Textured Decomposition Research Articles

A Parallel Implementation of the p-Version of the Finite Element Method

An iterative method based on the textured decomposition (TD) is developed in order to solve the systems of linear equations arising in the p-version of the finite element method. The iteration is used to implement the p-version in parallel on an MIMD computer NCUBEIsix. The objectives are twofold: to achieve high computational efficiency (that is, computational load should be balanced among the processors) and simultaneously to achieve rapid convergence. A superelement, consisting of four adjacent rectangular finite elements, is constructed for two-dimensional problems. Based on the structural property of the shape functions, each superelement is partitioned into three blocks in two different ways, and a two-leaf TD is used. Computations for a superelement associated with each leaf are assigned to two processors and are performed in parallel. A new preconditioner is introduced to accelerate convergence in a preconditioned textured decomposition (PTD). A special local communication strategy is used to avoid global assembly and global communication. Two model problems including a Laplace equation on a rectangular domain with a near singular solution and a Poisson equation on an L-shaped domain, are solved. The conjugate gradient (CG) method, the TD method, the recursive textured decomposition (RTD) method, both with and without preconditioning, and the classical iterative methods (Jacobi, Gauss–Seidel (GS), successive overrelaxation (SOR)), are used to solve both model problems. Load balance, speedup ratio, and spectral radii of the various iterations are studied The test results indicate that recursive PTD with a local communication strategy gives at least a $30\% $ improvement in computational time over the other methods.

Exact convergence of a parallel textured algorithm for data network optimal routing problems

In our earlier paper (1991), a textured decomposition based algorithm is developed to solve the optimal routing problem in data networks; a few examples were used to illustrate the speedup advantage and the convergence conditions for the textured algorithm to converge to a global minimum. The speedup advantage is investigated in Huang et al. (1993). However, the theoretical foundation is not provided. In this paper, we provide the foundation. First, we show that for any textured decomposition, the algorithm always converges to a stationary point, which may not be a global minimum. And then, we prove that if the conditions of the exact convergence theorem are satisfied, the textured algorithm will converge to a global minimum.

A parallel HAD-textured algorithm for constrained economic dispatch control problems

In our earlier papers we developed a parallel textured algorithm to solve the constrained economic dispatch control (CEDC) problems. The exact convergence theorem and its proof were provided to guarantee the convergence of the algorithm to the true solution; and some examples were given to show the impact of exact convergence conditions. In this paper, we incorporate the hierarchical aggregation-disaggregation (HAD) concept and the textured concept to solve the CEDC. The algorithm is then implemented on an nCUBE2 machine and tested on a modified IEEE 14-bus system, a modified IEEE 57-bus system, a 114-bus system, and a 228-bus system. Some test results are given to show the speedup advantage of the proposed algorithm over the one without textured decomposition. Even when the proposed algorithm is executed sequentially, the speedup is still essential.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>