High Dimensional Model Representation Research Articles

A novel kriging-improved high-dimensional model representation metamodelling technique for approximating high-dimensional problems

Aiming to address the issue of the ‘curse of dimensionality’ encountered in approximating high-dimensional problems using surrogate-based methods, high-dimensional model representation (HDMR), decomposing the high-dimensional problem into summands of different-order component functions, has been widely studied. To reduce the computational demands of the current HDMR metamodelling techniques, an improved high-dimensional model representation framework (iHDMR) is presented, taking full advantage of the relationships between the first-order and second-order component functions in the Cut-HDMR theory. then, a novel metamodelling technique, termed kriging (KRG)-iHDMR, is implemented by integrating the kriging technique and the suggested surface-axes alternating sampling strategy into the iHDMR framework. The accuracy and efficiency of KRG-iHDMR are demonstrated by various numerical and engineering examples with different dimensions. Finally, an optimization problem of the multi-stage solid launch vehicle propulsion system is introduced to verify the engineering feasibility of the KRG-iHDMR metamodelling technique.

Identification and modelling of parameters for the information-physical-social convergence characteristics of user-side flexible resources

The use of flexible resource information on the user side helps to increase system efficiency. Power system power variation becomes more pronounced with the access to renewable resources. Therefore, the study proposes a parameter identification and modeling method for the physical and social integration characteristics of flexible resource information on the user side. Taking the user’s air conditioning load as the object, the thermal dynamic model of the air conditioning building is constructed using equivalent thermal parameters, and the variable frequency air conditioning load is embedded in the battery model. The model parameter identification is carried out using high-dimensional model expression technology. According to the experimental data, in options 2 and 3, the system operator makes power purchases based on the storage status of the lithium battery or virtual battery, increasing the number of power purchases when the price of electricity is low and decreasing the number of power purchases when the price of electricity is high. This effectively reduces the system operator’s electricity costs. The error of multiple linear regression modelling varies widely, with relative errors up to 0.75 and an average relative error of 15.1%. The relative error of modelling based on the high-dimensional model expression technique is in the range of 0 to 0.2, with an average relative error of 5.5%. The results show that compared with multiple linear regression models, high-dimensional model representation technology has higher modeling accuracy and can accurately identify the parameters of the air conditioning load aggregation model, solving the problem of difficult parameter calculation in the practical application of the air conditioning load aggregation model, and providing technical support for power system regulation.

Stochastic analysis of thin beams and plates incorporating von-Kármán nonlinear strains using meshless method

The current paper proposes methods for stochastic meshless analysis of thin beams and plates, by considering von-Kármán nonlinear strains. Here, Young’s modulus is assumed to have a spatial variation over the domain and is modeled as a homogeneous random field. Further, it is discretized utilizing moving least square based shape functions. Gaussian and lognormal properties are presumed to model the randomness related to Young’s modulus. Element-free Galerkin method is used as the meshless tool. The present study suggests employing perturbation method if the quantities of interest are limited to the first or second probabilistic moments of the responses. The nonlinear equations in the proposed perturbation formulations are solved using Newton-Raphson iterative scheme for finding out the deterministic part of the response, as well as its initial two derivatives with respect to random variables. Additionally, if there is an interest in higher probabilistic moments, probability density functions, cumulative distribution functions, and so forth, the study proposes a method using high-dimensional model representation (HDMR) in the meshless framework. In HDMR, the nonlinear connection between input variables and the output response is represented in terms of hierarchical correlated function expansions. These hierarchical functions are approximated over the response values evaluated using deterministic analysis on selected number and combinations of control points of random variables involved. Lagrange interpolation method is employed for the construction of response surfaces. Unlike Monte Carlo simulation (MCS) which needs numerous deterministic nonlinear analyses to be performed during each simulation, HDMR requires only a few deterministic nonlinear analyses to construct the component functions. Hence, simulations performed on the reduced order models of high-dimensional response can be used to estimate the probabilistic moments as well as density functions and cumulative distributions of response, with high computational efficiency compared to MCS. The probabilistic moments and distributions produced for a range of coefficient of variation of input random field up to 30%, for both the normal and lognormal input random fields are compared with those provided by crude MCS on the system of equations and are observed to be matching well.

Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations.

We present a novel formulation of the vibrational density matrix renormalization group (vDMRG) algorithm tailored to strongly anharmonic molecules described by general, high-dimensional model representations of potential energy surfaces. For this purpose, we extend the vDMRG framework to support vibrational Hamiltonians expressed in the so-called n-mode second-quantization formalism. The resulting n-mode vDMRG method offers full flexibility with respect to both the functional form of the PES and the choice of the single-particle basis set. We leverage this framework to apply, for the first time, vDMRG based on an anharmonic modal basis set optimized with the vibrational self-consistent field algorithm on an on-the-fly constructed PES. We also extend the n-mode vDMRG framework to include excited-state-targeting algorithms in order to efficiently calculate anharmonic transition frequencies. We demonstrate the capabilities of our novel n-mode vDMRG framework for methyloxirane, a challenging molecule with 24 coupled vibrational modes.

Uncertainty assessment method for thermal runaway propagation of lithium-ion battery pack

To some extent, the thermal runaway issue pertaining to lithium-ion battery still present challenging problems, to put the safety-related issue under control in high-power and high-energy applications. One of these challenging issues relates to the underpinning stochastic uncertainties associated with thermal runaway propagation. Therefore, this study proposes a methodology to evaluate the uncertainty of thermal runaway propagation in Li-ion battery modules. Firstly, a numerical thermal runaway propagation model is developed for phase change material-based thermal management system and verified by experiments. Subsequently, the Adaptive Kriging High-dimensional model representation method is employed to construct a surrogate model, followed by global sensitivity analysis to evaluate the impact of six different influencing factors on thermal runaway propagation. Finally, the random fluctuation of the critical factors affecting thermal runaway propagation is integrated to analyze and quantify the uncertainty. The results show that the state of charge, the thermal conductivity of phase change material, and inter-cell spacing are critical factors affecting thermal runaway propagation. With the system parameters proposed in this paper, the critical state of charge range that significantly impacts the uncertainty of thermal runaway propagation is approximately 40% to 55%. The analysis of the propagation results and the cell temperature indicates that the highest level of uncertainty occurs at around 45% state of charge. In this case, the thermal runaway propagation is significantly impacted by changes in key factors, and the battery temperature exhibits high fluctuation levels. Furthermore, although there is a 59% likelihood of no propagation, the potential damage to all batteries becomes exceedingly significant once thermal runaway propagation occurs. The method proposed in this study effectively evaluates the uncertainty of thermal runaway propagation in battery systems. It identifies the causes of the uncertainty, which can be used to implement effective suppression measures to enhance safety.

Parametric study of square dome-shaped kirigami folded structure for blast mitigation

This paper explores the potential of using Square Dome-shaped Kirigami (SDK) structures as a core component in cladding systems for blast mitigation. The study aims to assess the effectiveness of the SDK foldcore in reducing the impact of blast loading and to determine the optimal geometry of the SDK foldcore for enhanced blast mitigation using High Dimensional Model Representations (HDMR). A nonlinear finite element analysis with the ABAQUS software package has been carried out to investigate the blast mitigation capacity. The model is subjected to a quasi-static compression test to simulate the crushing effect, and the results are used to calibrate the simulation with the experimental data from the literature. Four different levels of blast loading conditions are applied to evaluate the blast resistance of the system. The structural response of the SDK foldcore is compared to that of a traditional Square Honey Comb (SHC) core with the same relative density under the same blast loading conditions. The peak load transmitted by the SDK foldcore is much more consistent than the SHC core, even under different blast loading conditions. SDK foldcore significantly improves energy absorption, resulting in better performance in mitigating the effects of blast loading. Additionally, a mathematical model using HDMR is created to assess the performance of the SDK foldcore using geometric parameters as input variables. The HDMR model enables the generation of numerous alternative design solutions, allowing designers to select the most optimal geometry for the SDK foldcore.

A geometric sensitivity study for the aerodynamics of a strut-braced airframe

A global sensitivity analysis method, the adaptive-cut high-dimensional model representation method (HDMR) is considered to evaluate the impact of geometrical changes of a ultra-high aspect ratio strut-braced wing configuration on the aerodynamic performance. A data-driven reduced order model based on Proper Orthogonal Decomposition is used to keep the computational cost of the analysis at a manageable level. The airframe configuration is described using 7 geometrical parameters, which comprise the design space. The geometrical parameters are decomposed and analysed across their whole range of values by the HDMR to assess their influence on the Drag coefficient, Lift coefficient and Lift-to-Drag ratio of the aircraft. These results form the basis for a qualitative exploration of the aerodynamics of such airframes further augmented by high-fidelity CFD simulations. Results show that the sweep angle of the wing is the dominant parameter in terms of Drag due to changes in shock wave intensity. Changes in the wing root chord also implicitly affect the local geometry at the wing-strut junction, which influences the blockage effect and thus the intensity of the shock waves at the junction. Lift is affected primarily by the twist of the wing. Overall, the strut is shown to have significant effects on the performance of the aircraft design due to the presence of strong shock waves at the wing-strut junction and interference effects.

Experiment and simulation of hydrogen oxidation in a high-pressure turbulent flow reactor

Hydrogen oxidation has a complex dependence on pressure. Despite the well-known, reverse S-shape explosion limits, variation of hydrogen oxidation reactivity with pressure has rarely been investigated in a quantitative manner, particularly in the peninsula between the second and third ignition branches when the overall reactivity is low. This work investigates hydrogen oxidation in a turbulent flow reactor at 10 to 48 bar with an inlet temperature of 950 K and an equivalence ratio of 0.03–0.05, which complements our recent study at 1–8 bar in the same reactor (Lu et al. Proc. Combust. Inst. 2021, 243–250). The elevated pressures show a weak promoting effect, in contrast to the strong inhibiting effect at lower pressures in crossing the second ignition branch. Analyzing the hydrogen consumption rates across the pressure range revealed that the reactivity remains at a low level and is barely affected by pressure between 2 and 20 bar, suggesting the existence of a broad low-reactivity valley on the temperature–pressure plane which could be potentially important for practical applications.The measured species-time concentrations, combined with ignition delays selected from the literature, are used to develop a new hydrogen oxidation model focusing on hydrogen autoignition behavior. A comprehensive review of elementary reactions incorporating the latest updates for the rate coefficients is conducted to determine the prior uncertainties. A global optimization method incorporating random sampling and high-dimensional model representation is used to identify the most sensitive reactions and to constrain their uncertainty with the selected experimental targets. An optimized model is then obtained which shows excellent agreement with hydrogen autoignition experiments.

CFD Uncertainty Quantification using PCE–HDMR: Exemplary Application to a Buoyancy-Driven Mixing Process

For the investigation of uncertainties in high dimensional spaces of computationally expensive engineering applications, reliable Uncertainty Quantification (UQ) methods are needed. These methods should provide accurate and efficient High-Dimensional Model Representations of stochastic results using a reasonable number of calculations. Therefore, the PCE–HDMR approach (Polynomial Chaos Expansion–High-Dimensional Model Representation) is utilized to qualify appropriate UQ methods for large-scale computations in the field of Computational Fluid Dynamics. This technique is a combination of Cut-HDMR, a hierarchical decomposition modeling approach, with PCE. To demonstrate its effectiveness, the PCE–HDMR methodology in conjunction with complementary modeling techniques is applied for the UQ analysis of a buoyancy-driven mixing process between two miscible fluids within the Differentially Heated Cavity of aspect ratio 4. The results include a thorough probabilistic representation of time-dependent response quantities that comprehensively describe the mixing process. The stochastic models are derived from Large Eddy Simulations using PCE–HDMR and the Sparse Grid Method, which serves as a reference for the results from PCE–HDMR. The results show that PCE–HDMR provides accurate statistics of the modeled time-dependent stochastic processes and shows good agreement with the reference results. Thus, PCE–HDMR indicates great potential for UQ of technical-scale computations due to its efficiency and flexibility in the construction of stochastic models.

High Dimensional Model Representation Approach for Prediction and Optimization of the Supercritical Water Gasification System Coupled with Photothermal Energy Storage

Supercritical water gasification (SCWG) coupled with solar energy systems is a new biomass gasification technology developed in recent decades. However, conventional solar-powered biomass gasification technology has intermittent operation issues and involves multi-variable characteristics, strong coupling, and nonlinearity. To solve the above problems, firstly, a solar-driven biomass supercritical water gasification technology combined with a molten salt energy storage system is proposed in this paper. This system effectively overcomes the intermittent problem of solar energy and provides a new method for the carbon-neutral process of hydrogen production. Secondly, the high dimensional model representation (HDMR) approach, as a surrogate model, was used to predict the production and lower heating value of syngas developed in Aspen Plus, which were validated using experimental data obtained from the literature. The ultimate analysis of biomass, temperature, pressure, and biomass-to-water ratio (BWR) were selected as input variables for the model. The non-dominated sorted genetic algorithm II (NSGA II) was considered to maximize the gasification yield of H2 and the LHV of syngas in the SCWG process for five different types of biomass. Firstly, the results showed that HDMR models demonstrated high performance in predicting the mole fraction of H2, CH4, CO, CO2, gasification yield of H2, and lower heating value (LHV) with R2 of 0.995, 0.996, 0.997, 0.996, 0.999, and 0.995, respectively. Secondly, temperature and BWR were found to have significant effects on SCWG compared to pressure. Finally, the multi-objective optimization results for five different types of biomass are discussed in this paper. Therefore, these operating parameters can provide an optimal solution for increasing the economics and characteristics of syngas, thus keeping the process energy efficient.

Seismic fragility of non-ductile RC frames for pounding risk assessment

Building structures subjected to seismic excitation tend to vibrate with a unique set of dynamic characteristics, such as natural frequencies, fundamental mode shapes etc. In series-configured structural systems, a difference in such characteristics induces a phase difference between the structures’ oscillations, leading to structural collisions or pounding. Pounding often becomes a cause of catastrophic failures, hence an accurate estimation of structural integrity with respect to pounding is of paramount importance. This study aims to examine different fragility curve development methodologies available in the literature to assess the seismic performance of adjacent Reinforced Concrete (RC) structures subjected to structural pounding. The study primarily focuses on displacement-based fragility curves for the highly critical floor-to-column interactions of an eight-storey non-ductile RC frame, non-eccentrically adjacent to a three-storey rigid RC frame having different storey heights. Nonlinear time–history analyses have been performed to simulate pounding, and fragility curves have been generated using five methodologies. The direct Monte-Carlo Simulation technique has been used to benchmark against the obtained results to identify the suitability of each fragility assessment technique. Results indicate that even though there is no effect of the choice of assessment methodology on the estimated risk at low performance levels and intensity measures, the disparity becomes clearly visible at higher levels. The study presents alternate ways of developing accurate fragility curves by keeping in mind the need for computational efficiency without compromising the accuracy of results. The High Dimensional Model Representation meta-model is found to efficiently represent target pounding fragilities while incurring minimal computational costs.

Real-time reliability analysis of micro-milling processes considering the effects of tool wear

The development of micro-milling technology has promoted the high-precision machining of complex and small-size parts in modern manufacturing. Accurate cutting force control and reasonable machining parameters selection are conducive to the sustainable machining of micro-milling. In this work, a micro-milling mechanical model considering the change of tool wear, tool runout, and chip separation status with cutting time is proposed with the improved Gated Recurrent Unit with Multi-objective Dandelion Optimizer and mathematical modeling. The modeling accuracy is verified by experiments on Al6061. Combined with the established framework and the practical physical information, the posterior distribution of process parameters is inverted using the Bayesian updating method. Finally, the reliability calculation of machining system parameters is efficiently realized utilizing the posterior information and the High-dimensional Model Representation with the Stochastic configuration network method. The influence degree of the change of cutting parameters on the extreme value of micro-milling force is evaluated. The research results can provide analytical methods and theoretical guidance to ensure safe processing.

Surrogate-assisted evolutionary algorithm with hierarchical surrogate technique and adaptive infill strategy

Fitness functions of real-world optimization problems often need to be analyzed by expensive experiments or numerical simulations. Integrating these expensive simulations or experiments directly into optimization algorithms would result in substantial computational costs. Surrogate-assisted evolutionary algorithms (SAEAs) have attracted massive attention recently due to their high efficiency and applicability in solving real-world optimization problems. As the dimension of the optimization problem increases, the computational cost of constructing surrogates increases, and the surrogate model’s prediction accuracy may be severely degraded. High-dimensional model representation (HDMR) is a promising technique to partition a high-dimensional function into low-dimensional component functions. However, HDMR’s hierarchical structure limits its applicability in online SAEAs. To address these problems, this paper develops a surrogate-assisted evolutionary algorithm with hierarchical surrogate technique and adaptive infill strategy (SAEA-HAS). In this work, we propose a novel hierarchical surrogate technique, in which a composite surrogate model is constructed by the first-order HDMR model and an error value-based surrogate model, then, using the internal contrastive analysis method, a hierarchical surrogate model (HSM) combining the composite surrogate with the fitness value-based surrogate is established. In addition, an adaptive infill strategy is developed to balance the exploration and exploitation of the surrogate-assisted evolutionary search. Various test functions and an antenna optimization problem are employed to compare SAEA-HAS with several well-known SAEAs. The experimental results validate the effectiveness of SAEA-HAS.

Probabilistic Sensitivity Analysis With Dependent Variables: Covariance‐Based Decomposition of Hydrologic Models

AbstractVariance‐based analysis has emerged as method of choice for quantifying the sensitivity of the output, y, of a scalar‐valued square‐integrable function, f ∈ L2(), to its d ≥ 1 input variables, x = {x1, …, xd}, with support . The prototype of this approach, Sobol's method is a generalization of the analysis of variance (ANOVA) to d > 2 independent input variables and decomposes y, as sum of elementary functions of zeroth‐, first‐, second‐, up to dth‐order. This independence assumption is mathematically convenient but may not be borne out of the causal or correlational relationships between the x's. This paper is concerned with variance‐based sensitivity analysis (SA) for correlated input variables, for example, multivariate dependencies in a posterior parameter distribution. We use high‐dimensional model representation (HDMR) of Li et al. (2010, https://doi.org/10.1021/jp9096919), Li and Rabitz (2012, https://doi.org/10.1007/s10910-011-9898-0) and replace Sobol's elementary functions with so‐called component functions with unknown expansion coefficients to disentangle the structural, correlative and total contribution of input factors. We contrast the default HDMR methodology with cubic B‐splines and sequential coefficient estimation against its successor, HDMRext of Li and Rabitz (2012, https://doi.org/10.1007/s10910-011-9898-0), which uses polynomial component functions with an extended orthonormalized basis. Benchmark experiments confirm that HDMR and HDMRext parse out the structural and correlative contributions of input factors to the model output and infer an optimal experimental design with parameter correlation. Our last study applies HDMRext to probabilistic SA of a watershed model. The multivariate posterior parameter distribution supports model emulation and yields sensitivity indices that pertain to measured discharge data.

Cable force optimization of cable-stayed bridges: A surrogate model-assisted differential evolution method combined with B-Spline interpolation curves

Cable-stayed bridges have been widely constructed to cross large obstacles in transportation networks and metropolitan cities. As a highly redundant structure, the structural performance of a cable-stayed bridge can be significantly influenced by the cable prestressing forces. Therefore, the determination of optimal cable forces is necessary at the design stage. As computational mechanics matures, numerical computational methods such as high-fidelity finite element analyses facilitate accurate assessment of the bridge state, and evolutionary algorithms have been applied to enhance the global search ability. However, the combination of evolutionary algorithms and finite element model-based evaluations can substantially increase the computational burden, rendering the efficiency of optimization significantly time-consuming for practical engineering projects. To overcome this limitation, this work proposes a new surrogate model-assisted differential evolution method to facilitate the determination of optimal cable forces. Different from the existing methods, the proposed method develops a new individual selection strategy together with an additional archive to manage the diversity of population and further enhance the exploration ability. A new criterion is also proposed for the refinement of the surrogate model in the local search stage to avoid redundant information. Furthermore, B-Spline interpolation curve is integrated within the framework as a high-dimensional model representation technique for the design variables of cable forces. Five classic numerical examples and a practical engineering application validate the performance and advantages of the proposed method.

High dimensional model representation for flood fragility analysis of highway bridge

Climate changes and global warming have increased the frequency and severity of rains, hence the vulnerability of structures due to floods. Bridges are the most affected components of the transportation infrastructure affected by floods. Analysis in a probabilistic framework is required for reliable performance assessments of bridges. However, the detailed performance assessment of bridges requires consideration of the uncertainties in geometry, material properties, and flood loading. The random structural responses due to flood loading can be obtained by stochastic analysis; conventionally, this is carried out using Monte Carlo Simulations (MCS). Due to the high computational cost required for MCS, non-statistical methods like High Dimensional Model Representation (HDMR) can be a feasible alternative to MCS. The present study is an application of the HDMR method to develop metamodels for the flood response of the highway bridge, considering the uncertainties. The structural responses due to flood at the sampling points of the HDMR are obtained by conducting finite element analysis. The HDMR could predict the random bridge responses due to flood loading fairly well compared to other popular response surface methods (like Central Composite Design, Box Behnken Design, and Full Factorial Design) with significantly fewer simulations. HDMR provides a flood response metamodel, as a function of the sensitive random parameters, with a substantial reduction in the computational effort. The efficiency of the HDMR method can be attributed to the ability of HDMR to model the nonlinear flood responses in terms of the statistical properties of random variables. The metamodels for the pier drifts developed are further utilized to generate flood fragility curves. The HDMR metamodels are found to simplify the fragility computations.

High-Dimensional Model Representation-Based Surrogate Model for Optimization and Prediction of Biomass Gasification Process

Biomass gasification process has been predicted and optimized based on process temperature, pressure, and gasifying agent ratios by integrating Aspen Plus simulation with the high-dimensional model representation (HDMR) method. Results show that temperature and biomass to air ratio (BMR) have significant effects on gasification process. HDMR models demonstrated high performance in predicting H2, net heat (NH), higher heating value (HHV), and lower heating value (LHV) with coefficients of determination 0.96, 0.97, 0.99, and 0.99, respectively. HDMR-based single-objective optimization has maximum outputs for H2, HHV, and LHV (0.369 of mole fractions, 340 kJ/mol, and 305 kJ/mol, respectively) but NH would be negative at these conditions, which indicates that process is not energy-efficient. The optimal solution was determined by the multiobjective which produced 0.24 mole fraction of H2, 158.17 kJ/mol of HHV, 142.48 kJ/mol of LHV, and 442.37 kJ/s NH at 765°C, 0.59 BMR, and 1 bar. Therefore, these parameters can provide an optimal solution for increasing gasification yield, keeping process energy-efficient.

Comprehensive influence of uncertainty propagation of chemical kinetic parameters on laminar flame speed prediction: a case study of dimethyl ether

The uncertainties existing in the parameters of chemical kinetic models have a non-negligible influence on the model predictions. It is necessary to conduct a quantitative uncertainty analysis to explore the influence of each parameter on chemical mechanism predictions. To comprehensively consider the effect of the uncertainties of reaction rate parameters, thermodynamic parameters, and transport parameters on model predictions, local sensitivity analysis, local-sensitivity-based uncertainty analysis (LSUA), and random-sampling high dimensional model representation (RS-HDMR) method were coupled to investigate the uncertainty propagation of the chemical kinetic parameters to the calculated laminar flame speed of dimethyl ether under a wide range of conditions using a detailed mechanism. First, the uncertainty analysis was conducted using the local sensitivity analysis and the LSUA method under a wide range of operating conditions to identify the important operating conditions and chemical kinetic parameters. It is found that the prediction uncertainty of laminar flame speed is more obvious under the conditions of high dilution ratio, high pressure, and large equivalence ratio than that under other conditions. According to the results of LSUA, the prediction uncertainty is mainly from the reaction rate coefficients and thermodynamic data. Then, the uncertainty propagation from the significant parameters to the calculated laminar flame speed under important conditions was analysed using the RS-HDMR method. To reduce the huge computational cost of the RS-HDMR method, the backpropagation artificial neural network was employed. The RS-HDMR results indicate that the reaction H + O2 = O + OH has the highest sensitivity coefficient under the whole investigated conditions, which is different from the results using the LSUA method. The non-linear relationship between the rate coefficient and the predicted laminar flame speed is responsible for the discrepancy. Furthermore, it is found that the sensitivity coefficient of the input parameters strongly depends on the operating conditions.

A Scalable Unsupervised Feature Selection With Orthogonal Graph Representation for Hyperspectral Images

Feature selection is essential in various fields of science and engineering, from remote sensing to computer vision. Reducing data dimensionality by removing redundant features and selecting the most informative ones improves machine learning algorithms’ performance, especially in supervised classification tasks, while lowering storage needs. Graph-embedding techniques have recently been found efficient for feature selection since they preserve the geometric structure of the original feature space while embedding data into a low-dimensional subspace. However, the main drawback is the high computational cost of solving an eigenvalue decomposition problem, especially for large-scale problems. This paper addresses this issue by combining the graph embedding framework and representation theory for a novel feature selection method. Inspired by the high dimensional model representation, the feature transformation is assumed to be a linear combination of a set of univariate orthogonal functions carried out in the graph embedding framework. As a result, an explicit embedding function is created, which can be utilised to embed out-of-samples into low-dimensional space and provide a feature relevance score. The significant contribution of the proposed method is to divide an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional generalised eigenvalue problem into <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> small-sized eigenvalue problems. With this property, the computational complexity of the graph embedding is significantly reduced, resulting in a scalable feature selection method, which could be easily parallelized too. The performance of the proposed method is compared favourably to its counterparts in high-dimensional hyperspectral image processing in terms of classification accuracy, feature stability, and computational time.

Efficient inner-outer decoupling scheme for non-probabilistic model updating with high dimensional model representation and Chebyshev approximation

Interval arithmetic offers a powerful tool for structural model updating when uncertain-but-bounded parameters are considered. However, the application of interval model updating for practical engineering structure is hindered due to model complexity and huge computational burden involved in the repeated evaluations of non-probabilistic constraints. In this light, an efficient inner-outer decoupling scheme is proposed for non-probabilistic model updating in this study. The mathematical operation of interval model updating is decomposed into two layers labelled as inner layer with the operation of uncertainty propagation and outer layer with the operation of interval optimization. In the inner uncertainty propagation, the High Dimensional Model Representation (HDMR) is utilized to enable the decomposition of the model outputs in terms of multivariate inputs into the sum of multiple single-variate functions, which is further approximated by Chebyshev polynomials so that the stationary points of each function can be derived. In the outer layer, a fast-running optimization strategy based on the stationary points of Chebyshev polynomial approximation is proposed to accelerate tracking the bounds of model parameters by avoiding time-consuming brute-force interval optimization. As a result, the original non-probabilistic updating process with two interacted layers can be completely decoupled into two independent operations of the inner uncertainty propagation and outer interval optimization so as to enhance the search efficiency and convergence rate significantly. Two numerical case studies illustrate capability of the proposed method in updating the structural parameters intervals efficiently with the model outputs intervals agreeing well with the testing outputs intervals. Two experimental cases of steel plates and the Canton Tower also demonstrate the efficiency and advantages of the method in interval model updating.