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.

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.

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.