hybrid variable fidelity optimization by using a kriging based scaling function solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints variable fidelity methods address this issue by using lower fidelity models and a scaling function to approximate the higher fidelity models in a provably convergent framework in the past scaling functions have mainly been either first order multiplicative or additive corrections these are being extended to second order in this investigation variable metric approaches for calculating second order scaling information are developed a kriging based scaling function is introduced to better approximate the high fidelity response on a more global level an adaptive hybrid method is also developed in this investigation the adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization the methodologies developed in this research are compared to existing methods using two demonstration problems the first problem is analytic whereas the second involves the design of a supercritical high lift airfoil the results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high fidelity function calls required for convergence the results also indicate the hybrid method is both robust and effective

trust region model management in multidisciplinary design optimization a common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques the use of approximation strategies is designed to reduce the number of detailed costly computer simulations required during optimization while maintaining the pertinent features of the design problem this paper overviews the current state of the art in model management strategies for approximate optimization model management strategies coordinate the interaction between the optimization and the fidelity of the approximation models so as to ensure that the process converges to a solution of the original design problem approximations play an important role in multidisciplinary design optimization mdo by offering system behavior information at a relatively low cost most approximate mdo strategies are sequential in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence the move limits or trust region are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information in order to insure convergence of the sequence of approximate optimizations to a karush kuhn tucker solution a trust region model management or move limit strategy is required in this paper recent developments in approximate mdo strategies and issues of trust region model management in mdo are reviewed

the multiplicity of binary recurrences of the recurrence relation 0 1 it follows from the skolem mahler lech theorem that if u then one at least of and is a root of unity consequently we call um nondegenerate if none of these quantities is a root of unity in the thirties m ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity u 5 5 this conjecture was proved by kubota 4 kubota also proved that any nondegenerate binary recurrence um with elements um in a number eld k of degree d has multiplicity u 5 c d cf 5 beukers and tijdeman 2 established for this case the bound

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

implicit multigrid algorithms for incompressible turbulent flows on unstructured grids an implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described the foundation of the code is a backward euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual gmres technique for the gmres calculations several techniques are investigated for forming the matrix vector product convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four element airfoil viscous results which are compared with experimental data include the turbulent flow over a naca 4412 airfoil a three element airfoil for which mach number effects are investigated and three dimensional flow over a wing with a partial span flap

numerical solutions of two dimensional burgers equations by discrete adomian decomposition method in this paper the discrete adomian decomposition method adm is proposed to numerically solve the two dimensional burgers nonlinear difference equations two test problems are considered to illustrate the accuracy of the proposed discrete decomposition method it is shown that the numerical results are in good agreement with the exact solutions for each problem

numerical solution of one dimensional burgers equation using reproducing kernel function this paper presents a numerical method for one dimensional burgers equation by the hopf cole transformation and a reproducing kernel function abbreviated as rkf the numerical solution is given as explicit integral expressions with the rkf at each time step so that the computation is fully parallel the stability and error estimates are derived numerical results for some test problems are presented and compared with the exact solutions some numerical results are also compared with the results obtained by other methods the present method is easily implemented and effective

a predictor corrector compact finite difference scheme for burgers equation abstract in this paper a compact predictor corrector finite difference scheme is proposed to solve the burgers equation the scheme is based on compact derivatives approximation by which we get the spatial approximations of first order derivatives and second order derivatives with fourth order accuracy both for inner nodes and boundary nodes for the first time derivative item a two step predictor corrector method called maccormack method is used numerical experiments show the scheme is in good agreement with the exact solutions

combining global and local approximations a method based on a linear approximation to a scaling factor designated the global local approximation gla method is presented and shown capable of extending the range of usefulness of derivative based approximations to a more refined model the gla approach refines the conventional scaling factor by means of a linearly varying rather than constant scaling factor the capabilities of the method are demonstrated for a simple beam example with a crude and more refined fem model 6 refs

use of kriging models to approximate deterministic computer models the use of kriging models for approximation and metamodel based design and optimization has been steadily on the rise in the past decade the widespread usage of kriging models appears to be hampered by 1 the lack of guidance in selecting the appropriate form of the kriging model 2 computationally efficient algorithms for estimating the model s parameters and 3 an effective method to assess the resulting model s quality in this paper we compare 1 maximum likelihood estimation mle and cross validation cv parameter estimation methods for selecting a kriging model s parameters given its form and 2 and an r 2 of prediction and the corrected akaike information criterion for assessing the quality of the created kriging model permitting the comparison of different forms of a kriging model these methods are demonstrated with six test problems finally different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

sensitivity of complex internally coupled systems a method is presented for computing sensitivity derivatives with respect to independent input variables for complex internally coupled systems while avoiding the cost and inaccuracy of finite differencing performed on the entire system analysis the method entails two alternative algorithms the first is based on the classical implicit function theorem formulated on residuals of governing equations and the second develops the system sensitivity equations in a new form using the partial local sensitivity derivatives of the output with respect to the input for each part of the system a few application examples are presented to illustrate the discussion the method has potential to answer the what if questions by presenting engineers wigi sensitivity information on design tradeoffs to guide human judgment and formal optimization in addition the method is compatible with the modern technology of distributed computing as well as traditional division of design tasks among groups of specialists in the design process the capability to quantify the effects of proposed design changes may provide the basis for a mathematical model of design

gaussian functional regression for output prediction in this paper we introduce a gaussian functional regression gfr technique that integrates multi fidelity models with model reduction to efficiently predict the input output relationship of a high fidelity model the gfr method combines the high fidelity model with a low fidelity model to provide an estimate of the output of the high fidelity model in the form of a posterior distribution that can characterize uncertainty in the prediction a reduced basis approximation is constructed upon the low fidelity model and incorporated into the gfr method to yield an inexpensive posterior distribution of the output estimate as this posterior distribution depends crucially on a set of training inputs at which the high fidelity models are simulated we develop a greedy sampling algorithm to select the training inputs our approach results in an output prediction model that inherits the fidelity of the high fidelity model and has the computational complexity of the reduced basis approximation numerical results are presented to demonstrate the proposed approach

numerical solutions of nonlinear burgers equation with modified cubic b splines collocation method abstract in this paper a numerical method is proposed to approximate the solution of the nonlinear burgers equation the method is based on collocation of modified cubic b splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range we apply modified cubic b splines for spatial variable and derivatives which produce a system of first order ordinary differential equations we solve this system by using ssp rk43 or ssp rk54 scheme this method needs less storage space that causes less accumulation of numerical errors the numerical approximate solutions to the burgers equation have been computed without transforming the equation and without using the linearization illustrative eleven examples are included to demonstrate the validity and applicability of the technique easy and economical implementation is the strength of this method

higher order accurate numerical solution of unsteady burgers equation abstract higher order accurate finite difference schemes for solving the unsteady burgers equation which often arises in mathematical modeling used to solve problems in fluid dynamics are presented the unsteady burgers equation belongs to a few nonlinear partial differential equations which has an exact solution and it allows one to compare the numerical solution with the exact one and the properties of different numerical methods we propose an explicit finite difference scheme for a numerical solution of the heat equation with robin boundary conditions it has a sixth order approximation in the space variable and a third order approximation in the time variable as an application we developed numerical schemes for solving a numerical solution of burgers equation using the relationship between the heat and burgers equations this scheme has up to sixth order approximation in the space variables the main advantage of our approach is transition to one dimensional equation which essentially reduces the computation costs compared to other direct methods for solving the unsteady burgers equation the numerical results of test examples are found in good agreement with exact solutions for a wide range of reynolds number and confirm the approximation orders of the schemes proposed

numerical solution of nonlinear burgers equation using high accuracy multi quadric quasi interpolation in this paper a numerical method is presented to approximate the solution to nonlinear burgers equation which is related to many scientific research topics a numerical scheme by using high accuracy multi quadric quasi interpolation is presented in which a kind of multi quadric quasi interpolant is used to approximate the derivatives of the solution in spatial domain and finite difference is used to approximate the derivatives of the solution in temporal domain the advantage of the scheme is that it is mesh free and in each time step only a multi quadric quasi interpolant is employed so that the algorithm is easy to implement the numerical results of this scheme are also shown and compared with other numerical schemes

a finite difference approach for solution of burgers equation abstract in this paper we have applied restrictive pade approximation classical implicit finite difference method to the burgers equation with a set of initial and boundary conditions to obtain its numerical solution the stability region and truncation error of the new method are discussed the accuracy of the proposed method is demonstrated by the two test problems the numerical results obtained by this method for various values of viscosity have been compared with the exact solution to show the efficiency of the method the numerical results are found in good agreement with the exact solutions

finite difference schemes for three dimensional time dependent convection diffusion equation using full global discretization abstract the three dimensional time dependent convection diffusion equation cde is considered an exponential transformation is used to collectively transform the cde the idea of global discretization is used where attention is paid to the whole transformed cde but not to the individual spatial and temporal derivatives in the equation four finite difference schemes for both cde and transformed cde are established the modified partial differential equations of these schemes are obtained which indicate that the trunction errors of the schemes can be of second and fourth order depending on the prescription of the time step length some characteristic physical parameters i e local reynolds number local strouhal number and viscous diffusive length are introduced into the schemes and the viscous diffusive length is found to be a significant parameter in relating temporal discretisation with spatial discretisation a series of benchmark analytical solutions of navier stokes and burgers equations as well as the numerical solutions using the well known discretisation schemes are used to investigate the properties of the derived schemes the high order schemes achieve higher resolutions over the conventional schemes without decreasing much the sparsity of the matrix structures grid refinement studies reveal that the inverse exponential transformation of the finite difference schemes tends to destroy some resolution of the schemes especially for large local reynolds number

theory and application for the system of fractional burger equations with mittag leffler kernel abstract in this work the system of fractional burger differential equations are presented as a new fractional model for atangana baleanu fractional derivative with mittag leffler kernel the approximate consequences are analysed by applying an recurrent process the existence and uniquenes of solution for this system is discussed in order to appear the effects of several parameter and variables on the movement the approximate results are showed in graphics and are compared with obtained solutions for two different derivative in tables

a galerkin finite element approach to burgers equation a galerkin finite element method is presented for the numerical solution of burgers equation a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found via a crank nicolson approach involving a product approximation it is shown that this method is capable of solving burgers equation accurately for a wide range of viscosity values the results show that the new method performs better than the most of the methods available in the literature

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

the number of solutions of decomposable form equations has only finitely many solutions the diophantine approximation techniques of thue and mahler and improvements by siegel dyson roth and bombieri made it possible to derive good explicit upper bounds for the number of solutions of 1 1 the best such upper bound to date due to bombieri 1 is 2 12r bombieri assumed that f is irreducible and r 6 which was not essential in his proof for t 0 i e f x y 1 in x y z bombieri and schmidt 2 derived the upper bound constant r which is best possible in terms of r

a class of high order compact difference schemes for solving the burgers equations abstract in this paper a class of high order compact difference method is introduced for solving the burgers equations firstly a linear high order compact difference scheme is proposed to solve the one dimensional burgers equation the scheme is fourth order accurate in space and second order accurate in time linear stability analysis is conducted to show the scheme is conditionally stable because only three grid points are involved in each time level thomas algorithm can be directly used to solve the tridiagonal linear system then this method is extended to solve the two dimensional and three dimensional coupled burgers equations numerical experiments are carried out to demonstrate the accuracy and dependability of the present method

numerical solution of the burgers equation by automatic differentiation abstract we compute the solution of the one dimensional burgers equation by marching the solution in time using a taylor series expansion our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non linear algebraic equations instead we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the taylor series we illustrate the effectiveness of our method by solving four test problems with known exact solutions the numerical solutions we obtain are in excellent agreement with the exact solutions while being superior to other previously reported numerical solutions

multifidelity quasi newton method for design optimization multifidelity approaches are frequently used in design when high fidelity models are too expensive to use directly and lower fidelity models of reasonable accuracy exist in optimization corrected

2n order compact finite difference scheme with collocation method for solving the generalized burger s huxley and burger s fisher equations the generalized burger s huxley and burger s fisher equations are solved by fully different numerical scheme the equations are discretized in time by a new linear approximation scheme and in space by 2n order compact finite difference scheme after that a collocation method is applied also the two dimensional unsteady burger s equation is described by our proposed scheme numerical experiments and numerical comparisons are presented to show the efficiency and the accuracy of the proposed scheme

an approximation concepts approach to shape optimal design abstract in the past much of the work done in structural optimization consisted in resizing the members of fixed configuration models in that case a powerful design procedure has now emerged which is based on the coordinate use of explicit high quality approximations of the behavior constraints and dual methods of mathematical programming there is however a large class of problems for which the main degrees of freedom for the designer correspond to the shape of the structure itself the main objectives of this paper are to recall briefly a convenient geometric representation in which the boundaries of the structure are represented by bezier or b spline curves and then to discuss the choice of optimization algorithm it is shown that cost efficient methods for structural sizing may be advantageously extended to shape optimal design problems different approximation schemes are tested and a new general optimization algorithm is presented that combines mixed approximations and dual methods many large scale applications are treated to demonstrate the generality and the efficiency of the new formulation finally considerations are given about an integrated approach including cad computer codes and finite element optimization software

the modified local crank nicolson method for one and two dimensional burgers equations the modified local crank nicolson method is applied to solve one and two dimensional burgers equations new difference schemes that are explicit unconditionally stable and easy to compute are obtained numerical solutions obtained by the present method are compared with exact solutions and it is seen that they are in excellent agreement

parametric domain decomposition for accurate reduced order models applications of mp lrom methodology abstract the multivariate predictions of local reduced order models mp lrom methodology recently proposed by the authors moosavi et al 0000 uses machine learning based regression methods to predict the errors of reduced order models this study considers two applications of mp lrom first the error model is used in conjunction with a greedy sampling algorithm to generate decompositions of one dimensional parametric domains with overlapping regions such that the associated local reduced order models meet the prescribed accuracy requirements once a parametric domain decomposition is constructed any parametric configuration belongs to at least one of the partitions the local reduced order model associated with that partition approximates the full order model at the given parameters within an accuracy level that is estimated a priori the parameter domain decomposition creates a database of the available local bases local reduced order and high fidelity models and identifies the most accurate solutions for an arbitrary parametric configuration next this database is used to enhance the accuracy of the reduced order models using 1 lagrange interpolation of reduced bases in the matrix space 2 lagrange interpolation of reduced bases in the tangent space of the grassmann manifold 3 concatenation of reduced bases followed by a gram schmidt orthogonalization process and 4 lagrange interpolation of high fidelity model solutions numerical results with a viscous burgers model illustrate the potential of the mp lrom methodology to improve the design of parametric reduced order models

a sixth order compact finite difference scheme to the numerical solutions of burgers equation abstract a numerical solution of the one dimensional burgers equation is obtained using a sixth order compact finite difference method to achieve this a tridiagonal sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been combined the scheme is implemented to solve two test problems with known exact solutions comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort the present results are also seen to be more accurate than some available results given in the literature