Numerical methods, simulations and applications

Nonlinearity of mathematical models arising in almost all scientific areas, from social sciences and economics to engineering and medicine implies that solutions in closed form do not exist and numerical approach is necessary. Simulation techniques are now playing the key role both in model development and numerical solution if uncertainty or noise is present, as is the case with large number of applications. Applicability of numerical methods and simulations is thus the main force driving the rich mathematical area with research efforts concentrated both on theoretical challenges and computational implementation. Our research efforts will be concentrated in two direction - numerical optimization and numerical methods for singularly perturbed problems. Within numerical methods for continuous optimization we will consider the following topics: Continuos optimization problems of large scale; Nonsmooth problems with nonsmoothness present in objective function and/or constraints; Continuos optimization problems in noisy environment; Variabl? sample size methods; Optimization models in algorithmic trading; Optimization problems in medicine - cluster analysis. The planned research within the second research direction includes one dimensional and two dimensional problems with one or more perturbation parameters. The objective will be to derive and analyze robust numerical methods on layer adapted meshes resulting in uniform convergence.