Showing papers on "High-dimensional model representation published in 2007"

High dimensional model representation for piece-wise continuous function approximation

Abstract: High dimensional model representation (HDMR) approximates multivariate functions in such a way that the component functions of the approximation are ordered starting from a constant and gradually approaching to multivariance as we proceed along the terms like first-order, second-order and so on. Until now HDMR applications include construction of a computational model directly from laboratory/field data, creating an efficient fully equivalent operational model to replace an existing time-consuming mathematical model, identification of key model variables, global uncertainty assessments, efficient quantitative risk assessments, etc. In this paper, the potential of HDMR for tackling univariate and multivariate piece-wise continuous functions is explored. Eight numerical examples are presented to illustrate the performance of HDMR for approximating a univariate or a multivariate piece-wise continuous function with an equivalent continuous function. Copyright © 2007 John Wiley & Sons, Ltd.

Computational complexity investigations for high dimensional model representation algorithms used in multivariate interpolation problems

Abstract: In multivariate interpolation problems, increase in both the number of independent variables of the sought function and the number of nodes appearing in the data set cause computational and mathematical difficulties. It may be a better way to deal with less variate partitioned data sets instead of an N-dimensional data set in a multivariate interpolation problem. New algorithms such as High Dimensional Model Representation (HDMR), Generalized HDMR, Factorized HDMR, Hybrid HDMR are developed or rearranged for these types of problems. Up to now, the efficiency of the methods in mathematical sense were discussed in several papers. In this work, the efficiency of these methods in computational sense will be discussed. This investigation will be done by using several numerical implementations.

A novel hybrid high dimensional model representation (HHDMR) based on the combination of plain and logarithmic high dimensional model representations

Abstract: This paper focuses on a new version of Hybrid High Dimensional Model Representation for multivariate functions High Dimensional Model Representation (HDMR) was proposed to approximate the multivariate functions by the functions having less number of independent variables Towards this end, HDMR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in an ascending ordering of multivariance HDMR method is a scheme truncating the representation at a prescribed multivariance If the given multivariate function is purely additive then HDMR method spontaneously truncates at univariance, otherwise the highly multivariant terms are required On the other hand, if the given function is dominantly multiplicative then the Logarithmic HDMR method which truncates the scheme at a prescribed multivariance of the HDMR of the logarithm of the given function is taken into consideration In most cases the given multivariate function has both additive and multiplicative natures If so then a new method is needed Hybrid High Dimensional Model Representation method is used for these types of problems This new representation method joins both plain High Dimensional Model Representation and Logarithmic High Dimensional Model Representation components via an hybridity parameter This work focuses on the construction and certain details of this novel method

The influence of the peak location on the additivity of the high dimensional model representation

Abstract: Mathematical modeling of physical systems are commonly faced situations. Solution depends on the quality of the modeling directly. Among the many techniques, High Dimensional Model Representation (HDMR) is a new method that brings great efficiency to the modeling of a system. Although Interpolation, Splines, Finite differences etc. are useful methods, they do not often give as good results as HDMR does. These methods need more reference points or nodes and higher order polynomials for better results, and this means higher cost calculations. However, HDMR offers new expansions, truncation of intended order, needed for less sample points etc.. Since HDMR is a modeling method based on optimization and projection operator theory, solving problems with differential equations, input - output systems require less calculations with high effectiveness. In addition, HDMR contains analysis of variance calculations. Hence, HDMR is also an effective method for statistics.

Application of High Dimensional Model Representations (HDMR) and Interpolation to Optimize Nutrient Removal Under Uncertainty

Abstract: In order to circumvent difficulty caused by model uncertainty, high dimensional model representation (HDMR) and interpolation were employed to approximate model outputs with different combinations of manipulated variables and model parameters. The results showed that HDMR and interpolation could be successfully applied to optimize nutrient removal under uncertainty. INTRODUCTION Uncertainty is inevitable when using mathematical model to simulate real process that subject to both anthropentic and natural disturbance. If uncertainty associated with a model is neglected, an optimal solution induced from this model may be far from optimal when applied to reality. Some control strategies have been proposed to take model uncertainty into account. The most popular method is to perform Mante Carlo simulations. However, with so many parameters in ASM, the computation cost is prohibitively high. On the other hand, high dimensional model representation (HDMR) is a fast algorithm that can circumvent the apparent exponential difficulty of highdimensional mapping problem. It has been successfully applied in atmospheric chemistry modeling (Li et al, 2000). Interpolation is an algorithm used to estimate function values between data points. Here we adopted HDMR and interpolation to approximate model outputs under different combinations of manipulated variables and model parameters. PROCESS AND MODEL DESCRIPTION The basic process investigated here was an activated sludge process (AS) to which an anaerobic tank and anoxic tank were added to enhance nutrient removal (Figure 1). The dimensions of the units were listed in Table 1. Table 1. Main dimensions of units Construction Item Dimension Bioreactor Anaerobic tank Volume 884 m Anoxic tank Volume 1768 m Aerobic tank Volume 6375 m Clarifier Area 500 m Height 4m Table 2. Flux-based average influent characterization Parameters Unit Dimension Total COD mgCOD/l 404 Ortho-P mgP/l 1.94 Total phosphorus mgP/l 5.43 NH4-N mgN/l 12.72 TSS mgSS/l 261 INFLUENT LOAD The influent data were collected in Athens No. 2 wastewater treatment plant (WWTP) of Georgia in 1998 (Liu, 2000; Liu and Beck, 2000). Generally, the influent quality can be classified as medium. Its main characteristics were listed in Table 2. The influent COD was fractioned into its components as in ASM No. 2d (Henze et al, 1999). The range of components listed in Table 3 was induced from the results of previous model calibration. MODEL AND SIMULATION DESCRIPTION ASM 2d was selected as it includes both nitrogen and phosphorus removal. The model was calibrated with the data collected in Athens No. 2 WWTP in 1998. Totally, there were 41 parameters in this model. We selected 16 parameters according to their sensitivity, and included the fractionation of influent COD in the framework. Thus, totally 24 parameters were adjusted in each simulation. All simulations were performed on WEST simulation platform (Hemmis nv, Kortrijk, Belgium). The implementation of the process was shown in Figure 2. Figure 1. Flowchart of the process Figure 2. Implementation of process in WEST Table 3. Characterization of influent COD (Ratio of total COD) Component Definition Range (%) SI Inert soluble organic material 4.33~5.75 SA Fermentation products 4.59~6.31 SF Fermentable, readily biodegradable organic substrates 8.08~11.81 XI Inert particulate organic material 25.4~28.7 XS Slowly biodegradable substrates 36.5~51.8 XH Heterotrophic organisms 5.13~9.81 XAUT Nitrifying organisms 0.39~0.61 XPAO Phosphate-accumulating organisms 0.20~0.34 XPP Poly-phosphate 0 XPHA A cell internal storage product of PAO 0.08~0.15 CONTROL STRATEGY DESCRIPTION The control strategy used here was essentially stochastic optimization of manipulated variables under model uncertainty. The values of manipulated variables were chosen such that the expected objective function assumed a minimum (Infanger, 1993): z = min E f(x, ω) s/t x∈ C = ∩ω∈Ω Cω Where x – manipulated variables; ω model parameters; Ω set of possible realizations of ω; f(x, ω) – objective function; C – set of feasible solutions. The optimal solution represented the realistic solution of the stochastic optimization problem. X* ∈ arg min {Ef(x, ω) | x∈∩ω∈Ω C } Table 4. Range of model parameters Parame ter Definition Unit Range μPAO Maximum growth rate of PAO d 1.62~1.79 qPP Rate constant for storage of XPP gXppg XPAO d 1.98~2.16 YPHA PHA requirement for PP storage gCOD gP 0.06~0.16 KPS Saturation coefficient for phosphorus in storage of PP gPm 0.08~0.12 ηNO3_HE T Reduction factor for denitrification 0.33~0.49 KO2_AUT Saturation/inhibition coefficient for oxygen for XAUT gO2m 0.29~0.45 μAUT Maximum growth rate of XAUT d 0.87~1.04 bH Rate constant for lysis and decay d 0.33~0.45 bPP Rate for lysis of XPP d 0.0019~0.10 KNH4_AU T Saturation coefficient for ammonia for XAUT gNm 0.6~1.03 KA Saturation coefficient for acetate gCOD m 3~5 KF Saturation coefficient for growth on SF gCOD m 3~5 μH Maximum growth rate on substrate d 4.93~5.96 fXI Fraction of inert COD generated in biomass lysis 0.10~0.15 KX Saturation coefficient for particulate COD gCOD m 0.11~0.15 V0 Settling velocity of sludge m/d 617~916 For this research, 6 manipulated variables were employed, i.e. allocation of influent and return sludge among the three tanks, dissolved oxygen in aerobic tank, internal recirculation, outer recycle, and waste sludge. For each manipulated variable, we selected 5 values. Thus, totally we had 5 = 15,625 different combinations of manipulated variable values. Obviously, the simulations of all these combinations cannot be finished in a reasonable period of time. Alternatively, we used HDMR (Li et al, 2000) to approach model outputs as the following

A Practical Solution of the Random Eigenvalue Problems using Factorized Decomposition Technique

Abstract: This paper presents a practical solution for probabilistic characterization of real valued eigenvalues of positive semi-definite random matrices. The method involves a novel dimension reduction technique that facilitates a lower-dimensional approximation of a high dimensional problem. The present method is basically founded on the idea of high dimensional model representation (HDMR) technique. HDMR is a multivariate representation to capture the input-output relationship of a physical system with many variables. It is a very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Practically for most well-defined physical systems, only relatively low order correlations of the input variables are expected to have a significant effect on the overall response. HDMR expansion utilizes this property to present an accurate hierarchical representation of the physical system. The method involves multiplicative decomposition of a multivariate eigenfunction into multiple one-dimensional eigenvalues.