Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. First, we consider a linear constraint. The existence of optimal solutions for the optimization problem is shown as is its convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as is an eminently parallelizable gradient method for solving the optimality system. The linear constraint minimization problem is also recast as a linear least squares problem and is solved by a conjugate gradient method. The domain decomposition method can be extended to nonlinear problems such as the Navier-Stokes equations. This results from the fact that the objective functional for the minimization problem involves the jump in dependent variables across the interfaces between subdomains. Thus, the method does not require that the partial differential equations themselves be derivable through an extremal problem. An optimality system is derived by applying a Lagrange multiplier rule to a constrained optimization problem. Error estimates for finite element approximations are presented as is a gradient method to solve the optimality system. We also use a Gauss-Newton method to solve the minimization problem with the nonlinear constraint. / Ph. D.

The Sherman Morrison iteration method is developed to solve regularized least squares problems. Notions of pivoting and splitting are deliberated on to make the method more robust. The Sherman Morrison iteration method is shown to be effective when dealing with an extremely underdetermined least squares problem. The performance of the Sherman Morrison iteration is compared to classic direct methods, as well as iterative methods, in a number of experiments. Specific Matlab implementation of the Sherman Morrison iteration is discussed, with Matlab codes for the method available in the appendix. / Master of Science

The hierarchical (<i>H-</i>) matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Many operations from linear algebra like matrix-matrix and matrix-vector products, matrix inversion and LU decomposition can be implemented efficiently using the <i>H</i>-matrix format. Due to its importance in solving many problems in numerical linear algebra like least-squares problems, it is also desirable to have an efficient QR decomposition of <i>H</i>-matrices. In the past, two different approaches for this task have been suggested. We will review the resulting methods and suggest a new algorithm to compute the QR decomposition of an <i>H</i>-matrix. Like other <i>H</i>-arithmetic operations the <i>H</i>QR decomposition is of linear-polylogarithmic complexity. We will compare our new algorithm with the older ones by using two series of test examples and discuss benefits and drawbacks of the new approach.

The hierarchical (<i>H-</i>) matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Many operations from linear algebra like matrix-matrix and matrix-vector products, matrix inversion and LU decomposition can be implemented efficiently using the <i>H</i>-matrix format. Due to its importance in solving many problems in numerical linear algebra like least-squares problems, it is also desirable to have an efficient QR decomposition of <i>H</i>-matrices. In the past, two different approaches for this task have been suggested. We will review the resulting methods and suggest a new algorithm to compute the QR decomposition of an <i>H</i>-matrix. Like other <i>H</i>-arithmetic operations the <i>H</i>QR decomposition is of linear-polylogarithmic complexity. We will compare our new algorithm with the older ones by using two series of test examples and discuss benefits and drawbacks of the new approach.

Este trabalho aborda o problema de controle ótimo por modos deslizantes via função penalidade para sistemas de tempo discreto. Para resolver este problema será desenvolvido uma estrutura matricial alternativa baseada no problema de mínimos quadrados ponderados e funções penalidade. A partir desta nova formulação é possível obter a lei de controle ótimo por modos deslizantes, as equações de Riccati e a matriz do ganho de realimentação através desta estrutura matricial alternativa. A motivação para propormos essa nova abordagem é mostrar que é possível obter uma solução alternativa para o problema clássico de controle ótimo por modos deslizantes. / This work introduces a penalty function approach to deal with the optimal sliding mode control problem for discrete-time systems. To solve this problem an alternative array structure based on the problem of weighted least squares penalty function will be developed. Using this alternative matrix structure, the optimal sliding mode control law of, the matrix Riccati equations and feedback gain were obtained. The motivation of this new approach is to show that it is possible to obtain an alternative solution to the classic problem of optimal sliding mode control.

Este trabalho aborda o problema de controle ótimo por modos deslizantes via função penalidade para sistemas de tempo discreto. Para resolver este problema será desenvolvido uma estrutura matricial alternativa baseada no problema de mínimos quadrados ponderados e funções penalidade. A partir desta nova formulação é possível obter a lei de controle ótimo por modos deslizantes, as equações de Riccati e a matriz do ganho de realimentação através desta estrutura matricial alternativa. A motivação para propormos essa nova abordagem é mostrar que é possível obter uma solução alternativa para o problema clássico de controle ótimo por modos deslizantes. / This work introduces a penalty function approach to deal with the optimal sliding mode control problem for discrete-time systems. To solve this problem an alternative array structure based on the problem of weighted least squares penalty function will be developed. Using this alternative matrix structure, the optimal sliding mode control law of, the matrix Riccati equations and feedback gain were obtained. The motivation of this new approach is to show that it is possible to obtain an alternative solution to the classic problem of optimal sliding mode control.