A Geometric Optimization Algorithm With Applications To Computing Configuration Of Liquid Crystals

A large number of problems, such as computing configuration of liquid crystals, color image denoising etc. can be described as infinite-dimensional constrained variational problems. It is of theoretical and practical importance to find the numeral solution to those problems.In this thesis, for computing configuration of liquid crystals, the finite element method is used to discretize the problem involved, and a geometric optimization algorithm is borrowed to solve the finite-dimensional problem on some product manifold.Generally, a geometric optimization algorithm is proposed to solve a minimization problem on the unit sphere S N?1 in the Euclidean space R N. The continuous problem is discretized by finite elements and it is shown that any limiting point of a sequence of finite element solutions as the meshsize approaches zero is a solution of the continuous problem. The discretized problem is solved by the steepest descent algorithm with inexact Armijo's inexact line searches on some product manifold, the convergence of which is offered next. Programming the process above, then the geometric optimization algorithm is formed. The method is very easy to implement, and the iterative sequence always satisfies the constraint condition at the grid points as well as guarantees the decreasing of the objective function involved. A series of numerical results are included to show the efficiency of the method.