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Hessian Matrix (hessian + matrix)
Selected AbstractsGeometry optimization in density functional methodsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 9 2004J. Ulises Reveles Abstract The geometry optimization in delocalized internal coordinates is discussed within the framework of the density functional theory program deMon. A new algorithm for the selection of primitive coordinates according to their contribution to the nonredundant coordinate space is presented. With this new selection algorithm the excessive increase in computational time and the deterioration of the performance of the geometry optimization for floppy molecules and systems with high average coordination numbers is avoided. A new step selection based on the Cartesian geometry change is introduced. It combines the trust radius and line search method. The structure of the new geometry optimizer is described. The influence of the SCF convergence criteria and the grid accuracy on the geometry optimization are discussed. A performance analysis of the new geometry optimizer using different start Hessian matrices, basis sets and grid accuracies is given. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1109,1116, 2004 [source] Can error source terms in forecasting models be represented as Gaussian Markov noises?THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 609 2005C. Nicolis Abstract The repercussions of model error on the long term climatological means and on the variability around them are analysed. The extent to which a stochastic representation of error source terms provides a universal correcting mechanism is addressed. General relations are derived linking the model error to the climatological means and the variability properties of a forecasting model subjected to a correcting Gaussian Markov noise on the basis of moment equations associated with Fokker,Planck and Liouville type equations. These relations are implemented in a variety of models giving rise to regular and to chaotic solutions. As it turns out, forecasting models fall into distinct universality classes differing in their response to the effect of noise according to the structure of the Jacobian and the Hessian matrices of the model phase-space velocity. It is concluded that different trends may exist in which the ,correcting' noise tends to depress or, on the contrary, amplify the model error. Copyright © 2005 Royal Meteorological Society. [source] Empirical properties of duality theoryAUSTRALIAN JOURNAL OF AGRICULTURAL & RESOURCE ECONOMICS, Issue 1 2002Jayson L. Lusk This research examines selected empirical properties of duality relationships. Monte Carlo experiments indicate that Hessian matrices estimated from the normalised unrestricted profit, restricted profit and production functions yield conflicting results in the presence of measurement error and low relative price variability. In particular, small amounts of measurement error in quantity variables can translate into large errors in uncompensated estimates calculated via restricted and unrestricted profit and production functions. These results emphasise the need for high quality data when estimating empirical models in order to accurately determine dual relationships implied by economic theory. [source] Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2005Jean-François Remacle Abstract An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient two- and three-dimensional problems governed by Euler's equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain. Copyright © 2004 John Wiley & Sons, Ltd. [source] Application of second-order adjoint technique for conduit flow problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2007T. Kurahashi Abstract This paper presents the way to obtain the Newton gradient by using a traction given by the perturbation for the Lagrange multiplier. Conventionally, the second-order adjoint model using the Hessian/vector products expressed by the product of the Hessian matrix and the perturbation of the design variables has been researched (Comput. Optim. Appl. 1995; 4:241,262). However, in case that the boundary value would like to be obtained, this model cannot be applied directly. Therefore, the conventional second-order adjoint technique is extended to the boundary value determination problem and the second-order adjoint technique is applied to the conduit flow problem in this paper. As the minimization technique, the Newton-based method is employed. The Broyden,Fletcher,Goldfarb,Shanno (BFGS) method is applied to calculate the Hessian matrix which is used in the Newton-based method and a traction given by the perturbation for the Lagrange multiplier is used in the BFGS method. Copyright © 2007 John Wiley & Sons, Ltd. [source] Optimization of strong and weak coordinatesINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 12 2006Marcel Swart Abstract We present a new scheme for the geometry optimization of equilibrium and transition state structures that can be used for both strong and weak coordinates. We use a screening function that depends on atom-pair distances to differentiate strong coordinates from weak coordinates. This differentiation significantly accelerates the optimization of these coordinates, and thus of the overall geometry. An adapted version of the delocalized coordinates setup is used to generate automatically a set of internal coordinates that is shown to perform well for the geometry optimization of systems with weak and strong coordinates. For the Baker test set of 30 molecules, we need only 173 geometry cycles with PW91/TZ2P calculations, which compares well with the best previous attempts reported in literature. For the localization of transition state structures, we generate the initial Hessian matrix, using appropriate force constants from a database. In this way, one avoids the explicit computation of the Hessian matrix. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] Remarks on the updated Hessian matrix methodsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 6 2003Josep Maria Bofill Abstract Optimizing a function with respect to a set of variables using the quasi-Newton,Raphson method implies updating the Hessian matrix at each iteration. The Broyden,Fletcher,Goldfarb,Shanno update formula is used for minimization and the Murtagh,Sargent,Powell update formula for optimization of first-order saddle points. Two new formulae are proposed to update the Hessian matrix. One of these formulae is derived using exponential weights and should be used to locate first-order saddle points. The second formula is a modification of the TS,Broyden,Fletcher,Goldfarb,Shanno update and could used for both minimum and first-order saddle point optimizations. These two update Hessian matrix formulae present a performance that is the same and in many cases better that the Broyden,Fletcher,Goldfarb,Shanno and Murtagh,Sargent,Powell formulae. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 324,332, 2003 [source] Identification and thermodynamic treatment of several types of large-amplitude motionsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 14 2005Gernot Katzer Abstract We present a partially automated method for the thermodynamic treatment of large-amplitude motions. Starting from the molecular geometry and the Hessian matrix, we evaluate anharmonic partition functions for selected vibrational degrees of freedom. Supported anharmonic vibration types are internal rotation and inversion (oscillation in a double-well potential). By heuristic algorithms, we identify internal rotations in most cases automatically from the Hessian eigenvectors, and we also estimate the parameters of anharmonic partition functions (e.g., potential barrier, periodicity, and symmetry number) with thermodynamically sufficient precision. We demonstrate the validity of our schemes by comparison to pointwise calculated ab initio potential curves. © 2005 Wiley Periodicals, Inc. J Comput Chem 14: 1438,1451, 2005 [source] Calculation of the vibration frequencies of ,-quartz: The effect of Hamiltonian and basis setJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 15 2004C. M. Zicovich-Wilson Abstract The central-zone vibrational spectrum of ,-quartz (SiO2) is calculated by building the Hessian matrix numerically from the analytical gradients of the energy with respect to the atomic coordinates. The nonanalytical part is obtained with a finite field supercell approach for the high-frequency dielectric constant and a Wannier function scheme for the evaluation of Born charges. The results obtained with four different Hamiltonians, namely Hartree,Fock, DFT in its local (LDA) and nonlocal gradient corrected (PBE) approximation, and hybrid B3LYP, are discussed, showing that B3LYP performs far better than LDA and PBE, which in turn provide better results than HF, as the mean absolute difference from experimental frequencies is 6, 18, 21, and 44 cm,1, respectively, when a split valence basis set containing two sets of polarization functions is used. For the LDA results, comparison is possible with previous calculations based on the Density Functional Perturbation Theory and usage of a plane-wave basis set. The effects associated with the use of basis sets of increasing size are also investigated. It turns out that a split valence plus a single set of d polarization functions provides frequencies that differ from the ones obtained with a double set of d functions and a set of f functions on all atoms by on average less than 5 cm,1. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1873,1881, 2004 [source] Maximum likelihood estimation in space time bilinear modelsJOURNAL OF TIME SERIES ANALYSIS, Issue 1 2003YUQING DAI The space time bilinear (STBL) model is a special form of a multiple bilinear time series that can be used to model time series which exhibit bilinear behaviour on a spatial neighbourhood structure. The STBL model and its identification have been proposed and discussed by Dai and Billard (1998). The present work considers the problem of parameter estimation for the STBL model. A conditional maximum likelihood estimation procedure is provided through the use of a Newton,Raphson numerical optimization algorithm. The gradient vector and Hessian matrix are derived together with recursive equations for computation implementation. The methodology is illustrated with two simulated data sets, and one real-life data set. [source] A projected-steepest-descent potential-reduction algorithm for convex programming problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 10 2004Yixun Shi Abstract A recent work of Shi (Numer. Linear Algebra Appl. 2002; 9: 195,203) proposed a hybrid algorithm which combines a primal-dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. In this paper, we make efforts to further reduce the computational costs. We notice that in order to obtain the steepest descent direction of the potential function, the Hessian matrix of second order partial derivatives of the objective function needs to be computed. To avoid this, we in this paper propose another hybrid algorithm which uses a projected steepest descent direction of the objective function instead of the steepest descent direction of the potential function. The complexity of the original potential reduction algorithm still remains valid but the overall computational cost is further reduced. Our numerical experiments are also reported. Copyright © 2004 John Wiley & Sons, Ltd. [source] The Breakdown of the Minimum Polarizability Principle in Vibrational Motions as an Indicator of the Most Aromatic CenterCHEMISTRY - A EUROPEAN JOURNAL, Issue 20 2005Miquel Torrent-Sucarrat Dr. Abstract The vibrational motions that disobey the minimum polarizability principle (MPP) in ,-conjugated molecules are distortions of the equilibrium geometry that produce a reduction in the polarizability due to the localization of , electrons. For aromatic species, this electronic localization is responsible for the subsequent reduction in the aromaticity of the system. In the present work, we diagonalize the Hessian matrix of the polarizability with respect to the vibrational nontotally symmetric normal coordinates, to calculate the nontotally symmetric distortions that produce the maximum breakdown of the MPP in a series of twenty polycyclic aromatic hydrocarbons. It is shown that the nuclear displacements that break the MPP have larger components in those rings that possess the highest local aromaticity. Thus, these vibrational motions can be used as an indicator of local aromaticity. [source] | ||||||||||||||||