Construction of differentiable transformations

AbstractConstruction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method

Stochastic homogenization of elliptic equation and optimal control

In this thesis, we mainly study the numerical methods for stochastic homogenization of elliptic optimal control problem, where there is random variable involved in the constraint. We start with a simple one-dimensional optimal control problem and derive the effective equations of the original optimal control problem and the theory results regarding the convergence of solutions have been studied in \cite{kesavan}. Then an elliptic optimal control problem with coefficient having ergodicity is studied and convergence theorem is also given regarding the effective equations. Finally, another elliptic optimal control problem, where the normal product is replaced by the wick product, is discussed. An algorithm used to search the optimal solution is obtained. Numerical examples are given in each chapter