In recent years, extensive research work has been focused on the computation of exact and approximate inverse matrices for solving efficiently complex computational problems particularly on parallel computer systems [1] - [20] . In this article, a new class of Sparse Approximate Inverses matrices based on adaptive algorithmic modelling methods is presented. These adaptive algorithmic solution methods can be used for solving large sparse linear finite difference (FD) and finite element (FE) systems of irregular structures derived mainly from the discretization of parabolic and elliptic PDE’s in both two and three space dimensions [21] - [31] .

Let us consider a class of boundary value problems defined by the equation

subject to the general boundary conditions

where D is a closed bounded domain in R^N, with N ≤ 3 and ∂D the boundary of D, a predetermined singular perturbation parameter,; and the coefficients are continuous and differentiable functions in D,; f are sufficiently smooth functions on D and is the direction of the outward normal derivative.

The discrete analogue of Equation (1.1) leads to the solution of the general linear system

where the coefficient matrix A is a large sparse real (n ´ n) matrix of irregular structure. The structure of A is shown in the following Figure 1.

For solving the system (1.2), there is a choice between direct and iterative, assuming that there are no barriers due to memory requirements for the former or excessive runtimes (e.g. time dependent problems) for the latter. Note that for generality purposes the coefficient matrix is assumed to be unsymmetric (case occurring in the discretization of flow equations that arise in certain Hydrology studies [32] and of irregular non-zero structure (case resulting from the triangulation of irregular or regular domains into irregular elements in pipeline networks [33] . Algorithmic solution methods for the linear systems (1.2) applicable to both two and three space dimensions can be applied [22] , where the unsymmetric coefficient matrix, in which all the off-centre terms are grouped into regular bands, can be factorized exactly to yield direct algorithmic procedures for the FD or FE solution.

It should be noted that in the case of very large sparse linear and nonlinear systems with coefficients of irregular structure, the memory requirements and the corresponding computational work are prohibitively high and the use of exact inverse solvers is usually not recommended. In such cases, preconditioned iterative techniques for solving numerically the FD or FE linear systems (1.2) can be used by deriving semi-direct solution methods following the principle [27] [34] that implicit procedures based on approximately decomposing discrete operators into easily invertible factors facilitating the solution of (1.2). Sparse factorization procedures yield efficient

procedures for the FE or FD solution by manipulating the problem of the fill-in terms, which occur during the factorization [22] [35] . Note that simple compact storage schemes for the considered data can be used and preconditioned algorithmic solvers do not require any searching operations. An important feature of the proposed adaptive algorithmic methods is the provision of a class of iterative methods for solving large sparse unsymmetric systems of irregular non-zero structure, with additional computational facilities, i.e. the choice of fill-in parameters, rejection parameters, entropy-adaptivity-uncertainty (EAU) parameters [36] , by which the best method for a given problem can be selected. The proposed methods have a universal scope of application for numerically solving of elliptic and parabolic boundary value problems by either FD or FE discretization methods in both two and three space dimensions with the only restriction being that the coefficient matrix should be diagonally dominant.

The approximate factorization techniques and approximate inverse methodologies have been widely used for solving a large class of linear and nonlinear systems resulting in complex computational problems [12] [29] [31] [37] - [49] . The LU factorization of a given matrix is characterized as a high level algebraic description of Gaussian elimination and by expressing the outcomes of matrix algorithms in the language of matrix factorizations facilitates generalization and certain connections between algorithms that may appear different at scalar level [47] . The solution of linear system can be computed by a two-step triangular solving process, i.e.

For solving the symmetric problem, a variant of the LU factorization in which A is decomposed into three-matrix product, i.e., where D is diagonal and L, M are lower unit lower triangular. In this case the solution can be obtained in flops by solving (forward elimination), and (by substitution). Note that if then L = M and the computational work for the factorization is half of that required by Gaussian elimination. The factorization takes the form and can be used for solving symmetric problems, as well as the four-matrix product, i.e., where is the transpose of T (Varga, 1962). In the case of symmetric positive definite systems the factorization exists and is computationally stable. The factorization is known as Cholesky factorization and by solving the triangular system and, then. Note that the Gaxpy Cholesky version requires flops, where Gaxpy is a BLAS level-2 routine defined algebraically as requiring operations.

In the general case, such as the case of three space dimensions and the finite element discretization, the coefficient matrix has an irregular structure of the nonzero elements, where the non-diagonal elements can be grouped in regular bands of width and (width parameters) in distances m and p (semi-bandwidths) respectively.

The linear system (1.2) can be solved by direct (explicitly) or iterative (implicitly) methods depending on the availability of memory requirements. Several factorization/decomposition techniques can be used for facilitating the numerical solution of linear system (1.2), i.e. two, three, four term factorization schemes of the coefficient matrix A. Following an explicit solution of system (1.2) this system can equivalently written as

where M is the inverse matrix of A, i.e.. Since the computation of the exact inverse is a difficult computational problem particularly in the case of complex 3D problems, the approximate inverse matrix approach can be alternatively used.

Let us consider an approximate factorization of the coefficient matrix A,

where and, are lower and upper sparse triangular matrices of irregular structures of semi-bandwidths m and p retaining and fill-in terms respectively. The decomposition factors and are banded matrices with l₁ and l₂ the numbers of diagonals retained in semi-bandwidths m and p respectively (Figure 2 and Figure 3), of the following form.

The computation of the elements of the sparse decomposition factors has been presented in [23] [24] [27] .

The relationships of the elements of V matrix and the corresponding conventional (for and) is shown in the following Figure 4.

An analogue scheme can be obtained for matrix W, while the relationships of the elements of H matrix and the corresponding conventional (for and) is shown in the following Figure 5 (the same holds for the matrix F).

The (near) optimum values of fill-in parameters are mainly depended on the nature of the problem and structure of the coefficient matrix A [22] [47] - [49] .

The application of this storage scheme on the approximate inverse leads to the following optimized approximate inverse algorithm. Note that the computation of the approximate inverse algorithm pre-assumes the approximate factorization of the coefficient matrix A, i.e., where and are the lower and upper

sparse triangular decomposition factors [21] [23] - [25] [49] .

Algorithm OBAIM-1 (a, b, c, n, F, H, g, Γ, Ζ, ω, β, r₁, r₂, m, p, l₁, l₂, δl, M^*)

Purpose: This algorithm computes the elements of the approximate inverse of a given real (n ´ n) matrix of irregular structure

Input: diagonal elements a of matrix A; superdiagonal elements b, subdiagonal elements c, n order of A; submatrices F, H, of upper triadiagonal decomposition factor U, superdiagonal elements g of L; submatrices Γ, Z, of lower tridiagonal matrix L; diagonal elements ω of L; subdiagonal elements β of L; fill-in parameters r₁, r₂; semi-bandwidths m, p; l₁ and l₂ numbers of diagonals retained in semi-bandwidths m and p respectively, and are the numbers of diagonal retained in approximate inverse M^*/for simplicity reasons is chosen/

Output: elements of the approximate inverse

Computational Procedure:

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step 63: form the approximate inverse matrix

The subroutine mw (n, δl, s, q, x, y) performs the transformation in the indexes of the explicit approximate inverse matrix from its banded form to the optimized form. This routine has the following form:

Subroutine mw (n, δl, s, q, x, y)

If

Then

else

The computational work of the optimized OBAIM-1 algorithm is multiplica-

tions, while the memory requirements have been reduced down to words. It should be also noted that a class of approximate inverse matrix can be considered containing several sub-classes of approximate inverses according to memory requirements, computational work, accuracy, as indicated in the diagrammatic schemes (Figure 3 and Figure 7).

A class of Adaptive Iterative Schemes for solving large sparse linear systems includes the following adaptive preconditioned iterative methods:

where, α and β are predetermined acceleration parameters, and are sequences of preconditioned acceleration parameters and, i ≥ 0.

In this section a nonlinear case study by using approximate inverse preconditioned methods are presented.

The nonlinear case

Let us consider the nonlinear elliptic PDE

where subject to the Dirichlet boundary conditions

where ∂R is the exterior boundary of the domain R.

Equation (6.1) arises in magnetohydrodynamics (diffusion-reaction, vortex problems, electric space charge considerations) with its existence and uniqueness assured by the classical theory [32] [33] . The solution of Equation (6.1) can be obtained by the linearized Picard and quasi-linearized Newton iterative schemes as outer iterative schemes of the form:

and (6.2)

where δ denotes here the usual central difference operator.

The resulting large sparse nonlinear system is of the form

where Ω is a block tridiagonal matrix [23] .

Then, composite iterative schemes can be used, where Picard/Newton iterations are the outer iteration, while the inner iteration can be carried out either directly by an exact algorithm or by an approximate algorithm in conjunction with an explicit iterative method (6.3). The latter method can be written as

where the superscript l denotes the outer iteration index, the subscript I denotes the inner iteration and.

The outer iteration was terminated when the following criterion was satisfied

while the termination criterion of the inner iteration was

where ε₁ was taken initially as and then was decreased at each iterative step by 1/10 to 10⁻⁶, where it remained constant during the next iterative steps. Numerical experiments were carried out for nonlinear problem (6.4) with and the initial guesses when u was on the boundary 0.0, 5.0, 10.0 were chosen as 0.0, 4.0, 6.0 respectively. The performance of the composite schemes Newton-Explicit Preconditioned Simultaneous Displacement (EPSD) and Picard/Newton-EPCG are given in the following Table 2 and Table 3.

A class of exact and approximate inverse adaptive algorithmic procedures has been presented for solving numerically initial/boundary value problems. Several subclasses of optimized variants of these algorithms have been also proposed for solving economically highly nonlinear systems of irregular structure. It should be stated that the proposed explicit preconditioned iterative methods and their related variants can be efficiently used for solving large sparse nonlinear systems of irregular structure of complex computational problems and for the numerical solution of highly nonlinear initial/ boundary value problems in two and three space dimensions.

Future research work includes the parallelization of the proposed class of exact and approximate inverse matrices of irregular structure. These adaptive exact and approximate inverse algorithmic techniques can be used for solving efficiently highly nonlinear large sparse systems arising in the numerical solution of complex computational problems in parallel computer environments.

Anastasia-Dimitra Lipitakis, (2016) A Class of Generalized Approximate Inverse Solvers for Unsymmetric Linear Systems of Irregular Structure Based on Adaptive Algorithmic Modelling for Solving Complex Computational Problems in Three Space Dimensions. Applied Mathematics,07,1225-1240. doi: 10.4236/am.2016.711108