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A number of numerical experiments with a large-scale air pollution model (Unified Danish Eulerian Model, UNI-DEM) have been carried out to compute Sobol sensitivity measures. Sensitivity analysis of ozone concentrations according to variations of rates of chemical reactions is developed. The sensitivity study has been done for the areas of four European cities (Genova, Milan, Manchester, and Edinburgh) with different geographical locations. Sensitivity analysis of the output of UNI-DEM according to emission levels is also provided. The influence of emission levels on the concentrations of four important air pollutants (ammonia, ozone, ammonium sulphate and ammonium nitrate) over three European cities (Milan, Manchester, and Edinburgh) is considered. Variance-based methods are most often used for providing sensitivity analysis. Sobol variance-based approach for global sensitivity analysis has been applied to compute the corresponding sensitivity measures. Advanced stochastic methods based on modified Sobol sequences are used for computing the sensitivity measures. The numerical experiments show that the stochastic algorithms for the multidimensional integrals under consideration are one of the best stochastic techniques for computing the small in value sensitivity indices. [ABSTRACT FROM AUTHOR]

This study delves into the examination of Diophantine systems, which are sets of polynomial equations with rational coefficients involving multiple variables. Specifically, the focus is on understanding the concept of local and global triviality within these systems. A Diophantine system is considered "locally trivial" if it possesses a non-zero solution in p-adic rational numbers for all prime numbers, including infinity. On the other hand, a system is deemed "globally trivial" if it has a non-zero rational solution in Q. While local triviality is a prerequisite for global triviality, it is important to note that the converse is not universally valid, as exemplified by Selmer's counterexample involving the equation 3x3+4y3+5z3=0. The main inquiry of this research revolves around investigating the conditions under which Diophantine systems exhibit equivalent notions of local and global triviality, thereby satisfying the "local-to-global principle". By analyzing various classes of Diophantine systems, the study sheds light on the existence of symmetric systems that arise in problems of Euclidean geometry, wherein the lengths of geometric elements are represented by integers. This paper is a continuous work of the [7]. [ABSTRACT FROM AUTHOR]

In this paper for the first time a Hammersley and van der Corput sequences comparison for multidimensional sensitivity analysis has been done. Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is very significant for improving the reliability of these models. By definition, sensitivity analysis is a procedure for studying how sensitive are the output results of large-scale mathematical models to some uncertainties of the input data. The algorithms for multidimensional sensitivity analysis are applied to compute global Sobol sensitivity measures corresponding to the influence of several input parameters on the concentrations of important air pollutants. A deep experimental investigation is done to demonstrate the pros and cons of both approaches in the case study of the considered model. [ABSTRACT FROM AUTHOR]

A new Monte Carlo algorithm for solving systems of Linear Algebraic (LA) equations is presented and studied. The algorithm is based on the "Walk on Equations" Monte Carlo method recently developed by Ivan Dimov, Sylvain Maire and Jean Michel Sellier [4]. The advantageous of the new algorithm versus the older one is demonstrated especially for diagonally dominant matrix with dimension up to 1000. [ABSTRACT FROM AUTHOR]

Recently Monte Carlo and quasi-Monte Carlo approaches have become a very attractive and necessary computational tools in finance. The field of computational finance is becoming more complicated with increasing number of applications. The pricing of options is a very important in financial markets today and especially difficult when the dimension of the problem goes higher. Monte Carlo and quasi Monte Carlo methods are appropriate for solving multidimensional problems, since their computational complexity increases polynomially, but not exponentially with the dimensionality. A comprehensive experimental study based on scrambling of the Sobol sequence is applied for the first time to evaluate European style options. The Sobol scrambling method is not only one of the best available algorithms for high dimensional integrals but also one of the few possible methods, because in this work we show that the deterministic algorithms need a huge amount of time for the evaluation of the multidimensional integral, as it was discussed in this paper. The numerical tests show that the method is efficient for multidimensional integration and especially for computing multidimensional integrals of a very high dimension. [ABSTRACT FROM AUTHOR]

In this paper a study the sensitivity of concentration variations of some of the most dangerous air pollutants with respect to the anthropogenic emissions levels and with respect to some chemical reactions rates has been performed. Quasi-Monte Carlo algorithms based on Sobol, Faure, and different lattice rules are used in our sensitivity study of the Unified Danish Eulerian model. Different parts of the large amount of output data, produced by Unified Danish Eulerian model, were used in various practical applications, where the reliability of this data should be properly estimated. Another reason to choose this model as a case study here is its sophisticated chemical scheme, where all relevant chemical processes in the atmosphere are accurately represented. The numerical tests show that the stochastic algorithms under consideration, and especially the optimal lattice rule, are efficient for the multidimensional integrals under consideration and especially for computing small by value sensitivity indices. Clearly, the progress in the area of air pollution modeling, is closely connected with the progress in reliable algorithms for multidimensional integration. [ABSTRACT FROM AUTHOR]

The present study is based on The Unified Danish Eulerian Model (UNI-DEM) as one of the most advanced large- scale mathematical models that describes adequately all physical and chemical processes. One of the most attractive features of UNI-DEM is its advanced chemical scheme the Condensed CBM IV, which consider a large number of chemical species and numerous reactions between them, of which the ozone is one of the most important pollutants for its central role in many practical applications of the results. The calculations are done in a large spatial domain, which covers completely the European region and the Mediterranean, for certain time period. A comprehensive experimental study of Monte Carlo algorithm based on modifications of the Latin Hypercube Sampling Edge and Random Algorithms for multidimensional numerical integration has been done. Samplings with different seeds has been analyzed. This comparison has been made for the first time for sensitivity analysis of UNI-DEM. The algorithms have been successfully applied to compute global Sobol sensitivity measures corresponding to the six chemical reactions rates and four different groups of pollutants. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyse different stochastic methods for solving a class of integral equations, namely the second kind Fredholm integral equations. We study and compare different possible approaches to compute linear functionals of the integral under consideration. Error balancing of both stochastic and systematic errors has been discussed and applied during the numerical implementation of the algorithms. An almost optimal Monte Carlo algorithm for integral equations in a combination with the idea of balancing of both systematic and stochastic errors is analysed. Conclusions about the applicability and efficiency of the algorithms have been drawn. Meaningful numerical examples and experiments with experimental and theoretical relative errors are presented. It is shown that the balancing of errors reduce the computational complexity if the error is fixed. [ABSTRACT FROM AUTHOR]

Sensitivity studies are nowadays applied to some of the most complicated mathematical models from various intensively developing areas of application. Such a sophisticated model in the area of air pollution modeling is the Danish Eulerian Model, a powerful large scale air pollution model with a long development history. Over the years it was used successfully in different long-term environmental studies for the European region. In this talk we discuss some performance-critical achievements in the parallel optimization and a systematic approach for sensitivity analysis of the latest version of the Danish Eulerian Model, UNI-DEM. A comprehensive experimental study of Monte Carlo algorithms based on Latin Hypercube Sampling and Adaptive approach for multidimensional numerical integration has been done. The algorithms have been successfully applied to compute global Sobol sensitivity measures, corresponding to the influence of several input parameters on the concentrations of some of the air pollutants of highest importance. The numerical tests show that the Monte Carlo algorithms under consideration are efficient for the multidimensional integrals in this study. [ABSTRACT FROM AUTHOR]

Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is very significant for improving the reliability of these models. Several efficient quasi-Monte Carlo algorithms – the van der Corput sequence and lattice rules based on different generating vectors have been used in our sensitivity studies of the model output results for some air pollutants with respect to the emission levels and some chemical reactions rates. The algorithms have been successfully applied to compute global Sobol sensitivity measures corresponding to the influence of several input parameters (six chemical reactions rates and four different groups of pollutants) on the concentrations of important air pollutants. The numerical tests show that the van der Corput sequences under consideration are efficient for the multidimensional integrals under consideration and especially for computing small by value sensitivity indices. [ABSTRACT FROM AUTHOR]

Air pollution and meteorological models are examples of mathematical models with a lot of natural uncertainties in their input data sets and parameters. Sensitivity analysis is a powerful tool for studying and improving the reliability of such models. In this paper we present the results of a global sensitivity study of the Unified Danish Eulerian Model (UNI-DEM). One of the most important features of UNI-DEM is its advanced chemical scheme the Condensed CBM IV, which considers a large number of chemicals, and various reactions between them, of which the ozone is the most important pollutant because it is used in many practical applications. The stochastic methods based on Faure and Sobol sequences are used for computing the sensitivity measures. The numerical experiments show that the stochastic algorithms for the multidimensional integrals under consideration are efficient methods for computing the small value sensitivity indices. [ABSTRACT FROM AUTHOR]

In the present paper we consider European style options with an exponential payoff function. The problem is transformed to evaluation of multidimensional integrals of the exponential function over the unit cube where the values of the parameters involved in the formula depend the values of the European options. We compare the performance of quasi-Monte Carlo methods based on lattice rules for multiple integrals up to 30 dimensions. The performance of a lattice rule depends on the choice of the generator vectors. When the integrand is sufficiently regular the lattice rules outperform not only the standard Monte Carlo methods, but also other types of methods using low discrepancy sequences. We consider "rank 1" rules whose lattices have a a single generator vector. The advantages and disadvantages of the different quasi-Monte Carlo methods for multidimensional integrals related to evaluation of European options are studied in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem for the reconstruction of the right-hand side of parabolic systems of equations with overdeterminations, given by point measurements. The numerical algorithm is based on a special decomposition of the numerical solution first proposed for the heat equation in [9, 11]. Numerical experiments show the capacity of the numerical algorithm for parabolic systems. [ABSTRACT FROM AUTHOR]

Reliability of large-scale mathematical models is an important issue when such models are used to support decision makers. Sensitivity analysis of model outputs to variation or natural uncertainties of model inputs is crucial for improving the reliability of mathematical models. A comprehensive experimental study of Monte Carlo algorithms based on Sobol sequences for multidimensional numerical integration has been done. A comparison with Latin hypercube sampling and a particular quasi-Monte Carlo lattice rule based on generalized Fibonacci numbers has been presented. The algorithms have been successfully applied to compute global Sobol sensitivity measures corresponding to the influence of several input parameters (six chemical reactions rates and four different groups of pollutants) on the concentrations of important air pollutants. The concentration values have been generated by the Unified Danish Eulerian Model. The sensitivity study has been done for the areas of several European cities with different geographical locations. The numerical tests show that the stochastic algorithms under consideration are efficient for multidimensional integration and especially for computing small by value sensitivity indices. It is a crucial element since even small indices may be important to be estimated in order to achieve a more accurate distribution of inputs influence and a more reliable interpretation of the mathematical model results. [ABSTRACT FROM AUTHOR]

We propose sixth-order approximations for semilinear parabolic systems that are based on fourth-order compact difference schemes combined with Richardson extrapolation. Within this paper, Richardson extrapolation is applied directly to the solution procedures used within existing numerical algorithms for realization of compact fourth-order difference schemes. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed finite difference discretization strategy. [ABSTRACT FROM AUTHOR]

In this paper, we illustrate examples of highly accurate Monte Carlo and quasi-Monte Carlo methods for multiple integrals related to the evaluation of European style options. The idea is that the value of the option is formulated in terms of the expectation of some random variable; then the average of independent samples of this random variable is used to estimate the value of the option. First we obtain an integral representation for the value of the option using the risk neutral valuation formula. Then with an appropriations change of the constants we obtain a multidimensional integral over the unit hypercube of the corresponding dimensionality. Then we compare a specific type of lattice rules over one of the best low discrepancy sequence of Sobol for numerical integration. Quasi-Monte Carlo methods are compared with Adaptive and Crude Monte Carlo techniques for solving the problem. The four approaches are completely different thus it is a question of interest to know which one of them outperforms the other for evaluation multidimensional integrals in finance. Some of the advantages and disadvantages of the developed algorithms are discussed. [ABSTRACT FROM AUTHOR]