Linear System Solution with Iterative Methods

Large Linear System Solution with Iterative Methods 

Several methods have been proposed to provide an iterative solution to the linearized system derived from Newton s method. However, unlike symmetric positive definite matrices, these methods are sometimes inefficient. Particularly when the matrix is structured, it is generally preferable to adopt factorizations that exploit the system s structure. 

When problems rise from the simulation of a physical phenomenon, the resulting systems usually have a well-defined structure. 

Using them can be advantageous with very large sparse nonstructured matrices. 

---

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

It may appear as if this is no great improvement, since finding a solution to a linear equation system with direct methods requires about n3 operations, about half as many as the inversion. However, the solution of the linear equation system can be accomplished by iterative methods where, in each step, some product jv is formed. Superficially, this cuts down, the number of operations, but still requires the Jacobian to be computed and stored. However, for a very large class of important problems, such a product can be efficiently computed without the need of precalculating or storing the Jacobian. 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

In this paper, we shall touch the development of such numerical methods intended for the solution of the coupled evolution problems as e.g. thermoelasticity, which is described in Section 2. Here we also discuss the discretization of the evolution problems. As the computational demands are concentrated mainly in the solution of the arising linear systems, we shall focus on the application of suitable, efficient and parallelizable iterative solvers for these linear systems. Section 3 deals with some general techniques enhancing the efficiency of the iterative solution of discrete evolution problems. Section 4 is devoted to a short discussion of the numerical results. In Section 5, we shall describe solvers, which exploit the domain decomposition and parallel computations. Here we also mention another division techniques as displacement decomposition or composite grid methods. 

In Section 3, we discussed an acceleration of iterative solvers, now we shall touch the question of the choice of a suitable iterative methods and show that efficient methods can be found in the class of space decomposition - subspace conection methods, see Blaheta et. al. (2003). Remember that we are interested in the solution of linear systems with symmetric positive definite matrices and... 

An efficient and robust optimisation algorithm is primordial for this solution strategy. Rao Sawyer (1995) applied Powell s method to tackle the optimisation. Koyliioglu et al. (1995) defined a linear programming solution for this purpose. The input interval vector defines the number of constraints and, therefore, strongly influences the performance of the procedure. Also, because of the required execution of the deterministic FE analysis in each goal function evaluation, the optimisation approach is numerically expensive. Therefore, this approach is best suited for rather small FE models with a limited number of input uncertainties, unless approximate methods can be used that avoid the expensive iterative calculation of the entire FE system of equations. 

These methods can be applied both to the primitive model with constant e, and to models with space-dependent e(r) real charges dispersed in the whole dielectric medium. They aim at solving the Poisson equation (10) expressed as a set of finite-difference equations for each point of the grid. The linear system to be solved has elements depending both on e and on Pm. 3nd its solution is represented by a set of values in the grid points. These values have to be reached iteratively. 

With these basic definitions in hand, we now begin to consider the solution of the linear system Ax = b, in which x, b and is an A x A/ real matrix. We consider here elimination methods in which we convert the linear system into an equivalent one that is easier to solve. These methods are straightforward to implement and work generally for any linear system that has a unique solution however, they can be quite costly (perhaps prohibitively so) for large systems. Later, we consider iterative methods that are more effective for certain classes of large systems. 

---