Approximate Model Research Articles

Evolutionary multiobjective optimization assisted by scalarization function approximation for high-dimensional expensive problems

Surrogate-assisted evolutionary algorithms (SAEAs) are a promising approach for solving expensive multiobjective optimization problems, but they often cannot address high-dimensional problems. Although one common approach to overcoming this challenge is to construct reliable surrogates, their accuracy inevitably deteriorates in a high-dimensional search space. Thus, this paper presents a novel SAEA based on scalarization function approximation, which is designed to strengthen its robustness against this deterioration. The proposed algorithm constructs an approximation model for each scalarization function defined in a decomposition-based framework. Each decomposed problem is then solved using multiple independent models trained for its neighbor problems. The intent is to decrease the risk of search performance degradations caused by unreliable approximations and retain the redundancy of the surrogate-assisted search to hedge the risk of over-fitting. Furthermore, each approximation model is adapted to a promising region of its corresponding decomposed problem to reduce the complexity of model fitting given a limited number of training samples. Experimental results show that the proposed algorithm is competitive with state-of-the-art SAEAs adapted for high-dimensional problems.

Ameliorated Snake Optimizer-Based Approximate Merging of Disk Wang-Ball Curves.

A method for the approximate merging of disk Wang-Ball (DWB) curves based on the modified snake optimizer (BEESO) is proposed in this paper to address the problem of difficulties in the merging of DWB curves. By extending the approximate merging problem for traditional curves to disk curves and viewing it as an optimization problem, an approximate merging model is established to minimize the merging error through an error formulation. Considering the complexity of the model built, a BEESO with better convergence accuracy and convergence speed is introduced, which combines the snake optimizer (SO) and three strategies including bi-directional search, evolutionary population dynamics, and elite opposition-based learning. The merging results and merging errors of numerical examples demonstrate that BEESO is effective in solving approximate merging models, and it provides a new method for the compression and transfer of product shape data in Computer-Aided Geometric Design.

The Saturated Multiserver Job Queuing Model with Two Classes of Jobs: Exact and Approximate Results

We study a Multiserver Job Queuing Model (MJQM), i.e., a FIFO (First-In-First-Out) queue in which jobs request varying numbers of servers. Our investigation focuses on the saturation analysis of MJQM with two job classes. Jobs in the first class need only one server for an exponentially distributed service time, while jobs in the second class require either all servers or a fraction of them. We provide exact results for the scenario in which jobs in the second class require all servers and have generally distributed service times. Additionally, we develop an accurate approximate model of the MJQM behavior when jobs in the second class require a fraction of the available servers, and have exponentially distributed service times.

Optimizing the Value of a CO2 Water-Alternating-Gas Injection Project under Geological and Economic Uncertainties

Summary The numerical simulations required for the robust optimization (RO) of the alternating injection of water and CO2 in hydrocarbon reservoirs are computationally expensive due to engineering, geological, and economic uncertainties. Using approximation models of the desired objective function(s) can significantly decrease the cost associated with the optimization routines while providing an adequate sampling of the input variables or uncertainties. In this study, we optimized the value of a water-alternating-gas (WAG) injection project in a Niger-Delta oil reservoir by applying the Markowitz classical theory to a suitable approximation model of the objective function. Our RO methodology incorporated significant geological and economic uncertainties—such as uncertainties due to the upscaling of the coarse-scale reservoir model and those due to the lack of other relevant geological and economic data—to the optimization routine to create better operating strategies for the projects that are risk-quantified. To compute the objective function, a novel economic model for the CO2 sequestration processes in the Niger-Delta hydrocarbon basin was applied. The result obtained in the presented case showed that a net present value (NPV) of at least USD 65.98 million was derived from the project depending on the engineer’s or user’s confidence level. Overall, the applicability of proxy models to RO routines was demonstrated. As a result, investment decisions that accounted for uncertainties and any potential variation in the field’s operation and development were made.

Flow curve approximation of high-strength aluminium alloys in heat-assisted forming processes

Predicting the flow behaviour at high strain values beyond uniform elongation is a critical factor in sheet metal forming, especially in the context of heat-assisted forming processes of lightweight materials. Based on uniaxial tensile tests conducted at the temperature profiles of the Hotforming, W-Temper, and Warmforming process routes this study presents the determination of the flow curve for EN AW-7021 sheet material using five commonly used empirical approximation models. A support point obtained from layer compression tests was introduced to analyse the accuracy of the extrapolation models at higher strain values. All models considered were found to be in good agreement with the experimental tensile test data in the area of uniform elongation. However, the empirical approximation models show significant differences in strain hardening behaviour when higher strain values and varying temperature profiles are considered. For the Hotforming and W-Temper process routes, the Hockett-Sherby model estimates the flow behaviour with the highest accuracy. When using the Warmforming process, all approximation models considered seem to estimate the flow behaviour with reasonable accuracy, however the Ludwik-Hollomon modified equation shows the best correlation with the experimental data.

Batch fermentation kinetics study of biosynthesis lipopeptides by Bacillus altitudinis Q7 in 5 L fermenter

Optimization of fermentation parameters can effectively increase microbial metabolites. The fermentation conditions of biosynthesis lipopeptides by Bacillus altitudinis Q7 (B. altitudinis Q7) in 5 L reactor were optimized and the fermentation kinetics were monitored throughout the fermentation process. By design-response surface methodology, the results showed that under optimal conditions (72 h, pH 6.6, 31 °C, 3% inoculum, 200 rpm, 1.5 L/min aeration), the lipopeptide yield reached 1.689 mg/mL, 24.5% higher than under single-factor conditions, with maximum biomass dry weight of 1.650 g/L. According to the logistic and Luedeking-Piret approximate models, kinetic equation models of growth, product (lipopeptide) synthesis, and substrate (glucose) utilization during the fermentation process were simulated. This was done to fit variables and devise optimal control strategies. The research has improved the yield of target products in fermentation processes, reduced production costs and provided a theoretical basis for the simulation, optimisation and regulation of fermentation in modern industry.

Two-step AI-aided Bayesian source identification of urban-scale pollution

Poor air quality produces detrimental effects worldwide; hence, it is vital to thoroughly characterize air pollution sources and effectively address and mitigate such effects. Due to the complexity of the underlying physical processes, and uncertainties in the available observations, the air pollution source identification problem is typically cast within a Bayesian inversion framework. The latter incorporates prior knowledge and observations to characterize a release event through the posterior distribution of the source parameters. In this study, we rely on two-dimensional (2D) pollutant concentration distributions as observations, and adopt the Wasserstein (W2) distance to model the likelihood probability distribution for given emission parameters. Since the posterior distribution is estimated via random sampling that involves many forward model runs, the Bayesian framework can be computationally prohibitive for realistic urban air pollution problems that are driven by computationally demanding micro-scale flow simulations. Furthermore, computing the W2 distance is resource-intensive . In this context, we develop a computationally efficient Bayesian framework by following (i) a two-stage approach that reduces the cost of the Bayesian inversion, and (ii) an artificial intelligence (AI) approximation of the W2 distance. In the two-stage approach, a low-resolution dispersion model is run in the first stage to propose representative samples of emission parameters for final selection by the original high-resolution model in the second stage. In addition, we approximate the W2 distance using a deep neural network (DNN) to achieve an appreciable reduction in the computational cost with negligible loss in the inversion performance. We design numerical experiments to test the sensitivity of the inverse solution to the characteristics of the approximative model. The results indicate that pairing the two-stage approach with the DNN approximation of the W2 distance preserves the quality of the inverse solution, while achieving at least threefold reductions in the computational cost.

Optimization design on cooling structure of High-Temperature magnetic fluid dynamic seal device

In order to better sealing performance of magnetic fluid at high temperature, structural optimization design of cooling component in magnetic fluid dynamic sealing device at high temperature is studied in this paper. The magnetic field model of magnetic fluid sealing device at high-temperature is established to analyze the influence of temperature on the sealing performance of magnetic fluid, and magnetic field strength distribution is obtained. The simulated results show both magnetic induction strength difference and theoretical sealing pressure decrease gradually with the increase of the operating temperature. To reduce the temperature having great influence on sealing performance of magnetic fluid dynamic seal device, the temperature field model is constructed. Based on the Latin hypercube sampling method, the design matrix between cooling structural parameters and temperature is determined. In consideration of the neural network approach, the approximate agent model is built to analyze the correlation relationship between cooling structure parameters and cooling performance. The multi-island genetic algorithm is conducted on searching the optimal cooling structural parameters under global design space. The optimized results show that the average temperature of the permanent magnet is decreased by 27 °C, and the average temperature of the seal gap is reduced by 28 °C. The theoretical sealing differential pressure of the high-temperature magnetic fluid is improved by 49 KPa.

Regime Map of the Effective Medium Approximation Modelling of Micro-Rough Surfaces in Ellipsometry.

In this work, we discuss the precision of the effective medium approximation (EMA) model in the data analysis of spectroscopic ellipsometry (SE) for solid materials with micro-rough surfaces by drawing the regime map. The SE parameters ψ (amplitude ratio) and Δ (phase difference) of the EMA model were solved by rigorous coupled-wave analysis. The electromagnetic response of the actual surfaces with micro roughness was simulated by the finite-difference time-domain method, which was validated by the experimental results. The regime maps associated with the SE parameters and optical constants n (refractive index) and k (extinction coefficient) of the EMA model were drawn by a comparison of the actual values with the model values. We find that using EMA to model micro-rough surfaces with high absorption can result in a higher precision of the amplitude ratio and extinction coefficient. The precisions of ψ, Δ, n and k increase as the relative roughness σ/λ (σ: the root mean square roughness, λ: the incident wavelength) decreases. The precision of ψ has an influence on the precision of k and the precision of Δ affects the precision of n. Changing σ alone has little effect on the regime maps of the relative errors of SE parameters and optical constants. A superior advantage of drawing the regime map is that it enables the clear determination as to whether EMA is able to model the rough surfaces or not.

Error approximation and bias correction in dynamic problems using a recurrent neural network/finite element hybrid model

This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time-dependent, multi-fidelity problems, and use the trained hybrid models to perform bias correction of the low-fidelity models. The hybrid model uses FE basis functions as a spatial basis and RNNs for the approximation of the time dependencies of the FE basis' degrees of freedom. The training data sets consist of sparse, non-uniformly sampled snapshots of the discrepancy function, pre-computed from trajectory data of low- and high-fidelity dynamic FE models. To account for data sparsity and prevent overfitting, data upsampling and local weighting factors are employed, to instigate a trade-off between physically conforming model behavior and neural network regression. The proposed hybrid modeling methodology is showcased in three highly non-trivial engineering test-cases, all featuring transient FE models, namely, heat diffusion out of a heat sink, eddy-currents in a quadrupole magnet, and sound wave propagation in a cavity. The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models.

Rainfall Infiltration through Stratified Colluvial Deposits: Analytical Approach vs. Numerical Modelling

This work investigates the rainfall infiltration process within homogeneous and stratified colluvial deposits caused by short (1–3 h) and intense (40–90 mm/h) rainfall, using both analytical and numerical infiltration modelling. The findings of the investigation demonstrate that the classic Green–Ampt model can be employed effectively to study homogeneous colluvial covers with permeability equal to or lower than kw = 10−5 m/s and that are subject to a 1 h rainfall with intensity I ≥ 45–50 mm/h. In these circumstances, a top-down saturation front forms within the colluvial deposit, leading to the saturation of a 70–100 cm-thick layer. This critical condition occurs every 5–10 years in the mountain area of the Friuli Venezia Giulia Region (NE Italy), which corresponds to a lower return period of critical hydrologic events when compared with other mountain basins in the Alps due to the higher initial degree of saturation characterising colluvial covers in this area (70–95%). When analysing stratified colluvial covers, the Dagan–Bresler approximate model, as well as the numerical modelling, emphasised the strong influence that abrupt variations in the permeability of the various soil layers have on the infiltration process at depth. In particular, the presence of a top organic soil horizon that is rich in macro-pores and is characterised by a higher permeability (k = 10−4 m/s) actually reduces the possibility of surficial ponding, which is the basic condition of the “piston” models. The highly permeable top soil allows for a rapid downward infiltration up to contact with the underlying colluvial material, which is less permeable (k = 10−5 m/s). Therefore, a perched water table forms starting from the organic soil–colluvium interface, originating pore–water overpressures within the colluvial deposit, with maximum values in the order of 5–10 kPa.

Convex approximations of two-stage risk-averse mixed-integer recourse models

AbstractWe consider two-stage risk-averse mixed-integer recourse models with law invariant coherent risk measures. As in the risk-neutral case, these models are generally non-convex as a result of the integer restrictions on the second-stage decision variables and hence, hard to solve. To overcome this issue, we propose a convex approximation approach. We derive a performance guarantee for this approximation in the form of an asymptotic error bound, which depends on the choice of risk measure. This error bound, which extends an existing error bound for the conditional value at risk, shows that our approximation method works particularly well if the distribution of the random parameters in the model is highly dispersed. For special cases we derive tighter, non-asymptotic error bounds. Whereas our error bounds are valid only for a continuously distributed second-stage right-hand side vector, practical optimization methods often require discrete distributions. In this context, we show that our error bounds provide statistical error bounds for the corresponding (discretized) sample average approximation (SAA) model. In addition, we construct a Benders’ decomposition algorithm that uses our convex approximations in an SAA-framework and we provide a performance guarantee for the resulting algorithm solution. Finally, we perform numerical experiments which show that for certain risk measures our approach works even better than our theoretical performance guarantees suggest.

Maximizing the survival probability in a cash flow inventory problem with a joint service level constraint

This paper investigates a multi-period stochastic cash flow inventory problem with the aim of maximizing the long-term survival probability, which may be the objective of some retailers especially in periods of economic distress. Demand in each period is stochastic and can be non-stationary. In order to avoid too many lost sales under this objective, we introduce a joint chance constraint requiring the probability of no stockouts during the planning horizon to be higher than a specified service level. We develop a scenario-based model and a sample average approximation (SAA) model to solve the problem. A statistical upper bound on the survival probability based on SAA is provided and we discuss upper and lower bounds for the problem based on stochastic dynamic programming. We also propose a rolling horizon approach with service rate updating to test the out-of-sample performance of the two stochastic models and solve problems with long planning horizons. We test the two methods in large numerical tests and find that the rolling horizon approach together with the stochastic models can solve realistically sized problems in reasonable time.

Performance analysis of a complex process industrial unit utilizing intuitionistic fuzzy-based integrated framework

PurposeAn integrated intuitionistic fuzzy (IF) modelling-based framework for examining the performance analysis of a packaging unit (PU) in three different stages has been proposed.Design/methodology/approachFor the series and parallel configuration of PU, a mathematical model based on the intuitionistic fuzzy Lambda–Tau (IFLT) approach was developed in order to calculate various reliability parameters at various spreads. For determining membership and non-membership function-based reliability parameters for the top event, AND/OR gate transitions expression was employed.FindingsFor 15%–30% spread, unit’s availability for the membership function falls by 0.006442%, and it falls even more by 0.014907% with an increase in spread from 30% to 45%. In contrast, for 15%–30% spread, the availability of non-membership function-based systems reduces by 0.007491% and further diminishes. Risk analysis has presented applying an emerging approach called intuitionistic fuzzy failure mode and effect analysis (IFFMEA). For each of the stated failure causes, the output values of the intuitionistic fuzzy hybrid weighted Euclidean distance (IFHWED)-based IFFMEA have been tabulated. Failure causes like HP1, MT6, FB9, EL16, DR23, GR27, categorized under subsystems, namely hopper, motor, fluidized bed dryer, distributor, grader and bin, respectively, with corresponding IFFMEA output scores 1.0975, 1.0190, 0.8543, 1.0228, 0.9026, 1.0021, were the most critical one to contribute in the system’s failure.Research limitations/implicationsThe limitation of the proposed framework lies in the fact that the results obtained for both reliability and risk aspects mainly depend on the correctness of raw data provided by the experts. Also, an approximate model of PU is obtained from plant experts to carry performance analysis, and hence more attention is required in constructing the model. Under IFLT, reliability parameters of PU have been calculated at various spreads to study and analyse the failure behaviour of the unit for both membership and non-membership function in the IFS of [0.6,0.8]. For both membership- and non-membership-based results, availability of the considered system shows decreasing trend. To improve the performance of the considered system, risk assessment was carried using IFFMEA technique, ranking all the critical failure causes against IFHWED score value, on which more attention should be paid so as to avoid sudden failure of unit.Social implicationsThe livelihood of millions of farmers and workers depends on sugar industries. So perpetual running of these industries is very important from this viewpoint. On the basis of findings of reliability parameters, the maintenance manager could frame a correct maintenance policy for long-run availability of the sugar mills. This long-run availability will generate revenue, which, in turn, will ensure the livelihood of the farmers.Originality/valueMathematical modelling of the considered unit has been done applying basic expressions of AND/OR gate. IFTOPSIS approach has been implemented for ranking result comparison obtained under IFFMEA approach. Eventually, sensitivity analysis was also presented to demonstrate the stability of ranking of failure causes of PU.

Fourier series-based approximation of time-varying parameters in ordinary differential equations

Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.

Pulsed Electromagnetic Excitation of a Thin Wire—An Approximate Numerical Model Based on the Cagniard–DeHoop Method of Moments

Pulsed Electromagnetic Excitation of a Thin Wire—An Approximate Numerical Model Based on the Cagniard–DeHoop Method of Moments

The Physical Application of a Triangle Approximation Model

The Physical Application of a Triangle Approximation Model

The variation of growth in diameter and its relationship with the height of pedunculated oak (Quercus Robur L.) trees in the provenance cultures

In provenance cultures, studies were conducted on diameter growth dynamics in pedunculated oak (Quercus robur L.) trees and the height-diameter relationship was evaluated based on three data approximation models. The studies highlighted a signifi cant variation of the trunk diameter in trees of different origins and a similar growth reaction of oaks from four provenances (Hârjauca, Plaiul Fagului, Zloti, and Baimaclia) in relation to the environmental conditions of the common stand. In all the years of observation, the trees from Edineț had a rapid growth in diameter, thus demonstrating their ability to adapt through phenotypic plasticity. An uneven distribution of tree diameters was observed by provenance, indicating that dominant trees have an advantage in competition for light, slowing the growth of those with smaller diameters and causing a positive skew towards oaks with larger diameters. With the advancing age of the oaks, a tendency to decrease the values of the heritability coefficients for the trunk diameter was observed. At the age of 5 years, studies show a strong genetic control over the growth in diameter, a fact highlighted by the heritability coefficient value of 0,53 for this character. As the trees age, non-genetic factors such as environmental conditions become increasingly important in determining tree diameter growth. The comparative analysis of three statistical fitting models for data on pedunculate oak trees from different provenances, showed that the logarithmic model is the most appropriate to explain the relationship between their height and diameter.

Study on an Approximate Model for Predicting the Roll Force of an H-Beam Flange During Universal Finishing Rolling

Study on an Approximate Model for Predicting the Roll Force of an H-Beam Flange During Universal Finishing Rolling

Modeling multivariate positive‐valued time series using R‐INLA

AbstractIn this article, we describe fast Bayesian statistical analysis of vector positive‐valued time series, with application to interesting financial data streams. We discuss a flexible level correlated model (LCM) framework for building hierarchical models for vector positive‐valued time series. The LCM allows us to combine marginal gamma distributions for the positive‐valued component responses, while accounting for association among the components at a latent level. We introduce vector autoregression evolution of the latent states, deriving its precision matrix and enabling its estimation using integrated nested Laplace approximation (INLA) for fast approximate Bayesian modeling via the R‐INLA package, building custom functions to handle this setup. We use the proposed method to model interdependencies between intraday volatility measures from several stock indexes.