A Parallel-Vector Algorithm for Rapid Structural Analysis on High-Performance Computers

A parallel-vector algorithm for rapid structural analysis on high-performance computers

Pre-symmetric approach and applications to indefinite non-symmetric problems

: There has been no stable direct method for solving symmetric systems of linear equations which takes advantage of the symmetry. If the system is also positive definite, then fast, stable direct methods (e.g., Cholesky and symmetric Gaussian elimination) exist which preserve the symmetry. These methods are unstable for symmetric indefinite systems. Such systems often occur in the calculation of eigenvectors. Gaussian elimination with partial or complete pivoting is currently recommended for solving symmetric indefinite systems, and here symmetry is lost. A generalization of symmetric Gaussian elimination is presented here, called the diagonal pivoting method, in which pivots of order two as well as one are allowed in the decomposition. It is shown that the diagonal pivoting method for symmetric indefinite matrices takes advantage of symmetry so that only 1/6 n cubed multiplications, at most 1/3 n cubed additions, and 1/2 n squared storage locations are required to solve A x = b, where A is a non-singular symmetric matrix of order n. Furthermore, it is shown that the method is nearly as stable as Gaussian elimination with complete pivoting, while requiring only half the number of operations and half the storage.

SUMMARY Electric circuit models are constructed for general symmetric systems of linear algebraic equations. The modelling procedure is based on the node-voltage analysis of linear circuits under steady-state conditions. These electric circuit models can, in principle, be physically realized and used for the solution of the systems of equations by simply measuring the electric voltage at the circuit nodes. Copyright £: 2000 John Wiley & Sons, Ltd.

In this paper, we develop two algorithms to solve a nonlinear system of symmetric equations. The first is an algorithm based on modifying two Broyden–Fletcher–Goldfarb–Shanno (BFGS) methods. One of its advantages is that it is more suitable to effectively solve a small-scale system of nonlinear symmetric equations. In contrast, the second algorithm chooses new search directions by incorporating an approximation method of computing the gradients and their difference into the determination of search directions in the first algorithm. In essence, the second one can be viewed as an extension of the conjugate gradient method recently proposed by Lv et al. for solving unconstrained optimization problems. It was proved that these search directions are sufficiently descending for the approximate residual square of the equations, independent of the used line search rules. Global convergence of the two algorithms is established under mild assumptions. To test the algorithms, they are used to solve a number of benchmark test problems. Numerical results indicate that the developed algorithms in this paper outperform the other similar algorithms available in the literature.

We discuss the object-oriented design of a software package for solving sparse, symmetric systems of equations (positive definite and indefinite) by direct methods. At the highest layers, we decouple data structure classes from algorithmic classes for flexibility. We describe the important structural and algorithmic classes in our design, and discuss the trade-offs we made for high performance. The kernels at the lower layers were optimized by hand. Our results show no performance loss from our object-oriented design, while providing flexibility, ease of use, and extensibility over solvers using procedural design.

Interactively exploring elimination orderings in symbolic sparse Cholesky factorization

Cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods (IPMs) for linear programming (LP), its main advantages being numerical stability and efficiency for sparse systems. However in the presence of dense columns there is dramatic loss of efficiency. A typical approach to remedy this problem is to apply the Sherman-Morrison-Woodbury (SMW) update formula to the dense part. This approach while being very efficient, is not numerically stable. Here we propose using product-form Cholesky factorization to handle dense columns. The proposed approach is essentially as stable as the original Cholesky factorization and nearly as efficient as the SMW approach. We demonstrate these properties both theoretically and computationally. A key part of our theoretical analysis is the proof that the elements of the Cholesky factors of the matrices that arise in IPMs for LP are uniformly bounded as the duality gap converges to zero.

In this chapter, we describe the Hybrid Displacement Boundary Element Method for the time domain and derive formulations for elastodynamics and acoustics. These formulations lead to symmetric systems of equations with time-invariant mass and stiffness matrices. The symmetry is exploited to formulate a symmetric coupled system of equations for fluid-structure interaction in time domain.

In this paper, we propose a new parallel bidirectional algorithm, based on Cholesky factorization, for the solution of sparse symmetric system of linear equations. Unlike the existing algorithms, the numerical factorization phase of our algorithm is carried out in such a manner that the entire back substitution component of the substitution phase is replaced by a single step division. Since there is a substantial reduction in the time taken by the repeated execution of the substitution phase, our algorithm is particularly suited for the solution of systems with multiple b-vectors. The effectiveness of our algorithm is demonstrated by comparing it with the existing parallel algorithm, based on Cholesky factorization, using extensive simulation studies on two-dimensional problems discretized by FEM.

In Part I of this paper, we proposed a new parallel bidirectional algorithm, based on Cholesky factorization, for the solution of sparse symmetric system of linear equations. In this paper, we propose a new parallel bidirectional algorithm, based on LU factorization, for the solution of general sparse system of linear equations having non symmetric coefficient matrix. As with the sparse symmetric systems, the numerical factorization phase of our algorithm is carried out in such a manner that the entire back substitution component of the substitution phase is replaced by a single step division. However, due to absence of symmetry, important differences arise in the ordering technique, the symbolic factorization phase, and message passing during numerical factorization phase. The bidirectional substitution phase for solving general sparse systems is the same as that for sparse symmetric systems. The effectiveness of our algorithm is demonstrated by comparing it with the existing parallel algorithm, based on LU factorization, using extensive simulation studies.

We present in this paper two results concerning the convergence of the teo-grid algebraic algorithm for arbitrary symmetric systems of linear equations which are also positive definite.Ue obtain these results using a special construction of the interpolation operator based on Gaussian elimination on a sub-matrix of the original system matrix.At the end of the paper we make also some remarks concerning the symmetric indefinite systems.

We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners.

A parameterized SHSS iteration method for a class of complex symmetric system of linear equations

An absorbing boundary condition of second order is derived for mesh terminations conformal to the shape of the radiator or scatterer. The boundary condition is expressed entirely in terms of the principal curvatures of the termination surface and results in a symmetric system of equations when incorporated into the finite‐element functional.