Some Approximation Problems in Statistics and Probability
We review the results about the accuracy of approximations for distributions of functionals of sums of independent random elements with values in a Hilbert space. Mainly we consider recent results for quadratic and almost quadratic forms motivated by asymptotic problems in mathematical statistics. Some of the results are optimal and could not be further improved without additional conditions.
Particle transport in a porous medium occurs in environmental, chemical and industrial technologies. The transport of suspended concrete grains in a liquid grout through porous soil is used in construction industry to strengthen foundations. When particles are transported by a fluid flow in a porous medium, some particles are retained in the pores and form a deposit. The aim of the work is the construction and study of a one-dimensional mathematical model of particle transport and retention in the porous medium, taking into account the simultaneous action of several particle capture mechanisms. The model consists of mass balance equation and the kinetic equation of deposit growth. The deposit growth rate is proportional to the filtration function, which depends on the retained particles concentration, and the nonlinear concentration function, which depends on the concentration of suspended particles. The use of a new parameter, depending on the distance to the porous medium inlet allows to construct a global asymptotic solution in the entire area of the mathematical model. An explicit analytical solution is obtained as a series in two small parameters. The global asymptotics is close to the numerical solution at all points of the porous medium at any time.
This volume contains the extended version of selected talks given at the international research workshop "Coping with Complexity: Model Reduction and Data Analysis", Ambleside, UK, August 31 – September 4, 2009. The book is deliberately broad in scope and aims at promoting new ideas and methodological perspectives. The topics of the chapters range from theoretical analysis of complex and multiscale mathematical models to applications in e.g., fluid dynamics and chemical kinetics.