Showing papers on "High-dimensional model representation published in 2012"

General formulation of HDMR component functions with independent and correlated variables

Abstract: The High Dimensional Model Representation (HDMR) technique decomposes an n-variate function f (x) into a finite hierarchical expansion of component functions in terms of the input variables x = (x 1, x 2, . . . , x n ). The uniqueness of the HDMR component functions is crucial for performing global sensitivity analysis and other applications. When x 1, x 2, . . . , x n are independent variables, the HDMR component functions are uniquely defined under a specific so called vanishing condition. A new formulation for the HDMR component functions is presented including cases when x contains correlated variables. Under a relaxed vanishing condition, a general formulation for the component functions is derived providing a unique HDMR decomposition of f (x) for independent and/or correlated variables. The component functions with independent variables are special limiting cases of the general formulation. A novel numerical method is developed to efficiently and accurately determine the component functions. Thus, a unified framework for the HDMR decomposition of an n-variate function f (x) with independent and/or correlated variables is established. A simple three variable model with a correlated normal distribution of the variables is used to illustrate this new treatment.

Multicut-High Dimensional Model Representation for Structural Reliability Bounds Estimation Under Mixed Uncertainties

Abstract: In reliability analysis of structural systems involving both aleatory and epistemic uncertainties, in conjunction with multiple design points, every configuration of the interval variables is to be explored to determine the bounds on reliability. To reduce the computational cost involved, this article presents a novel uncertain analysis method for estimating the bounds on reliability of structural systems involving multiple design points in the presence of mixed uncertain (both random and fuzzy) variables. The proposed method involves Multicut-High Dimensional Model Representation (MHDMR) technique for the limit state/performance function approximation, the transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. The limit state function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at the most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by the 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 fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through four examples involving explicit/implicit performance functions.

Cut-HDMR-based fully equivalent operational model for analysis of unreinforced masonry structures

Abstract: Mesoscale models are highly competent for understanding behaviour of unreinforced masonry structures. Their only limitation is large computational expense. Fully Equivalent Operational Model forms an equivalent mathematical model to represent a particular phenomenon where explicit relationship between inputs and outputs are unknown. This paper explores the ability of a major variant of High Dimensional Model Representation (HDMR) technique, namely Cut-HDMR, to construct the most efficient Fully Equivalent Operational Model for nonlinear finite element analysis of mesoscale model of an unreinforced masonry structure. Conclusions are reached on various aspects such as, suitability of interpolation schemes and order of Cut-HDMR approximation.

Global Sensitivity Analysis of the Regional Atmospheric Chemical Mechanism: An Application of Random Sampling-High Dimensional Model Representation to Urban Oxidation Chemistry

Abstract: Chemical mechanisms play a crucial part for the air quality modeling and pollution control decision-making. Parameters in a chemical mechanism have uncertainties, leading to the uncertainties of model predictions. A recently developed global sensitivity analysis (SA) method based on Random Sampling-High Dimensional Model Representation (RS-HDMR) was applied to the Regional Atmospheric Chemical Mechanism (RACM) within a zero-dimensional photochemical model to highlight the main uncertainty sources of atmospheric hydroxyl (OH) and hydroperoxyl (HO2) radicals. This global SA approach can be applied as a routine in zero-dimensional photochemical modeling to comprehensively assess model uncertainty and sensitivity under different conditions. It also highlights the parameters to which the model is most sensitive during periods when the model/measurement OH and HO2 discrepancies are greatest. Uncertainties in 584 model parameters were assigned for measured constituents used to constrain the model, for photolysis...

Chebyshev high-dimensional model representation (Chebyshev-HDMR) potentials: application to reactive scattering of H2 from Pt(111) and Cu(111) surfaces

Abstract: Gas phase and surface reactions involving polyatomic molecules are of central importance to chemical physics, and require accurately fit potential energy surfaces describing the interaction in their systems. Here, we propose a method for generating a High Dimensional Model Representation (HDMR) of a multidimensional potential energy surface (PES) and apply it to reactive molecule–surface scattering problems. In the HDMR treatment, all N degrees of freedom (DOF) of an N-dimensional PES are represented but only n < N are explicitly coupled. The HDMR is obtained from Chebyshev polynomial expansions for each degree of freedom, where expansion coefficients are efficiently evaluated using discrete cosine transform (DCT) algorithms and properties of Chebyshev polynomials. HDMR surfaces constructed for the reactive scattering of H2 from Pt(111) and Cu(111) are used in quantum dynamics simulations; the resultant state-resolved reaction and scattering probabilities are compared to those from simulations using full (6D) PESs and n-mode PESs from previous work. The results are encouraging, and suggest that this method may be applicable to “late barrier” reactive systems for which the previously-used n-mode representation fails.

Application of HDMR method to reliability assessment of a single pile subjected to lateral load

Abstract: The paper presents an application of High Dimensional Model Representation (HDMR) to reliability assessment of a single pile subjected to lateral load. The purpose is to compare HDMR with some classical method based on response surface technique. First 3D numerical model of the problem for finite elements computations in the ABAQUS STANDARD program has been presented. The soil model is assumed to be linear elastic. However, contacts between the sidewall and the foundation of the pile and the soil are modelled as Coulomb one with friction and cohesion. Next the Response Surface Method is briefly reviewed in conjunction with reliability approach. Then the High Dimensional Model Representation approach is presented. In our approach the HDMR algorithm is based on polynomial of the second degree. Finally the numerical studies have been carried out. The first series of computations demonstrate the efficiency of HDMR in compari- son to neural network approach. The second series allows comparison of reliability indices resulting from three different approaches, namely neural network response surface, first-order HDMR and second-order HDMR. It has been observed that for increasing values of the length of the pile reli- ability indices reach similar values regardless of the method response surface applied.

State dependent parameter method for importance analysis in the presence of epistemic and aleatory uncertainties

Abstract: For the structure system with epistemic and aleatory uncertainties, a new state dependent parameter (SDP) based method is presented for obtaining the importance measures of the epistemic uncertainties. By use of the marginal probability density function (PDF) of the epistemic variable and the conditional PDF of the aleatory one at the fixed epistemic variable, the epistemic and aleatory uncertainties are propagated to the response of the structure firstly in the presented method. And the computational model for calculating the importance measures of the epistemic variables is established. For solving the computational model, the high efficient SDP method is applied to estimating the first order high dimensional model representation (HDMR) to obtain the importance measures. Compared with the direct Monte Carlo method, the presented method can considerably improve computational efficiency with acceptable precision. The presented method has wider applicability compared with the existing approximation method, because it is suitable not only for the linear response functions, but also for nonlinear response functions. Several examples are used to demonstrate the advantages of the presented method.

Adaptive Orthonormal Basis Functions for High Dimensional Metamodeling With Existing Sample Points

Abstract: High Dimensional Model Representation (HDMR) is a tool for generating an approximation of an input-output model for a multivariate function. It can be used to model a black-box function for metamodel-based optimization. Recently the authors’ team has developed a radial basis function based HDMR (RBF-HDMR) model that can efficiently model a high dimensional black-box function and, moreover, to uncover inner variable structures of the black-box function. This approach, however, requests a complete new, although optimized, set of sample points, as dictated by the methodology, while in engineering design practice one often has many existing sample data. How to utilize the existing data to efficiently construct a HDMR model is the focus of this paper. We first identify the Random-Sampling HDMR (RS-HDMR), which uses orthonormal basis functions as HDMR component functions and existing sample points can be used to calculate the coefficients of the basis functions. One of the important issues related to the RS-HDMR is that in theory the basis functions are obtained based on the continuous integrations related to the orthonormality conditions. In practice, however, the integrations are approximated by Monte Carlo summation and thus the basis functions may not satisfy the orthonormality conditions. In this paper, we propose new and adaptive orthonormal basis functions with respect to a given set of sample points for RS-HDMR approximation. RS-HDMR models are built for different test functions using the standard and new adaptive basis functions for different number of sample points. The relative errors for both models are calculated and compared. The results show that the models that are built using the new basis functions are more accurate.Copyright © 2012 by ASME

Feature selection by high dimensional model representation and its application to remote sensing

Abstract: As the number of feature increases, classification accuracy may decrease. Additionally, computational overload increases with a large number of features. For effective classification performance and shortened the training time, the redundant features should be eliminated before the classification process. In this paper, a new HDMR-based feature selection approach is presented, sorting the features with respect to their sensitivity coefficient calculated by HDMR sensitivity analysis. With the experiments conducted, the HDMR-based feature selection approach is competitive with sequential forward feature selection method and faster in terms of computational time, especially when dealing with datasets having a large number of features.

A novel approximation method for multivariate data partitioning

Abstract: Purpose – The plain High Dimensional Model Representation (HDMR) method needs Dirac delta type weights to partition the given multivariate data set for modelling an interpolation problem. Dirac delta type weight imposes a different importance level to each node of this set during the partitioning procedure which directly effects the performance of HDMR. The purpose of this paper is to develop a new method by using fluctuation free integration and HDMR methods to obtain optimized weight factors needed for identifying these importance levels for the multivariate data partitioning and modelling procedure.Design/methodology/approach – A common problem in multivariate interpolation problems where the sought function values are given at the nodes of a rectangular prismatic grid is to determine an analytical structure for the function under consideration. As the multivariance of an interpolation problem increases, incompletenesses appear in standard numerical methods and memory limitations in computer‐based appl...

Efficient GA-based electromagnetic optimization using HDMR-generated surrogate models

Abstract: An efficient surrogate modelling technique that supports the genetic algorithm (GA) -based optimization of electromagnetic (EM) devices is presented. The proposed method leverages high dimensional model representation (HDMR) expansions, which approximate observables or objective functions as series of iteratively constructed component functions involving only the most strongly interacting design variables. The contributions that feature in HDMR expansions are approximated via a multi-element probabilistic collocation (ME-PC) method. The proposed method is capable of generating surrogate models of rapidly varying observables or objective functions that involve a large number of design parameters. The efficiency and accuracy of the proposed method are demonstrated via its application to the placement of stacked patch antennas in a linear array.

Combined small scale high dimensional model representation

Abstract: Nowadays the utilization of High Dimensional Model Representation (HDMR), which is an algorithm for approximating multivariate functions, is becoming more pervasive in the applications of approximation theory. This extensive usage motivates new works on HDMR, to get better solutions while approximating to the multivariate functions. One of them is recently developed “Combined Small Scale High Dimensional Model Representation (CSSHDMR)". This new scheme not only optimises HDMR results but also provides good approximation with less terms than HDMR does. This paper presents the theory and the numerical results of the new method and shows that it is possible to apply approximation to multivariate functions by keeping only constant term of HDMR. From this aspect CSSHDMR can be used in any scientific problem which includes multivariate functions, from chemistry to statistics.

Fundamental elements of vector enhanced multivariance product representation

Abstract: A new version of High Dimensional Model Representation (HDMR) is presented in this work. Vector HDMR has been quite recently developed to deal with the decomposition of vector valued multivariate functions. It was an extension from scalars to vectors by possibly using matrix weights. However, that expansion is based on an ascending multivariance starting from a constant term via a set of appropriately imposed conditions which can be related to orthogonality in a conveniently chosen Hilbert space. This work adds more flexibility by introducing certain matrix valued univariate support functions. We assume weight matrices proportional to unit matrices. This work covers only the basic issues related to the fundamental elements of the new approach.

Multivariate data modelling through Piecewise generalized HDMR method

Abstract: This work aims to develop a new High Dimensional Model Representation (HDMR) based method which can construct an analytical structure for a given multivariate data modelling problem. Modelling multivariate data through a divide-and-conquer method stands for multivariate data partitioning process in which we deal with a number of less variate data sets instead of a single N dimensional problem. Generalized HDMR is one of these methods used to model a multivariate data set which has a number of scattered nodes with associated function values. However, Generalized HDMR includes a linear equation system with huge number of unknowns and equations to be solved. This equation sometimes has linearly dependent equations in it and this is an undesirable situation. This work offers a new method named Piecewise Generalized HDMR method which bypasses this disadvantage as well as reducing the mathematical complexity and CPU time needed to complete the algorithm of the previous method. Our new method splits the given problem domain into subdomains, applies the Generalized HDMR philosophy to each subdomain and superpositions the information coming from these subdomains. The algorithm of this new method and a number of numerical implementations are given in this paper.

A modified method of calculating High Dimensional Model Representation (HDMR) Terms for parallelization with MPI and CUDA

Abstract: If the values of a multivariate function f(x 1,x 2,?,x N ) are given at only a finite number of points in the space of its arguments and an interpolation which employs continuous functions is considered standard multivariate routines may become cumbersome as the dimensionality grows. This urges us to develop a divide-and-conquer algorithm which approximates the function. The given multivariate data are partitioned into low-variate data. This approach is called High Dimensional Model Representation (HDMR). However, the method in its current form is not applicable to problems having huge volumes of data. With the increasing dimension number and the number of the corresponding nodes, the volume of data in question reaches such a high level that it is beyond the capacity of any individual PC because huge volume of data requires much higher RAM capacity. Another aspect is that the structure of equalities used in the calculation of HDMR terms varies according to the dimension number of the problem. The number of loops in the algorithm increases with the increasing dimension number. In this work, as a first step, the equations used are modified in such a way that their structure does not depend on the dimension number. With the newly obtained equalities, the method becomes appropriate for parallelization. Due to the parallelization, the RAM problem arising from problems with high volume of data is solved. Finally, the performance of the parallelized method is analyzed.

Efficient Extraction of Vortex Structures by Coupling Proper Orthogonal Decomposition (POD) and High–Dimensional Model Representation (HDMR) Techniques

Abstract: Spatially evolving and temporally developing fluid flow patterns are encountered in wide-spectrum fluid thermal problems. The time and length scales of eddies involved may vary over a few orders of magnitude. Often, simultaneous detection, identification, and characterization of small- and large-scale eddies, and persistent structures such as coherent structures, are crucial. The amount of scientific data generated either in a wind tunnel or in a virtual numerical tunnel through computational fluid dynamics is too voluminous for both storage and handling. To this end, a variety of data analysis and feature extraction tools, which rely on model reduction and efficient on-the-fly reconstruction, are highly desirable. In this article, we introduce a novel and efficient yet cheaper reconstruction strategy that enables a symbiotic coupling between two such techniques, viz., proper orthogonal decomposition (POD) and high–dimensional model representation (HDMR).

Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation

Abstract: High Dimensional Model Representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on Hybrid HDMR which is composed of Plain HDMR and Logarithmic HDMR. The Plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the Logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the Hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of Plain and Logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness Approximation Theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.

Möbius transformational high dimensional model representation on multi-way arrays

Abstract: Transformational High Dimensional Model Representation has been used for continous structures with different transformations before. This work is inventive because not only for the transformation type but also its usage. Mobius Transformational High Dimensional Model Representation has been used at multi-way arrays, by using truncation approximant and inverse transformation an approximation has been obtained for original multi-way array.

Membership function of failure probability using multicut-high dimensional model representation

Abstract: The structural reliability analysis in the presence of mixed uncertain variables demands more computation as the entire configuration fuzzy variables needs to be explored. Moreover the existence of multiple design points deviate the accuracy of results as the optimization algorithms may converge to a local design point by neglecting the main contribution from the global design point. Therefore, in this paper a novel uncertainty analysis method for estimating the membership function of failure probability of structural systems involving multiple design points in the presence of mixed uncertain variables is presented. The proposed method involves Multicut-High Dimensional Model Representation technique for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. 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 analysis involving any number of fuzzy and random variables. The accuracy and efficiency of the proposed method is demonstrated through three examples.

Approximate Dynamic Programming based on High Dimensional Model Representation

Abstract: This article introduces an algorithm for implicit High Dimensional Model Representation (HDMR) of the Bellman equation. This approximation technique reduces memory demands of the algorithm considerably. Moreover, we show that HDMR enables fast approximate minimization which is essential for evaluation of the Bellman function. In each time step, the problem of parametrized HDMR minimization is relaxed into trust region problems, all sharing the same matrix. Finding its eigenvalue decomposition, we effectively achieve estimates of all minima. Their full-domain representation is avoided by HDMR and then the same approach is used recursively in the next time step. An illustrative example of N-armed bandid problem is included. We assume that the newly established connection between approximate HDMR minimization and the trust region problem can be beneficial also to many other applications.

Cubic transformational high dimensional model representation

Abstract: This work focuses on the development of a multivariate function approximating method by using cubic Transformational High Dimensional Model Representation (THDMR). The method uses the target function’s image under a cubic transformation for High Dimensional Model Representation (HDMR) instead of the function’s itself.