The Iterative Method For The Special Solution Of Some Nonlinear Problems

In the modern scientific computing and engineering, people often encounter many problems which have several solutions. For problems of this kind, what people are interested in are usually some special solutions. The problems we investigate in this paper belong to this kind of problems. For the nonsymmetric algebraic Riccati equation arising in nuclear physics, what people concern is its minimal positive solution. For the problems of the optimal shape design and the shape reconstruction problems, the goal is to find the optimal shape. How to obtain the solutions in which people are interested in an effective way, has attracted the attentions of many researchers.For the solutions of some nonlinear equations, there are many mature meth-ods such as Newton method, Euler-family and Halley-family. For the same prob-lem, due to the complexity of the computation of Jacobian matrix and the value of the function, the merits and demerits of different methods are different. Since the proposal of the King-Werner iteration method by King and Wiener, respec-tively, due to the advantage that the computation of the Jacobian matrix and the value of function is almost the same as the Newton iteration but the con-vergent order in the nonsingular case is high up to1+(?), it has attracted lots of attentions of many researchers. For the problems of the optimal shape design and the shape reconstruction, they can be transformed into the unconstrained optimization ones which can be solved by many methods such as the Lagrangian multiplier method, augmented Lagrangian method.In this paper, we will discuss the special solutions of three problems of this kind. The first problem is to seek the minimal positive solution of the nonsymmetric algebraic Riccati equation. The second one is to find the optimal shape of the head of an drum according to a certain criteria. The last problem is to seek the shape which is the most close to the actual shape of the crack or the inhomogeneity in the nonlinear magnetic material.In Chapter2, we will present the existing methods for the minimal positive solution of the nonsymmetric algebraic Riccati equation. We use the King-Werner iteration method to seek the minimal positive solution and give the convergence analysis and the error analysis. Numerical tests show King-Werner method has some advantage at the singular case. For the singular case, the existing methods converge slowly. By means of the shift technique proposed by Guo et al., the phenomenon that some existing methods converge slowly has been solved. In the shift technique, it will involve a positive real parameter. The value of this parameter has some effects on the convergence rates of some algorithms. By con-sidering the iterative sequences as the functions sequence of this parameter, we analyze the effects of the parameter on the convergence rates of these four meth-ods including the Simple Iteration method, Modified Simple; iteration method, nonlinear block Jaccobi iteration method and nonlinear block Gauss-Seidel iter-ation method. Numerical tests verified our theoretical analysis.In Chapter3, we solve a problem of finding the optimal shape of the head of an drum. We first introduce the Gatcaux derivative of the objective with respect to the density function, then design an algorithm by choosing a direction of updating the density function which can make the increment of the objective functional larger than0as soon as possible. We also do some numerical tests to show the efficiency and feasibility of the algorithm.In Chapter4. we want to find the crack or the inhomogeneity in the nonlinear magnetic material. This problem belongs to the shape reconstruction problem. We first use the piecewise constant level set method to represent the composition of the material, then transform this problem into a constrained optimization one and further transform the constrained optimization one into an unconstrained one by the Lagrangian multiplier method, finally solve the unconstrained opti-mization by the gradient method. Numerical tests show that the algorithm is not dependent on the initial choice of the level set function, and the algorithm is not only efficient, but also robust.