Showing papers on "High-dimensional model representation published in 2008"

A global sensitivity study of sulfur chemistry in a premixed methane flame model using HDMR

Abstract: The use of complex mechanisms is increasing within models describing a range of important chemical processes, including combustion. Parameters describing chemical reaction rates and thermodynamics can often be very uncertain. Highlighting the main parameters contributing to predictive uncertainty is an important part of model development. However, due to the computational cost of the models and their nonlinearity, traditional methods for sensitivity analysis are often not suitable. The high-dimensional model representation (HDMR) method was developed to express the input–output relationship of a complex model with a high-dimensional input space. A fully functional surrogate model can be constructed with low computational effort. First- and second-order sensitivity indices can then be calculated in an automatic way, over large input parameter ranges. These provide an importance ranking for the input parameters. An extension to the existing set of HDMR tools is developed in this work, where the number of HMDR component functions can be reduced to explore large dimensional input spaces very efficiently. The HDMR tools are demonstrated for a case study of a one-dimensional low-pressure premixed methane flame model doped with trace sulfur and nitrogen species. Uncertainties in rate constants and thermodynamic data are considered, leading to a study of 176 input parameters. Using the new HDMR tools, the use of screening methods such as the Morris method, which aim to identify unimportant parameters beforehand, can generally be avoided. However, in certain cases, a combination of a screening method and HDMR is computationally more efficient than using HDMR alone. The final ranking of important parameters is shown to be critically dependent on the uncertainty ranges chosen due to the nonlinearity of the model. The study demonstrates that the proposed HDMR method provides a powerful tool for general application to global sensitivity analysis of complex chemical mechanisms. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 40: 742–753, 2008

Regularized random-sampling high dimensional model representation (RS-HDMR)

Abstract: High Dimensional Model Representation (HDMR) is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. HDMR is based on a hierarchy of component functions of increasing dimensions. The Random-Sampling High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR utilizing random sampling of the input variables. To reduce the sampling effort, the RS-HDMR component functions are approximated in terms of a suitable set of basis functions, for instance, orthonormal polynomials. Oscillation of the outcome from the resultant orthonormal polynomial expansion can occur producing interpolation error, especially on the input domain boundary, when the sample size is not large. To reduce this error, a regularization method is introduced. After regularization, the resultant RS-HDMR component functions are smoother and have better prediction accuracy, especially for small sample sizes (e.g., often few hundred). The ignition time of a homogeneous H2/air combustion system within the range of initial temperature, 1000 < T 0 < 1500 K, pressure, 0.1 < P < 100 atm and equivalence ratio of H2/O2, 0.2 < R < 10 is used for testing the regularized RS-HDMR.

Probabilistic Analysis Using High Dimensional Model Representation and Fast Fourier Transform

Abstract: This paper presents an efficient probabilistic analysis method for predicting component reliability of structural/mechanical systems subject to random loads, material properties, and geometry. The proposed method involves High Dimensional Model Representation (HDMR) for the limit state/performance function approximation and fast Fourier transform for solving the convolution integral. The limit state/performance function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full-scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of random variabl...

A new approach for data partitioning through high dimensional model representation

Abstract: A multivariate function f(x1,..., xN) can be evaluated via interpolation if its values are given at a finite number nodes of a hyperprismatic grid in the space of independent variables x1, x2,..., xN. Interpolation is a way to characterize an infinite data structure (function) by a finite number of data approximately. Hence it leaves an infinite arbitrariness unless a mathematical structure with finite number of flexibilities is imposed for the unknown function. Imposed structure has finite dimensionality. When the dimensionality increases unboundedly, the complexities grow rapidly in the standard methods. The main purpose here is to partition the given multivariate data into a set of low-variate data by using high dimensional model representation (HDMR) and then, to interpolate each individual data in the set via Lagrange interpolation formula. As a result, computational complexity of the given problem and needed CPU time to obtain the results through a series of programs in computers decrease.

Factorized high dimensional model representation for structural reliability analysis

Abstract: Purpose – To develop a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry.Design/methodology/approach – High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then all right hand side components of HDMR must be used to be able to obtain the best result. However, if HDMR requires all components, which means 2N number of components, to get a desired accuracy, making the method very expensive in practice, then factorized HDMR (FHDMR) can be used. ...

On the approximation error in high dimensional model representation

Abstract: Mathematical models are often described by multivariate functions, which are usually approximated by a sum of lower dimensional functions. A major problem is the approximation error introduced and the factors that affect it. This paper investigates the error of approximating a multivariate function by a sum of lower dimensional functions in the setting of high dimensional model representations. Two kinds of approximations are studied, namely, the approximation based on the ANOVA (analysis of variance) decomposition and the approximation based on the anchored decomposition. We prove new theorems for the expected errors of approximations based on anchored decomposition when the anchor is chosen randomly and establish the relationship of the expected approximation errors with the global sensitivity indices of Sobol'. The expected approximation error give indications on how good or how bad could be the approximation based on anchored decomposition and when the approximation is good or bad. The influence of the anchor on the goodness of approximation is studied. Methods for choosing good anchors are presented.

A tool to improve the execution time of air quality models

Abstract: High dimensional model representation (HDMR) techniques are tools to improve mathematical modeling of physical systems. One of the main HDMR applications is the development and use of a fully equivalent operational model (FEOM) in the modeling of a physical system described by means of a set of ordinary differential equations. The accuracy of a system's output approximation obtained by using HDMR expansions is comparable with other approximation methods while the computational effort is much lower. We use FEOM to solve the initial-value problem for the stiff systems of ordinary differential equations which describe the chemical reactions' mechanisms used in the three-dimensional model of diffusion/advection of air pollutants. With the use of FEOM we achieved noticeable computational savings. We apply the developed FEOM to two different schemes of chemical reactions and give some general recommendations for the efficient FEOM implementation based on the obtained experience.

Fluctuationlessness Approximation Based Multivariate Integration in Hybrid High Dimensional Model Representation

Abstract: This paper focuses on multivariate integration via fluctuationlessness approximation in hybrid High Dimensional Model Representation (HHDMR) for multivariate functions. The basic idea here is to bypass the N—tuple integration with the help of the fluctuationlessness approximation as a quite powerful method for integral evaluations. This method decreases the computational complexity of the HHDMR method in computer based applications. Furthermore, by using this method, we are able to get rid of evaluating complicated integral structures.

Small scale high dimensional model representation

Abstract: HDMR, High Dimensional Model Representation is one of the recently developed tool in the approximation of multivariate functions. It is based on multivariate integration over orthogonal geometries under product type weight functions and uses a divide-and-conquer algorithm such that the target function is separated into components in ascending multivariance. In almost all applications HDMR is desired to be truncated at constant, univariate, or at most, bivariate components as approximations. The approximating quality of these components increases as the target functions additive nature dominates. Here, our purpose is to investigate the role of the geometric scale of HDMR be anticipating to get efficiency in the small scale geometries. The ultimate goal of this approach is to develop a new HDMR like finite elements.

Structural Failure Probability Estimation Using HDMR and FFT

Abstract: This paper presents a new and alternative method based on High Dimensional Model Representation (HDMR) and fast Fourier transform (FFT) to estimate the structural failure probability of structural systems subject to random loads, material properties and geometry. The proposed methodology is based on the limit state/performance function approximation and the convolution theorem to estimate the structural failure probability. The limit-state function is obtained by linear approximation of the first-order HDMR component functions at the most probable failure point, and the convolution integral is solved efficiently using the FFT technique. The proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The accuracy and efficiency of the proposed method is demonstrated through numerical examples involving implicit performance functions.

Introductory steps for an indexing based HDMR algorithm: lumping HDMR

Abstract: Generalized HDMR can be used to partition the given multivariate data set into less variate data sets to avoid the standard numerical methods' restrictions coming from the multivariance and to minimize the memory insufficiencies occured in the computer based applications when all nodes of the whole domain defined for an multivariate interpolation problem are not given, that is, randomly selected nodes are given. Hovewer, this method has some disadvantages to obtain acceptable approximations in engineering problems. For this purpose, High Dimensional Model Representation (HDMR) method can be used in these types of problems. In fact, this method needs an orthogonal geometry. The main purpose of this work is to reconstruct the given multivariate data set by imposing an indexing scheme and to obtain an orthogonal geometry for the given problem. The introductory steps of this new method called Lumping HDMR are given in this work. The remaining details and the maturization of this new method is still under intense study.

Fluctuationlessness approximation towards orthogonal hyperprismatic grid construction

Abstract: The main purpose of this work is to construct transformations which map the nodes created by the individual matrix representations N independent variables to the hypegrid nodes where the values of an N variate functions are given. Recent works of the Demiralp's group show that the matrix representation of a multivariate function can be approximated by the image of its independent variable matrix representations under that function at the fluctuationlessness limit. This brings the possibility of using only N dimensional cartesian space points which are characterized by N-tuples whose elements are the eigenvalues of the matrix representations of those independent variable. However, these points may not match the points where the values of the function under consideration are given. Hence, N-dimensional shifts of the eigenvalue based points to data given points is requiered. This can be done by using certain polynomial interpolations. This work aims at the evaluation of those polynomials.

High dimensional model representation (HDMR) and trigonometric transformational HDMR

Abstract: It is well known that mathematical modeling of physical systems is a must in most cases. Amongst many approximation techniques is the High Dimensional Model Representation Method. Here, preliminary analytical attemps are made for a newly developed version of HDMR the Trigonometric Transformational HDMR.

Numerical solution of ordinary differential equations via splines

Abstract: In this work a numerical solution for a given first order linear Ordinary Differential Equation is obtained by High Dimensional Model Representation (HDMR). Although HDMR is commonly used for multivariate functions, univariate functions are considered through the work to trace everything more easily. The numerical solution is written in the form of a linear combination of first linear and then quadratic functions chosen in Hilbert Space. The unknown constants in that combination are determined in such a way that the constancy measurer becomes maximum.

On the role of the weight function in the truncation quality of high dimensional model representation

Abstract: This paper investigates the role of the weight function in the quality of the High Dimensional Model Representation's (HDMR) truncations at different levels of multivariance. This is done by evaluating the additivity measurers at different levels of multivariance, for specified weight functions and targets for HDMR. The observations imply that the weight functions picking function values from regions where the addive nature of the target function dominates.

Data partitioning via high dimensional model representation by using paralel computing

Abstract: The multivariate functions become plague of plethora when the number of their arguments increases to high values as much as 2 digits numbers, because of the limitations of today's computers. Instead of computer programming directly, the mathematically efficient multivariate function representations must be developed before attempting computer programming. One of those methods is "High Dimensional Model Representation (HDMR)" Even when these types of methods are applied on the multivariate functions, the solution cost of the modified problem after the application of the method is still complicated for a stand-alone computer. In that case, the parallel programming helps us. Our main goal of this work is to modify the method for parallel computing.

How additive are the Fourier series in the sense of high Dimensional model representation

Abstract: The focus of this work is to investigate the quality of High Dimensional Model Representation (HDMR) to Fourier series Towards this end, we experimantate with various Fourier series which are constructed for known univariate functions Although the investigations are kept univariate, the extension that we obtain here to multivariate cases seems to be straightforward This is because we use the additivity measurers whose conceptual structures do not change from one multivariance to another The additiviy measurers are certain well-ordered functionals mapping from a Hilbert space of multi or univariate functions to the interval [0,1] and their close-to-one values mean certain level of additivity and therefore higher qualities of truncated HDMR approximants Hence, those entities are evaluated for certain known cases