# A tutorial on elliptic PDE solvers and their parallelization The intermediate case of the basis set with a local support, namely splines , results in the band blocks [3, 4] with the bandwidth comparable with that one from the finite-difference method.

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The direct method for solving the linear systems is the Gauss elimination. An adaptation of the Gauss elimination to the block-tridiagonal systems is known as the matrix Thomas algorithm  or as the matrix sweeping algorithm . In this algorithm, the idea of Gauss elimination is applied to the blocks themselves.

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The algorithm is well defined and robust for matrices with the diagonal dominance, but its sequential nature makes it difficult to be applied for parallel calculations. The present paper is focused on the developed arrowhead decomposition method ADM for efficient parallel solving the block-tridiagonal linear systems.

## Multigrid method - Wikipedia

In this method, the initial matrix is logically reduced to the "arrowhead" form, namely some new independent on-diagonal blocks, sparse off-diagonal blocks and a coupling matrix of much smaller size. The method originates from the domain decomposition [] and nested dissection [12, 13], where the idea. Any further distribution. In Ref. It is shown that this method is faster than the Wang algorithm [15, 16] and following ones.

In this paper, the computational speedup  of ADM with respect to the sequential matrix Thomas algorithm is analytically estimated based on the number of elementary multiplicative operations for the parallel and serial parts of the methods. A number of parallel processors required to achieve the maximum computational speedup is obtained. The analytical results are compared with the results of practical calculations. For the test linear systems, we use the discretized boundary value problems for the integro-differential Faddeev equations [, 4, 21, 22].

The unknown supervector X is composed of blocks Xi. The idea of the ADM is presented in Figure 1. The initial block-tridiagonal linear system 1 is rearranged into the equivalent "arrowhead" form which allows the parallel solving. The rearrangement is performed by interchanging the block-rows and block-columns.

The interchange of block-columns also leads to the change of the elements of the unknown vector and the RHS vector. The new structure of the matrix can be represented by the 2 x 2 block-matrix. Here, the unknown solution h corresponds to the moved part of the full solution. The notation is shown in Figure 1. The matrix element H is the bottom right coupling superblock. Other lateral superblocks WR, WL present additional blocks of the matrix. The solution of the system 2 is given by the relations.

These relations contain matrix products and inverses which can, to some extent, be done in parallel. Actually, instead of inverses we solve in parallel over k the linear systems for Sk. To solve the independent linear systems with Sk and the equation for h, one can apply any appropriate technique. Although the ADM can be used recursively nestedly for solving these linear systems , in the present paper, we employ the matrix Thomas algorithm.

Figure 1. A graphical scheme for rearrangement of the initial block-tridiagonal linear system into the equivalent form. The nonzero blocks and vectors are denoted by thick lines. New independent superblocks and corresponding vectors at each panel are denoted by thin lines. Top panel: the initial linear system with the marked interchanged blocks.

Central panel: the obtained rearranged system with the "arrowhead" matrix. Bottom panel: the notation of the matrix elements for the rearranged system. The computational speedup of the ADM with respect to the sequential matrix Thomas algorithm can be estimated as the ratio of the computation time by the matrix Thomas algorithm to that one of the ADM .

The computation time is directly related to the number of serial operations, namely products and divisions, of each algorithm. We calculated a number of serial multiplicative operations for both algorithms and, as a consequence, analytically estimated the computational speedup. The additive operations were not taken into account assuming that they require much less computational time. The time for the memory management is also considered to be negligible.

According to Ref.

Then, the number of multiplicative operations for the sequential matrix Thomas algorithm is . For the ADM, the total number of multiplicative operations was found to be given by the formula.

## Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.

In general, Nk may be different for different k, but should satisfy the relation:. The maximum performance is achieved if the computation time for solving the linear system with the block Sk is equal for each k. Therefore, for the sake of simplicity, we consider the case when all Nk are the same.

In such a case, if the number of parallel processors equals to the number of blocks M on the diagonal, then the number of serial operations of the ADM will be. As M increases further, the speedup flattens out, reaches its maximum and, then, decreases. A number of parallel processors required to achieve the maximum speedup is given as.

As a result, for using the ADM one can choose all independent blocks Sk of the same size. Smooth initial U vector to receive a new approximation U q 2. Restrict R 2 5x? Prolongate e q-1 to e q and add to U q. Geometric Classical MultiGrid. Danny Thorne.

Hitler writes a finite element solver

Total Recall Math, Part 2 Ordinary diff. Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation. Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones. Similar presentations. Upload Log in. My presentations Profile Feedback Log out. Log in.

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