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A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.

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 [

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,

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

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 [

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 [

procedures for the FE or FD solution by manipulating the problem of the fill-in terms, which occur during the factorization [

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 [

For solving the symmetric problem

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

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.

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

where _{1} and l_{2} the numbers of diagonals retained in semi-bandwidths m and p respectively (

The computation of the elements of the sparse decomposition factors has been presented in [

The relationships of the elements of V matrix and the corresponding conventional (for

An analogue scheme can be obtained for matrix W, while the relationships of the elements of H matrix and the corresponding conventional (for

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

An exact inverse algorithm based on adaptive algorithmic methodologies for solving linear unsymmetric systems of irregular structure arising in FD/FE discretization of boundary-value problems in three space dimensions has been recently presented [

It should be also noted that in the case that only

parabolic boundary values, can be obtained. Such algorithms are described in the next sections.

A class of optimized approximate inverse variants can be obtained by considering a (near) optimized choice of the approximate inverse M depends on the selection of related parameters, i.e. fill-in parameters r_{1}, r_{2}, retention parameters δl_{1}, δl_{2} and entropy-adaptivity-uncertainty (EAU) parameters [

where ^{*}

Algorithmic solution methods for the linear systems (1.2) applicable to both two and three space dimension can be applied [

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 be diagonally dominant.

Let us assume that

having main disadvantages, i.e. high storage requirements and computational work involved particularly in the case of solving very large unsymmetric linear systems. A class of approximate inverses

Let us consider now the approximate inverse of A with the form

where

Then, by post-multiplying Equation (3.2) by

where_{r.}, by considering the equations in the analytical form for i-row with

where

The elements of the approximate inverse for

where

Let us consider the exact inverse M of the original coefficient matrix A in equation (2.1). Note that the computation of the inverse is indicated in the following characteristic diagram (

It should be noted that the diagonal elements (in bold) are firstly computed (starting from the last element of the inverse, i.e.

the last diagonal element ^{*} of sub-class IV, is almost the half of that required by the approximate inverse of sub-class III. Note diagrammatically, as it is shown in ^{*} are computed.

By generalized this storage saving computational technique, we consider the above DS technique can be replaced by k-sweep (KS) technique, i.e. after the computation of the last diagonal element^{*} of this sub-class by using the KS-storage technique is considerably smaller than that required by the approximate inverse resulting from the application of the DS storage technique. In the case of k = 2 the KS-storage technique reduces to the example shown in

An optimized explicit banded approximate inverse by minimizing the memory requirements of EBAIM-1 algorithm

In order to minimize the memory requirements of EBAIM-1 algorithm, which in particular in the case of very large matrices of irregular structure can be prohibitively high, we consider the inverse M of equation (5.4) revolving its elements by 180˚ about the anti-diagonal removing the diagonal and the (δl-1) super diagonals in the first δl columns, while the rest δl sub-diagonals in the rest δl columns, then results the following form of the inverse (

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.

sparse triangular decomposition factors [

Algorithm OBAIM-1 (a, b, c, n, F, H, g, Γ, Ζ, ω, β, r_{1}, r_{2}, m, p, l_{1}, l_{2}, 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 r_{1}, r_{2}; semi-bandwidths m, p; l_{1} and l_{2} numbers of diagonals retained in semi-bandwidths m and p respectively, ^{*}/for simplicity reasons

Output: elements

Computational Procedure:

step 1: let_{ }

step 2: for

step 3: for

step 4: if

step 5: if

step 6: if

step 7:

step 8: else

step 9:

step 10:

step 11: else

step 12: _{ }

Step 13: else

if

step 14: if

step 15:

step 16:

step 17:

step 18: _{ }

step 19: else

if

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step 21:

step 22:

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step 25:

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step 27: else

step 28:

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step 31:

step 32:

step 33: else

if

step 34: if

step 35:

step 36:

step 37:

step 38:

step 39:

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step 41: else

step 42:

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step 45:

step 46:

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step 50:

step 51: else

step 52:

step 53:

step 54:

step 55:

step 56:

step 57:

step 58:

step 59:

step 60:

step 61: for

step 62:

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

tions, while the memory requirements have been reduced down to

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

where

During the last decades extensive research work has been focused in the preconditioned approach and preconditioned iterative methods for solving large linear and nonlinear problems in sequential and parallel environments [^{*} due to its superior convergence rate for solving very large complex computational problems [

Algorithm EPCG-1 (A, n, s, u_{0}, u, r)

Purpose: This algorithm computes the solution vector of the linear system

Input: A given matrix, n order of A, s known rhs vector, u_{0} initial guess

Output: solution vector u, residual r

Computational Procedure:

Step 1: let

Step 2: set

Step 3: for

compute

//compute scalar quantities

Step 4: form

Step 5: evaluate

Step 6: compute

Step 7: compute

Step 8: compute

Step 9: if there is no convergence go to step 3,

Step 10: else

print the approximate solution _{ }

Note that a good approximant M^{*} leads obviously to an improved EPCG method. The effectiveness of the explicit preconditioned iterative methods for solving certain classes of elliptic boundary value problems on regular domains is related to the fact that the exact inverse of A (although is full) exhibits a similar fuzzy structure around the principal diagonal and m-diagonals [

In the case of symmetric coefficient matrix by using the four-matrix decomposition [

where 1) the elements of exact inverse of subclass I are obtained after the exact decomposition ^{+}, with excessive memory and computational requirements, 2) the elements of the inverse ^{S}^{2 }of subclass III have been computed from the approximate inverse, while the exact decomposition

Note that the largest elements of inverse matrix are mainly gathered around the main diagonal in distances

An indication of the sparsity and memory requirements of optimized versions of approximate inverses is given in the following

It should be noted that in the case that δl_{2} = 0 the approximate inverse algorithm is reduced to an algorithm for solving FE systems in two space dimensions of semi-bandwidth m, while if _{1} = 1 and δl_{2} = 0, then the approximate inverse reduces to one for solving linear FD systems in two space dimensions of semi-bandwidth m [

δl = 2 | δl = m | δl = 2m | δl = p | δl = 2p | δl = 4p | |
---|---|---|---|---|---|---|

Diagonal vectors | 3 | 41 | 83 | 801 | 1603 | 3207 |

Spasity | 99.9 | 99.5 | 99 | 90 | 80 | 59.9 |

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

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 [

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 [

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 ε_{1} was taken initially as ^{−6}, where it remained constant during the next iterative steps. Numerical experiments were carried out for nonlinear problem (6.4) with

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.

Picard-EPSD | Newton-EPSD | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

δl | 40 | 100 | 180 | 40 | 100 | 180 | |||||||

B.C. U ≡ 0.0 | |||||||||||||

a | Inner Iterat | Outer Iterat | Inner Iterat | Outer Iterat | Inner Iterat | Outer Iterat | Inner Iterat | Outer Iterat | Inner Iterat | Outer Iterat | Inner Iterat | Outer Iterat | |

1.20 | 72 | 27 | 56 | 23 | 55 | 23 | 79 | 30 | 61 | 25 | 59 | 24 | |

1.30 | 73 | 31 | 52 | 22 | 51 | 22 | 74 | 29 | 56 | 23 | 55 | 23 | |

1.40 | >200 | - | 50 | 21 | 47 | 20 | >200 | - | 52 | 22 | 50 | 21 | |

1.50 | - | - | 45 | 20 | 44 | 20 | 48 | 21 | 47 | 21 | |||

1.60 | - | - | 42 | 19 | 41 | 19 | 45 | 20 | 45 | 21 | |||

B.C. U ≡ 2.0 | |||||||||||||

1.20 | 188 | 9 | 142 | 9 | 131 | 9 | 58 | 7 | 48 | 7 | 37 | 6 | |

1.30 | 173 | 9 | 130 | 9 | 122 | 9 | >200 | - | 44 | 7 | 38 | 7 | |

1.40 | >200 | 123 | 9 | 114 | 9 | 41 | 7 | 42 | 6 | ||||

1.50 | - | 118 | 9 | 117 | 9 | 42 | 7 | 38 | 6 | ||||

1.60 | 132 | 10 | 130 | 10 | >200 | - | 44 | 6 | |||||

Picard -EPCG | Newton-EPCG | |||||||
---|---|---|---|---|---|---|---|---|

Overall iterations | Outer iterations | Overall iterations | Outer iterations | |||||

δl | r = 1 | r = 2 | r = 4 | r = 1 | r = 2 | r = 4 | ||

B.C. | U ≡ 0.0 | |||||||

20 | 103 | 96 | >150 | 6 | 85 | 94 | 161 | 6 |

40 | 75 | 70 | 64 | 6 | 68 | 54 | 55 | 6 |

60 | 71 | 53 | 57 | 6 | 65 | 49 | 49 | 6 |

B.C. | U ≡ 10.0 | |||||||

20 | * | * | * | - | 138 | 127 | 131 | 6 |

40 | * | * | * | - | 115 | 112 | 92 | 6 |

60 | * | * | * | - | 114 | 85 | 84 | 6 |

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