This paper presents an optimization method based on a particular polynomial lattice rule with interlaced factor two for estimating sensitivity indices in global sensitivity analysis, focusing on total, first-order and second-order Sobol indices.A comparison with one of the best available methods the Modified Sobol Sequence and component by component construction polynomial lattice rule have been done.Relative errors for key output quantities are analyzed and compared.Our results show that the proposed optimization method consistently outperforms other methods in accurately estimating both first-order and total-order sensitivity indices, especially for parameters with smaller effects.These findings highlight the strengths and limitations of each method, providing guidance for selecting appropriate stochastic sampling strategies in computational sensitivity analysis.

The focus of this study is on the optimal use of high performance computing in the area of environmental security (air pollution transport, in particular).Contemporary mathematical models of air pollution transport should include a fairly large set of chemical and photochemical reactions to be established as a reliable simulation tool.The investigations and the numerical results reported in this paper have been obtained by using a large-scale mathematical model called the Danish Eulerian Model (DEM).For optimization of some applications of the Danish Eulerian Model in various important scientific, social and economic areas, it is of great importance to simplify the model as much as possible, preserving the high reliability of its output results.A careful sensitivity analysis is needed in order to decide how to do such simplifications.On the other hand, it is important to analyze the influence of variations of the initial conditions, the boundary conditions, the rates of some chemical reactions, etc. on the model results in order to make right assumptions about the possible simplifications, which could be done.The sensitivity analysis version of the Danish Eulerian Model was created for these purposes.Its complexity is of higher order, a real challenge for the top performance supercomputers nowadays.The sensitivity analysis version of DEM (SA-DEM) has been implemented on the new Bulgarian petascale supercomputer DISCOVERER.It is a part of the European High Performance Computing Joint Undertaking (EuroHPC), which is building a network of 8 powerful supercomputers across the European Union (3 pre-exascale and 5 petascale).The results of some scalability experiments with SA-DEM on the new Bulgarian petascale supercomputer DISCOVERER are presented here.They are compared with similar experiments performed on the Mare Nostrum III supercomputer at Barcelona Supercomputing Centre -the most powerful supercomputer in Spain by that time, upgraded currently to the pre-exascale Mare Nostrum V, also part of the EuroHPC JU infrastructure.

This paper introduces a sophisticated multidimensional sensitivity analysis, incorporating cutting-edge stochastic methods for air pollution modeling.The study focuses on a large-scale long-distance transportation model of air pollutants, specifically the Unified Danish Eulerian Model (UNI-DEM).This mathematical model plays a pivotal role in understanding the detrimental impacts of heightened levels of air pollution.With this research, our intent is to employ it to tackle crucial questions related to environmental protection.We suggest advanced Monte Carlo and quasi-Monte Carlo methods, leveraging specific lattice and digital sequences to enhance the computational effectiveness of multi-dimensional numerical integration.Moreover, we further refine the existing stochastic methodologies for digital ecosystem modeling.The main aspect of our investigation is to analyze the sensitivity of the UNI-DEM model output to changes in the input emissions of human-induced pollutants and the rates of a number of chemical reactions.The developed algorithms are utilized to calculate global Sobol sensitivity measures for various input parameters.We also assess their influence on key air pollutant concentrations in different European cities, considering the diverse geographical locations.The overarching goal of this research is to broaden our understanding of the elements influencing air pollution and inform potent strategies to alleviate its negative impacts on the environment.The work is supported by the Project BG05M2OP001-1.001-0004UNITe,

The purpose of this article is to consider a special class of combinatorial problems, the so called Prouhet-Tarry-Escot problem, solution of which is realized by constructing finite sequences of ±1.For example, for fixed p ∈ N, is well known the existence of np ∈ N with the property: any set of np consecutive integers can be divided into 2 sets, with equal sums of its p thpowers.The considered property remains valid also for sets of finite arithmetic progressions of complex numbers.

In this work we investigate advanced stochastic methods for solving a specific multidimensional problem related to neural networks.Monte Carlo and quasi-Monte Carlo techniques have been developed over many years in a range of different fields, but have only recently been applied to the problems in neural networks.As well as providing a consistent framework for statistical pattern recognition, the stochastic approach offers a number of practical advantages including a solution to the problem for higher dimensions.For the first time multidimensional integrals up to 100 dimensions related to this area will be discussed in our numerical study.

AbstractÐRecently stochastic methods have become very important tool for high performance computing of very high dimensional problems in computational finance.The advantages and disadvantages of the different highly efficient stochastic methods for multidimensional integrals related to evaluation of European style options will be analyzed.Multidimensional integrals up to 100 dimensions related to European options will be computed with highly efficient optimized lattice rules.

An optimized Monte Carlo approach (OPTIMIZED MC) for a Fredholm integral equations of the second kind is presented and discussed in the present paper.Numerical examples and results are discussed and MC algorithms with various initial and transition probabilities are compared.

We study an optimized Adaptive Monte Carlo algorithm for the Wigner kernel -an important problem in quantum mechanics.We will compare the results with the basic adaptive approach and other stochastic approaches for computing the Wigner kernel represented by difficult multidimensional integrals in dimension d up to 12.The higher cases d > 12 will be considered for the first time.A comprehensive study and an analysis of the computational complexity of the optimized Adaptive MC algorithm under consideration has also been presented.

Stochastic techniques have been developed over many years in a range of different fields, but have only recently been applied to the problems in machine learning.A fundamental problem in this area is the accurate evaluation of multidimensional integrals.An introduction to the theory of the stochastic optimal generating vectors has been given.A new optimized lattice sequence with a special choice of the optimal generating vector has been applied to compute multidimensional integrals up to 30-dimensions.Clearly, the progress in the area of machine learning is closely related to the progress in reliable algorithms for multidimensional integration.

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 algorithm is optimized by choosing the appropriate values for the relaxation parameters which leads to dramatic reduction in time and lower relative errors for a given number of iterations.Numerical tests are performed for examples with matrices of different size and on a system coming from a finite element approximation of a problem describing a beam structure in constructive mechanics.