Rass sand. The feature friction angles increased from 38.73° to 43.01° (+4.28°), and after the treatment, clinker residue content of soil evaluated to 5% (WRC = 5%)

Solvers for Systems of Nonlinear Algebraic Equations - Their Sensitivity to Starting Vectors

In this note we compare the sensitivity of six advanced solvers for systems of nonlinear algebraic equations to the choice of starting vectors. We will report on results of our experiments in which, for each test problem, the calculated solution was used as the center from which we have moved away in various directions and observed the behavior of each solver attempting to find the solution. We are particularly interested in determining the best global starting vectors. Experimental results are presented and discussed

First-principles computational study on structural, elastic, magnetic, electronic, and thermoelectric properties of Co

In this research work, first-principles computational study is performed on the structural, elastic, thermal, magnetic, electronic, and thermoelectric properties of the ternary Heusler compound Co2MnGe in its cubic phase. For this purpose, the “full potential linearized augmented plane-wave FP-L(APW + lo)” approach realized in the WIEN2k code is employed. To determine total energy, the exchange–correlation energy/potential part is treated within the “Perdew–Burke–Ernzerhof (PBE)” parameterized approach of “generalized gradient approximation (GGA) and modified Becke–Johnson (mBJ)” schemes. The magnetic phase stability was predicted via quantum mechanically total energy calculations for both non-magnetic and magnetic phases. Our obtained results for total energy show that the title material is stable in the ferromagnetic phase. The analysis of the profile of density of states (DOS), band structure plots, and the calculations of spin magnetic moment endorse the semi-metallic nature of the title compound. Calculations of the elastic constants, Cij, and results of the elastic moduli, such as bulk modulus (B), shear modulus (G), Young modulus (E), Poisson ratio (ν), and ratio B/G, are reported and analyzed as well. Gibbs computational code based on the “quasi-harmonic Debye model” is used to explore thermal properties, whereas parameters to understand the thermoelectric behavior, BoltzTrap code based on Boltzmann theory for transport properties is applied. Besides that, the chemical potential effect on the Seebeck coefficient and power factor is also analyzed at temperatures 300, 600, and 900 K. The results of thermoelectric parameters of the title Heusler compound, for the spin-down channel, are found good; hence, the obtained results highlight the title compound as a potential candidate for thermoelectric devices

Bibliography

frontal Package, technical report TR-93-020, Computer and Information Sciences Dept., University of Florida, June 1993. [12] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li and J. W. H. Liu, A Supernodal Approach to Sparse Partial Pivoting, preprint, Xerox Palo Alto Research Center, 1995. [13] I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, London, 1986. 102 APPENDIX A. A.2. SOFTWARE 101 ` indicator for the choice of t T the execution trace t numerical threshold U right upper triangle matrix V set of intermediate variables v i node; variable ¯ v i to v i corresponding vector of adjoints v c(i) c(i)th critical var