In this thesis, we study some problems on the Navier-Stokes equations and Boltzmann- Enskog equation which are in the active research area of applied mathematics. The system of Navier-Stokes equations is a typical example of the conservation laws. In the first part of the thesis, we study the global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity.
This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied.In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements.Papers II, III, IV, and V are [...]
Stochastic approximation is one of the oldest approaches for solving stochastic optimization problems. In the first part of the dissertation, we study the convergence and asymptotic normality of a generalized form of stochastic approximation algorithm with deterministic perturbation sequences. Both one-simulation and two-simulation methods are considered. Assuming a special structure on the deterministic sequence, we [...]