Sergei S. Kucherenko

Estimating the approximation error when fixing unessential factors in global sensitivity analysis

One of the major settings of global sensitivity analysis is that of fixing non-influential factors, in order to reduce the dimensionality of a model. However, this is often done without knowing the magnitude of the approximation error being produced. This paper presents a new theorem for the estimation of the average approximation error generated when fixing a group of non-influential factors. A simple function where analytical solutions are available is used to illustrate the theorem. The numerical estimation of small sensitivity indices is discussed.

Monte Carlo evaluation of derivative-based global sensitivity measures

Abstract A novel approach for evaluation of derivative-based global sensitivity measures (DGSM) is presented. It is compared with the Morris and the Sobol’ sensitivity indices methods. It is shown that there is a link between DGSM and Sobol’ sensitivity indices. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is many orders of magnitude lower than that for estimation of the Sobol’ sensitivity indices. It is also lower than that for the Morris method. Efficiencies of Monte Carlo (MC) and quasi-Monte Carlo (QMC) sampling methods for calculation of DGSM are compared. It is shown that the superiority of QMC over MC depends on the problems effective dimension, which can also be estimated using DGSM.

Estimation of global sensitivity indices for models with dependent variables

Abstract A novel approach for estimation variance-based sensitivity indices for models with dependent variables is presented. Both the first order and total sensitivity indices are derived as generalizations of Sobolʼ sensitivity indices. Formulas and Monte Carlo numerical estimates similar to Sobolʼ formulas are derived. A copula-based approach is proposed for sampling from arbitrary multivariate probability distributions. A good agreement between analytical and numerical values of the first order and total indices for considered test cases is obtained. The behavior of sensitivity indices depends on the relative predominance of interactions and correlations. The method is shown to be efficient and general.

Application of Deterministic Low-Discrepancy Sequences in Global Optimization

It has been recognized through theory and practice that uniformly distributed deterministic sequences provide more accurate results than purely random sequences. A quasi Monte Carlo (QMC) variant of a multi level single linkage (MLSL) algorithm for global optimization is compared with an original stochastic MLSL algorithm for a number of test problems of various complexities. An emphasis is made on high dimensional problems. Two different low-discrepancy sequences (LDS) are used and their efficiency is analysed. It is shown that application of LDS can significantly increase the efficiency of MLSL. The dependence of the sample size required for locating global minima on the number of variables is examined. It is found that higher confidence in the obtained solution and possibly a reduction in the computational time can be achieved by the increase of the total sample size N. N should also be increased as the dimensionality of problems grows. For high dimensional problems clustering methods become inefficient. For such problems a multistart method can be more computationally expedient.

A new derivative based importance criterion for groups of variables and its link with the global sensitivity indices

A new derivative based criterion τ y for groups of input variables is presented. It is shown that there is a link between global sensitivity indices and the new derivative based measure. It is proved that small values of derivative based measures imply small values of total sensitivity indices. However, for highly nonlinear functions the ranking of important variables using derivative based importance measures can be different from that based on the global sensitivity indices. The computational costs of evaluating global sensitivity indices and derivative based measures, are compared and some important tests are considered.

Comparison of deterministic and stochastic approaches to global optimization

In this paper, we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial Branch-and-Bound algorithm, whereas the second approach is a Quasi Monte Carlo (QMC) variant of a stochastic multi level single linkage (MLSL) algorithm. Both algorithms apply to problems in a very general form and are not dependent on problem structure. The test suite we chose is fairly extensive in scope, in that it includes constrained and unconstrained problems, continuous and mixed-integer problems. The conclusion of the tests is that in general the QMC variant of the MLSL algorithm is generally faster, although in some instances the Branch-and-Bound algorithm outperforms it.

On global sensitivity analysis of quasi-Monte Carlo algorithms

Different Quasi-Monte Carlo algorithms corresponding to the same Monte Carlo algorithm are considered. Even in the case when their constructive dimensions are equal and the same quasi-random points are used, the efficiencies of these algorithms may differ. Global sensitivity analysis provides an insight into this situation. As a model problem two well-known approximations of a Wiener integral are considered: the standard one and the Brownian bridge. The advantage of the Brownian bridge is confirmed.

Metamodelling with independent and dependent inputs

a b s t r a c t In the cases of computationally expensive models the metamodelling technique which maps inputs and outputs is a very useful and practical way of making computations tractable. A number of new techniques which improve the efficiency of the Random Sampling-High dimensional model representation (RS-HDMR) for models with independent and dependent input variables are presented. Two different metamodelling methods for models with dependent input variables are compared. Both techniques are based on a Quasi Monte Carlo variant of RS-HDMR. The first technique makes use of transformation of the dependent input vector into a Gaussian independent random vector and then applies the decomposition of the model using a tensored Hermite polynomial basis. The second approach uses a direct decomposition of the model function into a basis which consists of the marginal distributions of input components and their joint distribution. For both methods the copula formalism is used. Numerical tests prove that the developed methods are robust and efficient.

Calculating First-order Sensitivity Measures: A Benchmark of Some Recent Methodologies

Abstract This work compares three different global sensitivity analysis techniques, namely the state-dependent parameter (SDP) modelling, the random balance designs, and the improved formulas of the Sobol’ sensitivity indices. These techniques are not yet commonly known in the literature. Strengths and weaknesses of each technique in terms of efficiency and computational cost are highlighted, thus enabling the user to choose the more suitable method depending on the computational model analysed. Two test functions proposed in the literature are considered. Computational costs and convergence rates for each function are compared and discussed.

A flexible and generic approach to dynamic modelling of supply chains

In this paper, we present a new modelling approach for realistic supply chain simulation. The model provides an experimental environment for informed comparison between different supply chain policies. A basic simulation model for a generic node, from which a supply chain network can be built, has been developed using an object-oriented approach. This generic model allows the incorporation of the information and physical systems and decision-making policies used by each node. The object-oriented approach gives the flexibility in specifying the supply chain configuration and operation decisions, and policies. Stochastic simulations are achieved by applying Latin Supercube Sampling to the uncertain variables in descending order of importance, which reduces the number of simulations required. We also present a case study to show that the model is applicable to a real-life situation for dynamic stochastic studies.