cavity approach to the spectral density of sparse symmetric random matrices the spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method we consider two cases matrices whose associated graphs are locally treelike and sparse covariance matrices we derive a closed set of equations from which the density of eigenvalues can be efficiently calculated within this approach the wigner semicircle law for gaussian matrices and the marcenko pastur law for covariance matrices are recovered easily our results are compared with numerical diagonalization showing excellent agreement

an evaluation of back propagation neural networks for the optimal design of structural systems part ii numerical evaluation abstract in the companion paper the neural network training procedures used in the present evaluation were described in detail also in the companion paper two new training methods that used the derivative information to build global approximations and a mixed local global approximation scheme were proposed in addition to the common function based training method in this paper we carry out an exhaustive numerical evaluation of the proposed and existing methods using five test problems with varying complexity approximately 1100 test cases were run as part of the numerical evaluations first the efficiency of three different optimization algorithms used for training including a sequential quadratic programming algorithm a levenberg marquardt algorithm and a genetic algorithm is evaluated the developed training schemes were next evaluated by comparing with the classical structural optimization methods the two new derivative based training methods were found to be more accurate but less efficient than function based training further compared with classical structural optimization by non linear programming algorithms including design sensitivity analysis global approximation and mixed local global approximation using neural networks were found to be far less efficient with careful training the neural networks were found capable of predicting the exact solution but this ability depended considerably on the complexity of the original problem in general the accuracy of the network which is a function of the training effort suffered considerably in large problems the most surprising observation of the present study was that the training process required over 90 of the total solution time even when considerable effort was exercised at using the most efficient training procedure as expected the use of neural network approximations once training was complete speeded up the optimal design process significantly

stabilization of parameter estimation for kriging based approximation with empirical semivariogram abstract this paper discusses stabilization of an optimization procedure to determine an appropriate semivariogram parameter using the empirical semivariogram for kriging based approximation from a viewpoint of a computational cost for constructing a surrogate model a semivariogram fitting approach using the empirical semivariogram can be usable for the parameter estimation of a semivariogram function however instability of the optimization procedure for determination of the semivariogram parameter may be observed in some cases and it causes to generate an invalid surrogate model for this problem a simple technique for stabilization of the optimization for the parameter determination is proposed in this paper the proposed approach employs a normalization technique of input data with respect to values of variables and an objective function the proposed method is applied to some numerical examples and the numerical results illustrate validity and effectiveness of the proposed method

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

derivative free optimization of expensive functions with computational error using weighted regression we propose a derivative free algorithm for optimizing computationally expensive functions with computational error the algorithm is based on the trust region regression method by conn scheinberg and vicente a r conn k scheinberg and l n vicente ima j numer anal 28 2008 pp 721 748 but uses weighted regression to obtain more accurate model functions at each trust region iteration a heuristic weighting scheme is proposed that simultaneously handles i differing levels of uncertainty in function evaluations and ii errors induced by poor model fidelity we also extend the theory of lambda poisedness and strong lambda poisedness to weighted regression we report computational results comparing interpolation regression and weighted regression methods on a collection of benchmark problems weighted regression appears to outperform interpolation and regression models on nondifferentiable functions and functions with deterministic noise

an efficient algorithm for kriging approximation and optimization with large scale sampling data abstract this paper describes an algorithm to improve a computational cost for estimation using the kriging method with a large number of sampling data an improved formula to compute the weighting coefficient for kriging estimation is proposed the sherman morrison woodbury formula is applied to solving an approximated simultaneous equation to determine a weighting coefficient a profile of the matrix is reduced by sorting of given data applying the proposal formula to several examples indicates its characteristics as a numerical example layout optimization of a beam structure for eigenfrequency maximization is solved the results show an applicability and effectiveness of the proposed method

on the convergence of the multidirectional search algorithm this paper presents the convergence analysis for the multidirectional search algorithm a direct search method for unconstrained minimization the analysis follows the classic lines of proofs of convergence for gradient related methods the novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary although it is assumed that the function is continuously differentiable over some subset of the domain the proof can be extended to treat most nonsmooth cases of interest the argument breaks down only at points where the derivative exists but is not continuous finally it is shown how a general convergence theory can be developed for an entire class of direct search methods which includes such methods as the factorial design algorithm and the pattern search algorithm that share a key feature of the multidirectional search algorithm

reduced aliasing formulations of the convective terms within the navier stokes equations for a compressible fluid the effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized navier stokes equations is examined in the presence of a non trivial density spectrum fourier analysis shows that the existing skew symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form several formulations of the convective term including a new formulation proposed for cubically nonlinear terms are tested in direct numerical simulation dns of decaying compressible isotropic turbulence both in chemically inert small density fluctuations and reactive cases large density fluctuations and for different degrees of resolution in the dns of reactive turbulent flow the new cubic skew symmetric form gives the most accurate results consistent with the spectral error analysis and at the lowest cost in marginally resolved dns and les poorly resolved by definition the new cubic skew symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high order dissipative filters

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

on trust region methods for unconstrained minimization without derivatives we consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function then a new vector of variables is calculated by minimizing the current model within a trust region techniques are described for adjusting the trust region radius and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model particular attention is given to quadratic models with diagonal second derivative matrices because numerical experiments show that they are often more efficient than full quadratic models for general objective functions finally some recent research on the updating of full quadratic models is described briefly using fewer interpolation equations than before the resultant freedom is taken up by minimizing the frobenius norm of the change to the second derivative matrix of the model a preliminary version of this method provides some very promising numerical results

constrained optimization involving expensive function evaluations a sequential approach abstract this paper presents a new sequential method for constrained nonlinear optimization problems the principal characteristics of these problems are very time consuming function evaluations and the absence of derivative information such problems are common in design optimization where time consuming function evaluations are carried out by simulation tools e g fem cfd classical optimization methods based on derivatives are not applicable because often derivative information is not available and is too expensive to approximate through finite differencing the algorithm first creates an experimental design in the design points the underlying functions are evaluated local linear approximations of the real model are obtained with help of weighted regression techniques the approximating model is then optimized within a trust region to find the best feasible objective improving point this trust region moves along the most promising direction which is determined on the basis of the evaluated objective values and constraint violations combined in a filter criterion if the geometry of the points that determine the local approximations becomes bad i e the points are located in such a way that they result in a bad approximation of the actual model then we evaluate a geometry improving instead of an objective improving point in each iteration a new local linear approximation is built and either a new point is evaluated objective or geometry improving or the trust region is decreased convergence of the algorithm is guided by the size of this trust region the focus of the approach is on getting good solutions with a limited number of function evaluations

a numerical study of sparse random matrices a numerical study is presented for the eigensolution statistics of largen n real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row the model shows classical percolation and quantum localization transitions atp c 1 andp q 1 respectively in the rigid limitp n we demonstrate that the averaged density of states follows the wigner semicircle law and the corresponding nearest energy level spacing distribution functionp s obeys the wigner surmise in the very sparse matrix limitp n withp p q a singularity e 1 e is found as e 0 and exponential tails develop in the high e regions but thep s distribution remains consistent with level repulsion the localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations the valuep q 1 4 is roughly estimated in agreement with previous studies of the anderson transition in dilute bethe lattices

kriging hyperparameter tuning strategies response surfaces have been extensively used as a method of building effective surrogate models of high fidelity computational simulations of the numerous types of response surface models kriging is perhaps one of the most effective due to its ability to model complicated responses through interpolation or regression of known data while providing an estimate of the error in its prediction there is however little information indicating the extent to which the hyperparameters of a kriging model need to be tuned for the resulting surrogate model to be effective the following paper addresses this issue by investigating how often and how well it is necessary to tune the hyperparameters of a kriging model as it is updated during an optimization process to this end an optimization benchmarking procedure is introduced and used to assess the performance of five different tuning strategies over a range of problem sizes the results of this benchmark demonstrate the performance gains that can be associated with reducing the complexity of the hyperparameter tuning process for complicated design problems the strategy of tuning hyperparameters only once after the initial design of experiments is shown to perform poorly

optimal design of frame structures using multivariate spline approximation the objective of this work is to demonstrate the efficiency and applicability of the optimization algorithm using a multivariate spline approximation in the present algorithm based on the function values and first order derivatives of the constraints available at the intermediate points of optimization an explicit approximation of constraint functions is created by using the least squares spline algorithm the nonlinearity of the function is adaptively updated using feedback information from the previous two iterations for finding the order of spline approximation in addition constraint deletion and design variable linking concepts are employed in solving the approximate problem by using the quasianalytical sequential quadratic programming and dual methods the behavior constraints include stresses displacements and local buckling in the optimum design of frame structures to demonstrate the broad applicability of the optimization algorithm the cross sectional dimensions are directly selected as the design variables for frame problems having thin walled rectangular and tube cross sectional members and the cross sectional areas are selected as design variables for an section problem

random symmetric matrices with a constraint the spectral density of random impedance networks we derive the mean eigenvalue density for symmetric gaussian random n x n matrices in the limit of large n with a constraint implying that the row sum of matrix elements should vanish the result is shown to be equivalent to a result found recently for the average density of resonances in random impedance networks y v fyodorov j phys a 32 7429 1999 in the case of banded matrices the analytical results are compared with those extracted from the numerical solution of kirchhoff equations for quasi one dimensional random impedance networks

an enhanced hybrid method for the simulation of highly skewed non gaussian stochastic fields abstract in this paper an enhanced hybrid method ehm is presented for the simulation of homogeneous non gaussian stochastic fields with prescribed target marginal distribution and spectral density function the presented methodology constitutes an efficient blending of the deodatis micaletti method with a neural network based function approximation precisely the function fitting ability of neural networks based on the resilient back propagation rprop learning algorithm is employed to approximate the unknown underlying gaussian spectrum the resulting algorithm can be successfully applied for simulating narrow banded fields with very large skewness at a fraction of the computing time required by the existing methods its computational efficiency is demonstrated in three numerical examples involving fields that follow the beta and lognormal distributions

a multiscale strategy for structural optimization this paper presents a multiscale strategy dedicated to structural optimization the applications concern the study of geometric details such as holes surface profiles etc within the structures with frictional contacts the first characteristic of the method is that it uses a micro macro approach this approach is based on a domain decomposition into substructures and interfaces which involves the resolution of independent micro problems in each substructure and transfers macro information only through the interfaces the second characteristic is the use of a multiresolution strategy in order to reduce the computation cost for problems with evolving design parameters the last characteristic is the capability to model the geometry of details without remeshing thanks to two features the use of a local enrichment method and the use of level set functions to easily modify the boundary of the detail during the optimization process

bayesian regularization neural networks for optimizing fluid flow processes this paper details the application of neural networks and evolutionary strategies for shape optimization problems these commonly grouped under the name soft computing methods are quite recent methods and in demand for application to engineering optimization tasks for complex problems like in non convex optimization such heuristic techniques are able to outperform conventional optimization methods we show that the use of progressive network models can yield satisfactory results when optimizing engineering relevant applications numerical examples are given

on the density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

localization in random bipartite graphs numerical and empirical study we investigate adjacency matrices of bipartite graphs with a power law degree distribution motivation for this study is twofold first vibrational states in granular matter and jammed sphere packings second graphs encoding social interaction especially electronic commerce we establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph at the jamming threshold where the two parts have the same size localization vanishes we found that the multifractal spectrum is nontrivial in the delocalized phase but still near the mobility edge we also study an empirical bipartite graph namely the amazon reviewer item network we found that in this specific graph the mobility edge disappears and we draw a conclusion from this fact regarding earlier empirical studies of the amazon network

thb splines multi patch parameterization for multiply connected planar domains via template segmentation abstract given a planar multiply connected domain we provide a multi patch parameterization exploiting a segmentation algorithm and the construction of templates the segmentation is a fundamental step which allows to split the input domain into patches topologically equivalent to quadrilaterals templates are prototypes of segmentation that will be chosen according to the shape of the given domain we will compute a bijective harmonic mapping h defined between the input and the chosen template the segmentation will be transferred from the prototype template to the input by approximating the inverse mapping templates are equipped with a multi patch structure and each patch i is parametrized individually by a geometry mapping g i the final multi patch parameterization of is achieved exploiting the combination of each geometry mapping g i and the inverse of the mapping h obtained by projection into the thb spline space

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

orbit optimization by radial basis function interpolation in trust regions we present a new derivative free algorithm orbit for unconstrained local optimization of computationally expensive functions a trust region framework using interpolating radial basis function rbf models is employed the rbf models considered often allow orbit to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques approximation guarantees are obtained by ensuring that a subset of the interpolation points is sufficiently poised for linear interpolation the rbf property of conditional positive definiteness yields a natural method for adding additional points we present numerical results on test problems to motivate the use of orbit when only a relatively small number of expensive function evaluations are available results on two very different application problems calibration of a watershed model and optimization of a pde based bioremediation plan are also encouraging and support orbit s effectiveness on blackbox functions for which no special mathematical structure is known or available

finding local optima of high dimensional functions using direct search methods this paper focuses on a subclass of box constrained non linear optimization problems we are particularly concerned with settings where gradient information is unreliable or too costly to calculate and the function evaluations themselves are very costly this encourages the use of derivative free optimization methods and especially a subclass of these referred to as direct search methods the thrust of our investigation is twofold first we implement and evaluate a number of traditional direct search methods according to the premise that they should be suitable as local optimizers when used in a metaheuristic framework second we introduce a new direct search method based on scatter search designed to remedy the lack of a good derivative free method for solving problems of high dimensions our new direct search method has convergence properties comparable to those of existing methods in addition to being able to solve larger problems more effectively

stochastic field representation using bi fidelity combination of proper orthogonal decomposition and kriging abstract in the current paper efficient uncertainty quantification uq of high dimensional stochastic fields is performed via a bi fidelity surrogate model the method is based on combination of proper orthogonal decomposition pod and kriging method in the developed method the trend function of the kriging model is estimated via a low cost pod whilst the stochastic gaussian contribution is tuned using a limited number of high fidelity computations the proposed method is applied to three test cases including a highly nonlinear test function thermally driven flow in a cavity with stochastic wall temperatures and turbulent heat transfer in a ribbed channel with stochastic heat flux boundary condition implementing both pod and pod kriging methods leads to significant reduction of computational cost in comparison to the classical full polynomial chaos expansion pce with least cost saving of more than 60 furthermore in all cases combined pod kriging method performs either superior or at least similar to the pod method more specifically in surrogate models with lower number of low and high fidelity samples greater gains in accuracy are obtained consequently for a specific level of accuracy pod kriging method requires lower computational cost

finitely coordinated models for low temperature phases of amorphous systems we introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite coordination effects on the low temperature behaviour of finite dimensional systems our models use a description in terms of small deviations of particle coordinates from a set of reference positions particularly appropriate for the description of low temperature phenomena a born von karman type expansion with random coefficients is used to model effects of frozen heterogeneities the key quantity appearing in the theoretical description is a full distribution of effective single site potentials which needs to be determined self consistently if microscopic interactions are harmonic the effective single site potentials turn out to be harmonic as well and the distribution of these single site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels for structural glasses characterized by frustration and anharmonicities in the microscopic interactions the distribution of single site potentials involves anharmonicities of all orders and both single well and double well potentials are observed the latter with a broad spectrum of barrier heights the appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single site potentials as previously observed for fully connected systems double well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account

hybrid approximation algorithm with kriging and quadratic polynomial based approach for approximate optimization this paper describes a new hybrid algorithm that uses a kriging and quadratic polynomial based approach for approximate optimization the kriging method is used for generating a global approximation model and the polynomial based approximation method is used for generating a local approximation model the kriging system is only used to construct a polynomial based locally approximate model by estimating some function values and hessian components of an estimated surface the number of kriging estimations can be reduced in comparison with direct kriging based optimization and a local optimum solution on an approximated surface can be clearly estimated without use of an optimization procedure based on a local appropriate quadratic polynomial model numerical examples of engineering optimization using the proposed method illustrate validity and effectiveness of the proposed method copyright 2006 john wiley sons ltd

sobolev seminorm of quadratic functions with applications to derivative free optimization this paper studies the h 1 sobolev seminorm of quadratic functions the research is motivated by the least norm interpolation that is widely used in derivative free optimization we express the h 1 seminorm of a quadratic function explicitly in terms of the hessian and the gradient when the underlying domain is a ball the seminorm gives new insights into least norm interpolation it clarifies the analytical and geometrical meaning of the objective function in least norm interpolation we employ the seminorm to study the extended symmetric broyden update proposed by powell numerical results show that the new thoery helps improve the performance of the update apart from the theoretical results we propose a new method of comparing derivative free solvers which is more convincing than merely counting the numbers of function evaluations

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

parallel radial basis function methods for the global optimization of expensive functions abstract we introduce a master worker framework for parallel global optimization of computationally expensive functions using response surface models in particular we parallelize two radial basis function rbf methods for global optimization namely the rbf method by gutmann gutmann h m 2001a a radial basis function method for global optimization journal of global optimization 19 3 201 227 gutmann rbf and the rbf method by regis and shoemaker regis r g shoemaker c a 2005 constrained global optimization of expensive black box functions using radial basis functions journal of global optimization 31 153 171 cors rbf we modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel we compare the performance of the two parallel rbf methods with a parallel multistart derivative based algorithm a parallel multistart derivative free trust region algorithm and a parallel evolutionary algorithm on eleven test problems and on a 6 dimensional groundwater bioremediation application the results indicate that the two parallel rbf algorithms are generally better than the other three alternatives on most of the test problems moreover the two parallel rbf algorithms have comparable performances on the test problems considered finally we report good speedups for both parallel rbf algorithms when using a small number of processors

truncated newton training algorithm for neurocomputational viscoplastic model we present an estimate approach to compute the viscoplastic behavior of a polymer matrix composite under different thermomechanical environments this investigation incorporates computational neural network as the tool for determining the creep behavior of the composite we propose a new second order learning algorithm for training the multilayer networks training in the neural network is generally specified as the minimization of an appropriate error function with respect to parameters of the network weights and learning rates corresponding to excitory and inhibitory connections we propose here a technique for error minimization based on the use of the truncated newton tn large scale unconstrained minimization technique with quadratic convergence rate this technique offers a more sophisticated use of the gradient information compared to simple steepest descent or conjugate gradient methods in this work we briefly specify the necessary details for implementing the tn method for training the neural networks that predicts the viscoplastic behavior of the polymeric composite we provide comparative experimental results and explicit model results to verify the effectiveness of the neural networks based model these results verify the superiority of the present approach compared to the explicit modeling scheme moreover the present study demonstrates for the first time the feasibility of introducing the tn method with quadratic convergence rate to the field of neural networks

objective derivative free methods for constrained optimization we propose feasible descent methods for constrained minimization that do not make explicit use of the derivative of the objective function the methods iteratively sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective function value is not sampled the search arcs are obtained by projecting search direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone of first order feasible variations at the current iterate we show that these methods have desirable convergence properties under certain regularity assumptions on the constraints in the case of linear constraints the projections are redundant and the regularity assumptions hold automatically numerical experience with the methods in the linearly constrained case is reported

an evaluation of back propagation neural networks for the optimal design of structural systems part i training procedures design optimization using approximations based on feed forward back propagation neural network is the topic of much recent research the neural network schemes that have been proposed in the literature for optimal design of structural systems differ in their architecture and training procedures furthermore their utility vis a vis classical optimization techniques is not always clear a systematic comparison of the efficiency and accuracy of the neural network based solution schemes to classical structural optimization techniques is the aim of this and the companion paper in this paper the neural network training procedures used in the present evaluation are described in detail when using first order nonlinear programming algorithms with neural networks the ability to approximate derivatives is important therefore mainly for completeness of evaluation two new training methods that use the derivative information are proposed in addition to the now common function based training method the first method uses the derivatives to create additional training points in the vicinity of the original points based on taylor s series expansion the second method attempts to minimize the error in derivatives while imposing the error in output functions as constraint expressions for analytical derivatives are derived for both function based and derivative based training significant savings in computational time are reported when calculating derivatives using built in analytical derivatives instead of using finite difference derivatives in the companion paper the proposed methods are applied to solve five optimization problems with varying degree of complexity approximately 1100 test cases are executed in the companion paper to compare the accuracy and efficiency of neural network based optimization with the classical approaches

stability analysis for different formulations of the nonlinear term in p n p n 2 spectral element discretizations of the navier stokes equations we show that for the pn pn 2 spectral element method in which velocity and pressure are approximated by polynomials of order n and n 2 respectively numerical instabilities can occur in the spatially discretized navier stokes equations both a staggered and nonstaggered arrangement of the n 2 pressure points are considered these instabilities can be masked by viscous damping at low reynolds numbers we demonstrate that the instabilities depend on the formulation of the nonlinear term the numerical discretization is stable for the convective form but unstable for the divergence and the skew symmetric form further numerical analysis indicates that this instability is not caused by nonlinear effects since it occurs for linearized systems as well an eigenvalue analysis of the fully discretized system shows that an instability is introduced by the formulation of the nonlinear term we demonstrate that the instability is related to the divergence error of the computed solution at those velocity points at which the continuity equation is not enforced

global convergence of radial basis function trust region derivative free algorithms we analyze globally convergent derivative free trust region algorithms relying on radial basis function interpolation models our results extend the recent work of conn scheinberg and vicente siam j optim 20 2009 pp 387 415 to fully linear models that have a nonlinear term we characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first order critical points for the orbit algorithm of wild regis and shoemaker siam j sci comput 30 2008 pp 3197 3219 using orbit we present numerical results for different types of radial basis functions on a series of test problems we also demonstrate the use of orbit in finding local minima on a computationally expensive environmental engineering problem

sparse random matrices the eigenvalue spectrum revisited we revisit the derivation of the density of states of sparse random matrices we derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit using the iterative scheme introduced by biroli and monasson 1999 j phys a math gen 32 l255 we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration the combination of the two methods yields a very simple geometric interpretation of the tails of the spectrum we test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

structural optimization using kriging approximation an optimization method using kriging approximation is applied to a structural optimization problem the method involves two main processes the first is a space estimation process that uses the kriging method and the second is an optimization process the use of the kriging method makes it easier to perform the approximation optimization as an example of the estimation performed as part of structural optimization a response surface for layout optimization of beam reinforcement is estimated to evaluate the applicability of the kriging method kriging estimation is compared with neural network approximation as a numerical example the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated the obtained results clearly show the applicability of the method

condor a new parallel constrained extension of powell s uobyqa algorithm experimental results and comparison with the dfo algorithm this paper presents an algorithmic extension of powell s uobyqa algorithm unconstrained optimization by quadratical approximation we start by summarizing the original algorithm of powell and by presenting it in a more comprehensible form thereafter we report comparative numerical results between uobyqa dfo and a parallel constrained extension of uobyqa that will be called in the paper condor constrained non linear direct parallel optimization using trust region method for high computing load function the experimental results are very encouraging and validate the approach they open wide possibilities in the field of noisy and high computing load objective functions optimization from 2 min to several days like for instance industrial shape optimization based on computation fluid dynamic codes or partial differential equations solvers finally we present a new easily comprehensible and fully stand alone implementation in c of the parallel algorithm

a numerical approach for shape optimization of fluid flow domains abstract a numerical method for the shape optimization of fluid flow domains is presented and analyzed the procedure is based on a flow solver a mathematical optimization tool and a technique for shape variation which are combined into an integrated procedure the flow solver relies on the discretization of the incompressible navier stokes equations by means of the finite volume method for block structured boundary fitted grids with multi grid acceleration the optimization tool is an implementation of a trust region based derivative free method it is designed to minimize smooth functions whose evaluations are considered expensive and whose derivatives are not available or not desirable to approximate the shape variation is obtained by deforming the computational grid employed by the flow solver for this purpose displacement fields scaled by the design variables are added to the initial grid the displacement vectors are computed once before starting the optimization cycle by using a free form deformation technique applications illustrating the functionality and the properties of the method are presented for some examples of engineering interest such as the minimization of a pressure drop the maximization of a lift force and the optimization of a wall temperature

incompressibility and no slip boundaries in the chebyshev tau approximation i present a minor but important correction to kleiser and schumann s influence matrix solution for numerical simulation of 3d time dependent incompressible flow between no slip plane boundaries l kleiser and u schumann in proc 3rd gamm conf numerical methods in fluid mechanics edited by e h hirschel vieweg braunschweig 1980 p 165 the solution technique i present is for the most part identical to kleiser and schumann s the only noteworthy difference being my treatment of the truncation or tau errors though this improvement appears to be but a slight modification to their original algorithm the error encountered when attempting to satisfy the incompressibility constraint is reduced dramatically by a factor of 1011 producing velocity fields which are divergence free to within machine precision

least frobenius norm updating of quadratic models that satisfy interpolation conditions quadratic models of objective functions are highly useful in many optimization algorithms they are updated regularly to include new information about the objective function such as the difference between two gradient vectors we consider the case however when each model interpolates some function values so an update is required when a new function value replaces an old one we let the number of interpolation conditions m say be such that there is freedom in each new quadratic model that is taken up by minimizing the frobenius norm of the second derivative matrix of the change to the model this variational problem is expressed as the solution of an m n 1 m n 1 system of linear equations where n is the number of variables of the objective function further the inverse of the matrix of the system provides the coefficients of quadratic lagrange functions of the current interpolation problem a method is presented for updating all these coefficients in m n 2 operations which allows the model to be updated too an extension to the method is also described that suppresses the constant terms of the lagrange functions these techniques have a useful stability property that is investigated in some numerical experiments

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

a cascade optimization methodology for automatic parameter identification and shape process optimization in metal forming simulation computer simulations of metal forming processes using the finite element method fem are today well established this form of simulation uses an increasing number of sophisticated geometrical and material models relying on a certain number of input data which are not always readily available the aim of inverse problems which will be considered here is to determine one or more of the input data relating to these forming process simulations thereby leading to a desired result in this paper we will focus on two categories of such inverse problems the first category consists of parameter identification inverse problems these involve evaluating the material parameters for material constitutive models that would lead to the most accurate results with respect to physical experiments i e minimizing the difference between experimental results and fem simulations the second category consists of shape process optimization inverse problems these involve determining the initial geometry of the specimen and or the shape of the forming tools as well as some parameters of the process itself in order to provide the desired final geometry after the forming process these two categories of inverse problems can be formulated as optimization problems in a similar way i e by using identical optimization algorithms in this paper we intend firstly to solve these two types of optimization problems by using different non linear gradient based optimization methods and secondly to compare their efficiency and robustness in a variety of numerical applications

new sequential and parallel derivative free algorithms for unconstrained minimization this paper presents sequential and parallel derivative free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest it is proved that under mild assumptions a sufficient decrease condition holds for a nonsmooth function based on this property the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value if the function is strictly differentiable at its limit points a sub sequence of points generated by the algorithm converges to a first order stationary point nabla f x 0 if the function is convex around its limit points convergence of a subsequence to a point with nonnegative directional derivatives on a set of search directions is ensured preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods

a derivative free nonmonotone line search technique for unconstrained optimization a tolerant derivative free nonmonotone line search technique is proposed and analyzed several consecutive increases in the objective function and also nondescent directions are admitted for unconstrained minimization to exemplify the power of this new line search we describe a direct search algorithm in which the directions are chosen randomly the convergence properties of this random method rely exclusively on the line search technique we present numerical experiments to illustrate the advantages of using a derivative free nonmonotone globalization strategy with approximated gradient type methods and also with the inverse sr1 update that could produce nondescent directions in all cases we use a local variation finite differences approximation to the gradient

short note extension of kleiser and schumann s influence matrix method for generalized velocity boundary conditions in 1980 kleiser and schumann introduced a novel influence matrix method to treat the incompressibility and no slip boundary conditions when solving the navier stokes equations they also outlined the related tau error correction technique which is essential for the high accuracy direct numerical simulation dns of turbulent flows however their method is not valid for robin type velocity boundary conditions i e b u email protected email protected email protected 0 in this note a new influence matrix method is introduced where the boundary condition and tau correction are enforced in one step using an extended influence matrix the new method is simple and easy to be implemented it broadens the applicability of the kleiser and schumann method examples with the new method show excellent agreement with data in the literature and the velocity field is divergence free up to machine precision

a derivative free multidimensional optimal search method using lucas number we consider a unimodal continuous function of n variables over an n dimensional rectangular parallelopiped and explain a search method to find the domain of uncertainty which contains the point that maximizes the function our search method does not require differentiability assumption the function is projected along each dimension an algorithm is developed to eliminate the subset of the domain at each step which does not contain the solution lucas numbers are used in the elimination process

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

uobyqa unconstrained optimization by quadratic approximation uobyqa is a new algorithm for general unconstrained optimization calculations that takes account of the curvature of the objective function f say by forming quadratic models by interpolation therefore because no first derivatives are required each model is defined by n 1 n 2 values of f where n is the number of variables and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them a typical iteration of the algorithm generates a new vector of variables widetilde underline x t say either by minimizing the quadratic model subject to a trust region bound or by a procedure that should improve the accuracy of the model then usually f widetilde underline x t is obtained and one of the interpolation points is replaced by widetilde underline x t therefore the paper addresses the initial positions of the interpolation points the adjustment of trust region radii the calculation of widetilde underline x t in the two cases that have been mentioned and the selection of the point to be replaced further uobyqa works with the lagrange functions of the interpolation equations explicitly so their coefficients are updated when an interpolation point is moved the lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation to f which allows the algorithm to achieve a fast rate of convergence these features are discussed and a summary of the algorithm is given finally a fortran implementation of uobyqa is applied to several choices of f in order to investigate accuracy robustness in the presence of rounding errors the effects of first derivative discontinuities and the amount of work the numerical results are very promising for n 20 but larger values are problematical because the routine work of an iteration is of fourth order in the number of variables

geometric filtration using proper orthogonal decomposition for aerodynamic design optimization when carrying out design searches traditional variable screening techniques can find it extremely difficult to distinguish between important and unimportant variables this is particularly true when only a small number of simulations is combined with a parameterization which results in a large number of variables of seemingly equal importance here the authors present a variable reduction technique which employs proper orthogonal decomposition to filter out undesirable or badly performing geometries from an optimization process unlike traditional screening techniques the presented method operates at the geometric level instead of the variable level the filtering process uses the designs which result from a geometry parameterization instead of the variables which control the parameterization the method is shown to perform well in the optimization of a two dimensional airfoil for the minimization of drag to lift ratio producing designs better than those resulting from traditional kriging based surrogate model optimization and with a significant reduction in surrogate tuning cost

kriging based approximate stochastic homogenization analysis for composite materials abstract this paper describes a novel method for a stochastic analysis of a multiscale homogenization problem an inhomogeneous material such as a composite material has a complex microstructure and sometimes it has an uncertainty in geometry or a material property of a microstructure this microscopic uncertainty will cause dispersion of a macroscopic homogenized material property of a composite material in order to analyze this problem an approximation based stochastic analysis approach is developed the proposed method uses a flexible approximation technique and variable transformation of a probabilistic density function in this paper the kriging method is used for approximation and integral estimation of a probabilistic density function as a numerical example a stochastic response analysis for a homogenized elastic tensor and homogenized elastic constants of a unidirectional fiber reinforced composite material is performed using the monte carlo simulation the perturbation based homogenization method and the proposed method uncertainties in material properties and geometry of a microstructure of a unidirectional fiber reinforced composite plate are taken into account from the numerical results validity and effectiveness of the proposed method are shown

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

global convergence of radial basis function trust region algorithms for derivative free optimization we analyze globally convergent derivative free trust region algorithms relying on radial basis function interpolation models our results extend the recent work of conn scheinberg and vicente siam j optim 20 2009 pp 387 415 to fully linear models that have a nonlinear term we characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first order critical points for the orbit algorithm of wild regis and shoemaker siam j sci comput 30 2008 pp 3197 3219 using orbit we present numerical results for different types of radial basis functions on a series of test problems we also demonstrate the use of orbit in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater

derivative free optimization methods for optimizing stirrer configurations abstract in this paper a numerical approach for the optimization of stirrer configurations is presented the methodology is based on a flow solver and a mathematical optimization tool which are integrated into an automated procedure the flow solver is based on the discretization of the incompressible navier stokes equations by means of a fully conservative finite volume method for block structured boundary fitted grids for allowing a flexible discretization of complex stirrer geometries two derivative free optimization algorithms the dfo and condor are considered they are implementations of trust region based derivative free methods using multivariate polynomial interpolation both are designed to minimize smooth functions whose evaluations are considered to be expensive and whose derivatives are not available or not desirable to approximate an exemplary application for a standard stirrer configuration illustrates the functionality and the properties of the proposed methods it also gives a comparison of the two optimization algorithms

self correcting geometry in model based algorithms for derivative free unconstrained optimization several efficient methods for derivative free optimization are based on the construction and maintenance of an interpolation model for the objective function most of these algorithms use special geometry improving iterations where the geometry poisedness of the underlying interpolation set is made better at the cost of one or more function evaluations we show that such geometry improvements cannot be completely eliminated if one wishes to ensure global convergence but we also provide an algorithm where such steps occur only in the final stage of the algorithm where criticality of a putative stationary point is verified global convergence for this method is proved by making use of a self correction mechanism inherent to the combination of trust regions and interpolation models this mechanism also throws some light on the surprisingly good numerical results reported by fasano morales and nocedal optim methods softw 24 2009 pp 145 154 for a method where no care is ever taken to guarantee poisedness of the interpolation set

sensitivity of optimal shapes of artificial grafts with respect to flow parameters abstract the difficulties arising in the numerical solution of pde constrained shape optimization problems are manifold key ingredients are the optimization strategy and the shape deformation method furthermore the robustness of the optimal shape with respect to simulation parameters is of great interest in this paper we consider fluid flows described by the incompressible navier stokes equations previous studies on artificial bypass grafts indicated the need for specific constitutive models to account for the non newtonian nature of blood in particular the constitutive model was shown to affect the solution of the shape optimization problem we employ a shape optimization framework that couples a finite element solver with quasi newton type optimizers and a bezier spline shape parametrization to compute derivatives of the optimal shapes with respect to viscosity we transform the entire optimization framework by combining the automatic differentiation tools adifor2 and tapenade we demonstrate the impact of the geometry parametrization and of geometric constraints on the optimization outcome finally we employ the transformed framework to compute the sensitivity of the optimal shape of bypass grafts with respect to kinematic viscosity the resulting sensitivities predict very accurately the influence of viscosity changes on the optimal shape the proposed methodology provides a powerful tool to further investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters

tetrahedralhpfinite elements we present a new discretisation for the incompressible navier stokes equations that extends spectral methods to three dimensional complex domains consisting of tetrahedral subdomains the algorithm is based on standard concepts ofhpfinite elements as well as tensorial spectral elements this new formulation employs a hierarchical modal basis constructed from a new apex co ordinate system which retains a generalised tensor product these properties enable the development of computationally efficienct algorithms for use on standard finite volume unstructured meshes a detailed analysis is presented that documents the stability and exponential convergence of the method and several flow cases are simulated and compared with analytical and experimental results

truncation error analysis of the rotational form for the convective terms in the navier stokes equation the truncation error of the rotational form for the convective terms in the navier stokes equation is examined in the direct numerical simulation dns of the fully developed turbulent channel flow in which the low order finite difference method was used for the partial derivatives in the wall normal direction an estimate of the truncation error using the taylor expansion revealed that this truncation error term is comparable to the rotational stress generation term represented by the nonlineark model in the reynolds averaged turbulence models in the governing equations for the reynolds shear stress and the normal stress due to the coriolis force term acting in a channel flow rotating about the spanwise axis the effective angular velocity due to the truncation error term was dependent on the distance from the wall and the turbulence was reduced on both sides of the walls analogously to the laminarization of turbulence on the suction side of the conventional rotating channel flow this analogy was further assessed in the dns of the channel flow rotating with this effective angular velocity the time development of the wall friction velocity was similar to that obtained using the rotational form

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

the correlated knowledge gradient for simulation optimization of continuous parameters using gaussian process regression we extend the concept of the correlated knowledge gradient policy for the ranking and selection of a finite set of alternatives to the case of continuous decision variables we propose an approximate knowledge gradient for problems with continuous decision variables in the context of a gaussian process regression model in a bayesian setting along with an algorithm to maximize the approximate knowledge gradient in the problem class considered we use the knowledge gradient for continuous parameters to sequentially choose where to sample an expensive noisy function in order to find the maximum quickly we show that the knowledge gradient for continuous decisions is a generalization of the efficient global optimization algorithm proposed in d r jones m schonlau and w j welch j global optim 13 1998 pp 455 492

finitely connected vector spin systems with random matrix interactions we use finite connectivity equilibrium replica theory to solve models of finitely connected unit length vectorial spins with random pair interactions which are of the orthogonal matrix type finitely connected spin models although still of a mean field nature can be regarded as a convenient level of description in between fully connected and finite dimensional ones since the spins are continuous and the connectivity c remains finite in the thermodynamic limit the replica symmetric order parameter is a functional the general theory is developed for arbitrary values of the dimension d of the spins and arbitrary choices of the ensemble of random orthogonal matrices we calculate phase diagrams and the values of moments of the order parameter explicitly for d 2 finitely connected xy spins with random chiral interactions and for d 3 finitely connected classical heisenberg spins with random chiral interactions numerical simulations are shown to support our predictions quite satisfactorily

pattern search methods for user provided points application to molecular geometry problems this paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry the goal is to find a configuration of a cluster or a molecule with minimum total energy the minimization problems in this class of molecular geometry problems have no constraints and the objective function is smooth the difficulties arise from the existence of several local minima and especially from the expensive function evaluation total energy and the possible nonavailability of first order derivatives we introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives numerical results for a particular instance of this new class of pattern search methods are presented showing the promise of our approach the new pattern search methods can be used in any other context where there is a user provided scheme to generate points leading to a potential objective function decrease