Limited Number Of Model Evaluations Research Articles

Variational Autoencoders for Dimensionality Reduction of Automotive Vibroacoustic Models

<div class="section abstract"><div class="htmlview paragraph">In order to predict reality as accurately as possible leads to the fact that numerical models in automotive vibroacoustic problems become increasingly high dimensional. This makes applications with a large number of model evaluations, e.g. optimization tasks or uncertainty quantification hard to solve, as they become computationally very expensive. Engineers are thus faced with the challenge of making decisions based on a limited number of model evaluations, which increases the need for data-efficient methods and reduced order models.</div><div class="htmlview paragraph">In this contribution, variational autoencoders (VAEs) are used to reduce the dimensionality of the vibroacoustic model of a vehicle body and to find a low-dimensional latent representation of the system. Autoencoders are neural networks consisting of an encoder and a decoder network and they are trained in order to learn the identity mapping between a reduced approximation and the initial input while enforcing a dimensionality reduction in the latent space. This allows decoding the hidden data generating structure behind the data and enables an interpretation based on the latent variables, which is extremely valuable in the engineering design process. The performance of the VAE approach is compared to a conventional principal component analysis. Finally, the trained VAE is used as a deep generative model and it is investigated to which extent the pre-trained decoder network can be used to generate new artificial realizations at low costs. These artificially generated samples can then be used to enhance the training data basis for other neural network approaches or data-driven surrogate models.</div></div>

Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

A novel sparse polynomial chaos expansion (PCE) is proposed in this paper for global sensitivity analysis (GSA). The proposed model combines variational Bayesian (VB) inference and automatic relevance determination (ARD) with the PCE model. The VB inference is utilized to compute the PCE coefficients. The PCE coefficients are obtained through a simple optimization procedure in the VB framework. On the other hand, the curse of dimensionality issue of PCE model is tackled using the ARD which reduces the number of polynomial bases significantly. The applicability of the proposed approach is illustrated by performing GSA on five numerical examples. The results show that the proposed approach outperforms a similar state-of-art surrogate model in obtaining an accurate sensitivity index using limited number of model evaluations. For all the examples, the PCE models are highly sparse, which require very few polynomial bases to assess an accurate sensitivity index.

Derivative-based global sensitivity measure using radial basis function

Global sensitivity analysis (GSA) is always used to measure the contribution of input variables on the variation of model output. In the fields of GSA, derivative-based global sensitivity measure (DGSM) has gained much attention because of its high efficiency in the identification of insignificant input variables. The commonly used approach to estimate DGSM is the Morris method, which requires a large number of model evaluations to compute the gradient of model output. Therefore, it leads to a great challenge in the computational costs, particularly for expensive problems. To alleviate the intensive computation, an estimator based on Gaussian radial basis function (RBF) metamodel is proposed to compute DGSM. With the aid of RBF, the DGSM can be formulated in an explicit expression of one-dimensional Gaussian integrals. For the case of uniformly and normally distributed variables, the analytical expressions of the integral can be derived while for other cases with general distribution, the numerical approach with the quasi-Monte Carlo method is developed to approximate the integrals. The performance of the proposed method is validated by five numerical examples, which illustrate the proposed RBF-based estimator has the capability to yield desirable accuracy and convergence with a limited number of model evaluations.

Value of Information Analysis via Active Learning and Knowledge Sharing in Error-Controlled Adaptive Kriging

Large uncertainties in many phenomena have challenged decision making. Collecting additional information to better characterize reducible uncertainties is among decision alternatives. Value of information (VoI) analysis is a mathematical decision framework that quantifies expected potential benefits of new data and assists with optimal allocation of resources for information collection. However, analysis of VoI is computational very costly because of the underlying Bayesian inference especially for equality-type information. This paper proposes the first surrogate-based framework for VoI analysis. Instead of modeling the limit state functions describing events of interest for decision making, which is commonly pursued in surrogate model-based reliability methods, the proposed framework models system responses. This approach affords sharing equality-type information from observations among surrogate models to update likelihoods of multiple events of interest. Moreover, two knowledge sharing schemes called model and training points sharing are proposed to most effectively take advantage of the knowledge offered by costly model evaluations. Both schemes are integrated with an error rate-based adaptive training approach to efficiently generate accurate Kriging surrogate models. The proposed VoI analysis framework is applied for an optimal decision-making problem involving load testing of a truss bridge. While state-of-the-art methods based on importance sampling and adaptive Kriging Monte Carlo simulation are unable to solve this problem, the proposed method is shown to offer accurate and robust estimates of VoI with a limited number of model evaluations. Therefore, the proposed method facilitates the application of VoI for complex decision problems.

Efficient uncertainty quantification method applied to structural fire engineering computations

Probabilistic Risk Assessment methodologies are gaining traction in fire engineering practice as a (necessary) means to demonstrate adequate safety for uncommon buildings. This induces a need to apply uncertainty quantification to structural fire engineering problems. Yet, the combination of probabilistic methods and advanced numerical fire engineering tools has been limited due to the absence of a methodology which is both efficient (i.e. requires a limited number of model evaluations) and unbiased (i.e. without prior assumptions regarding the output distribution type). In this paper, the recently proposed MaxEnt method is combined with the dedicated structural fire engineering software SAFIR to evaluate the ability of the method to achieve efficient, unbiased assessments of structural fire performance. The case studies include the probability density function (PDF) of (i) the standard fire resistance of a composite column; (ii) the load bearing capacity of a composite floorplate exhibiting tensile membrane action, after 90 min of standard fire exposure; (iii) the load bearing capacity of the same composite floorplate, considering a parametric fire exposure including cooling phase; and (iv) the maximum temperature reached in a protected steel element under realistic fire exposure. In the first application, the MaxEnt PDF correctly and efficiently captures the distribution obtained using Monte Carlo Simulations. The floorplate example under parametric fire exposure shows the true strength of the MaxEnt method as an unbiased assessment, as different failure modes are observed in the cooling phase resulting in an irregular shape of the load-bearing capacity PDF. For this case, reliance on a traditional assumption of lognormality for the capacity would result in an overestimation of the capacity at lower quantiles. The last case study yields a bi-modal output due to the physics-based duality between localized (traveling) and post-flashover fire development. While the MaxEnt captures this bi-modality, it does not accurately reproduce the distribution obtained by Monte Carlo Simulation. Limitations of the MaxEnt method and needs for further research are discussed at the end of the paper.

Uncertainty quantification and global sensitivity analysis for economic models

We present a global sensitivity analysis that quantifies the impact of parameter uncertainty on model outcomes. Specifically, we propose variance‐decomposition‐based Sobol' indices to establish an importance ranking of parameters and univariate effects to determine the direction of their impact. We employ the state‐of‐the‐art approach of constructing a polynomial chaos expansion of the model, from which Sobol' indices and univariate effects are then obtained analytically, using only a limited number of model evaluations. We apply this analysis to several quantities of interest of a standard real‐business‐cycle model and compare it to traditional local sensitivity analysis approaches. The results show that local sensitivity analysis can be very misleading, whereas the proposed method accurately and efficiently ranks all parameters according to importance, identifying interactions and nonlinearities. Computational techniques uncertainty quantification sensitivity analysis polynomial chaos expansion C60 C63

Global sensitivity analysis using support vector regression

• An orthogonal polynomial kernel function is proposed for SVR model. • An iteration scheme is introduced to optimize the proposed kernel function. • Sobol’ indices are obtained by post-processing the coefficients of the SVR model. • The proposed method is efficient for complex problems. Global sensitivity analysis (GSA) plays an important role in exploring the respective effects of input variables on response variables. In this paper, a new kernel function derived from orthogonal polynomials is proposed for support vector regression (SVR). Based on this new kernel function, the Sobol’ global sensitivity indices can be computed analytically by the coefficients of the surrogate model built by SVR. In order to improve the performance of the SVR model, a kernel function iteration scheme is introduced further. Due to the excellent generalization performance and structural risk minimization principle, the SVR possesses the advantages of solving non-linear prediction problems with small samples. Thus, the proposed method is capable of computing the Sobol’ indices with a relatively limited number of model evaluations. The proposed method is examined by several examples, and the sensitivity analysis results are compared with the sparse polynomial chaos expansion (PCE), high dimensional model representation (HDMR) and Gaussian radial basis (RBF) SVR model. The examined examples show that the proposed method is an efficient approach for GSA of complex models.

An Unbiased Method for Probabilistic Fire Safety Engineering, Requiring a Limited Number of Model Evaluations

The rise of Performance Based Design methodologies for fire safety engineering has increased the interest of the fire safety community in the concepts of risk and reliability. Practical applications have however been severely hampered by the lack of an efficient unbiased calculation methodology. This is because on the one hand, the distribution types of model output variables in fire safety engineering are not known and traditional distribution types as for example the normal and lognormal distribution may result in unsafe approximations. Therefore unbiased methods must be applied which make no (implicit) assumptions on the PDF type. Traditionally these unbiased methods are based on Monte Carlo simulations. On the other hand, Monte Carlo simulations require a large number of model evaluations and are therefore too computationally expensive when large and nonlinear calculation models are applied, as is common in fire safety engineering. The methodology presented in this paper avoids this deadlock by making an unbiased estimate of the PDF based on only a very limited number of model evaluations. The methodology is known as the Maximum Entropy Multiplicative Dimensional Reduction Method (ME-MDRM) and results in a mathematical formula for the probability density function (PDF) describing the uncertain output variable. The method can be applied with existing models and calculation tools and allows for a parallelization of model evaluations. The example applications given in the paper stem from the field of structural fire safety and illustrate the excellent performance of the method for probabilistic structural fire safety engineering. The ME-MDRM can however be considered applicable to other types of engineering models as well.

Uncertainty propagation using infinite mixture of Gaussian processes and variational Bayesian inference

Uncertainty propagation in flow through porous media problems is a challenging problem. This is due to the high-dimensionality of the random property fields, e.g. permeability and porosity, as well as the computational complexity of the models that are involved. The usual approach is to construct a surrogate response surface and then use it instead of the expensive model to carry out the uncertainty propagation task. However, the construction of the surrogate surface is hampered by various aspects such as the limited number of model evaluations that one can afford, the curse of dimensionality, multi-variate responses with non-trivial correlations, potential localized features of the response and/or discontinuities. In this work, we extend upon the concept of the Multi-output Gaussian Process (MGP) to effectively deal with all of these difficulties simultaneously. This non-trivial extension involves an infinite mixture of MGP's that is trained using variational Bayesian inference. Prior to observing any data, a Dirichlet process is used to generate the components of the MGP mixture. The Bayesian nature of the model allows for the quantification of the uncertainties due to the limited number of simulations, i.e., we can derive error bars for the statistics of interest. The automatic detection of the mixture components by the variational inference algorithm is able to capture discontinuities and localized features without adhering to ad hoc constructions. Finally, correlations between the components of multi-variate responses are captured by the underlying MGP model in a natural way.

Sobol’ sensitivity analysis of a complex environmental model

Complex environmental models are controlled by a high number of parameters. Accurately estimating the values of all these parameters is almost impossible. Sensitivity analysis (SA) results enable the selection of the parameters to include in a calibration procedure, but can also assist in the identification of the model processes. Additionally, a sensitivity analysis can yield crucial information on the use and meaning of the model parameters. This paper presents a Sobol’ sensitivity analysis for flow simulations by a SWAT model of the river Kleine Nete, with the objective to assess the first order, second order and total sensitivity effects. Confidence intervals for the resulting sensitivity indices are inferred by applying bootstrapping. The results indicate that the curve number value (CN2) is the most important parameter of the model and that no more than 9 parameters (out of 26) are needed to have an adequate representation of the model variability. The convergence of the parameter ranking for total sensitivity effects is relatively fast, which is promising for factor fixing purposes. It is also shown that the Sobol’ sensitivity analysis enhances the understanding of the model, by e.g. pointing out 3 significant pairwise interactions. In general, it can be concluded that the Sobol’ sensitivity analysis can be successfully applied for factor fixing and factor prioritization with respect to the input parameters of a SWAT model, even with a limited number of model evaluations. The analysis also supports the identification of model processes, parameter values and parameter interaction effects.

Probabilistic uncertainty analysis of an FRF of a structure using a Gaussian process emulator

This paper introduces methods for probabilistic uncertainty analysis of a frequency response function (FRF) of a structure obtained via a finite element (FE) model. The methods are applicable to computationally expensive FE models, making use of a Bayesian metamodel known as an emulator. The emulator produces fast predictions of the FE model output, but also accounts for the additional uncertainty induced by only having a limited number of model evaluations. Two approaches to the probabilistic uncertainty analysis of FRFs are developed. The first considers the uncertainty in the response at discrete frequencies, giving pointwise uncertainty intervals. The second considers the uncertainty in an entire FRF across a frequency range, giving an uncertainty envelope function. The methods are demonstrated and compared to alternative approaches in a practical case study.

Extension of Latin hypercube samples with correlated variables

A procedure for extending the size of a Latin hypercube sample (LHS) with rank correlated variables is described and illustrated. The extension procedure starts with an LHS of size m and associated rank correlation matrix C and constructs a new LHS of size 2 m that contains the elements of the original LHS and has a rank correlation matrix that is close to the original rank correlation matrix C. The procedure is intended for use in conjunction with uncertainty and sensitivity analysis of computationally demanding models in which it is important to make efficient use of a necessarily limited number of model evaluations.

Novel Cellular Automata Approach to Optimal Water Distribution Network Design

This paper proposes a novel heuristic-based and cellular automata-inspired approach to the optimal design of water distribution networks. The design of water distribution networks is of central importance to the water industry, but many networks cannot be optimally designed by traditional techniques due to their complexity. Genetic algorithms have become a state-of-the-art technique for this purpose but are hampered by the fact that they are population based and require a large number of model evaluations to achieve good solutions. The proposed approach uses a parallel, localist, heuristic-based algorithm to optimally design water distribution networks requiring only a limited number of model evaluations. The algorithm is applied to a well-known simple test network and two real water distribution systems in the U.K. The results indicate that the proposed cellular approach is a viable alternative to genetic algorithm approaches while using only a fraction of the computational time required by its evolutionary counterpart.

Application of the Taguchi method to the design of a robot sensor

SummaryThis paper considers the use of the Taguchi method as a means to identify the optimum design of a robot sensor. The sensor is modelled by a complex set of equations which makes optimisation using traditional calculusbased methods difficult. The Taguchi method is employed as a systematic method to understand the performance of the sensor whilst using a limited number of model evaluations. The Taguchi method is based on the experimental design technique and the results of using a full factorial design, 1/2 and 1/4 fractional factorial designs are compared. The advantage of using the Taguchi method over traditional experimental techniques is also discussed.