Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors

Computational Efficiency in Bayesian Model and Variable Selection

Large scale Bayesian model averaging and variable selection exercises present, despite the great increase in desktop computing power, considerable computational challenges. Due to the large scale it is impossible to evaluate all possible models and estimates of posterior probabilities are instead obtained from stochastic (MCMC) schemes designed to converge on the posterior distribution over the model space. While this frees us from the requirement of evaluating all possible models the computational effort is still substantial and efficient implementation is vital. Efficient implementation is concerned with two issues: the efficiency of the MCMC algorithm itself and efficient computation of the quantities needed to obtain a draw from the MCMC algorithm. We evaluate several different MCMC algorithms and find that relatively simple algorithms with local moves perform competitively except possibly when the data is highly collinear. For the second aspect, efficient computation within the sampler, we focus on the important case of linear models where the computations essentially reduce to least squares calculations. Least squares solvers that update a previous model estimate are appealing when the MCMC algorithm makes local moves and we find that the Cholesky update is both fast and accurate.Bayesian Model Averaging; Sweep operator; Cholesky decomposition; QR decomposition; Swendsen-Wang algorithm

Computational Efficiency in Bayesian Model and Variable Selection

This paper is concerned with the efficient implementation of Bayesian model averaging (BMA) and Bayesian variable selection, when the number of candidate variables and models is large, and estimation of posterior model probabilities must be based on a subset of the models. Efficient implementation is concerned with two issues, the efficiency of the MCMC algorithm itself and efficient computation of the quantities needed to obtain a draw from the MCMC algorithm. For the first aspect, it is desirable that the chain moves well and quickly through the model space and takes draws from regions with high probabilities. In this context there is a natural trade-off between local moves, which make use of the current parameter values to propose plausible values for model parameters, and more global transitions, which potentially allow exploration of the distribution of interest in fewer steps, but where each step is more computationally intensive. We assess the convergence properties of simple samplers based on local moves and some recently proposed algorithms intended to improve on the basic samplers. For the second aspect, efficient computation within the sampler, we focus on the important case of linear models where the computations essentially reduce to least squares calculations. When the chain makes local moves, adding or dropping a variable, substantial gains in efficiency can be made by updating the previous least squares solution.

Efficient strategies for deriving the subset VAR models

Abstract.: Algorithms for computing the subset Vector Autoregressive (VAR) models are proposed. These algorithms can be used to choose a subset of the most statistically-significant variables of a VAR model. In such cases, the selection criteria are based on the residual sum of squares or the estimated residual covariance matrix. The VAR model with zero coefficient restrictions is formulated as a Seemingly Unrelated Regressions (SUR) model. Furthermore, the SUR model is transformed into one of smaller size, where the exogenous matrices comprise columns of a triangular matrix. Efficient algorithms which exploit the common columns of the exogenous matrices, sparse structure of the variance-covariance of the disturbances and special properties of the SUR models are investigated. The main computational tool of the selection strategies is the generalized QR decomposition and its modificatio

Computing 3SLS Solutions of Simultaneous Equation Models with a Possible Singular Variance-Convariance Matrix

Algorithms for computing the three-stage least squares (3SLS) estimator usually require the disturbance convariance matrix to be non-singular. However, the solution of a reformulated simultaneous equation model (SEM) results into the redundancy of this condition. Having as a basic tool the QR decomposition, the 3SLS estimator, its dispersion matrix and methods for estimating the singular disturbance covariance matrix and derived. Expressions revealing linear combinations between the observations which become redundant have also been presented. Algorithms for computing the 3SLS estimator after the SEM have been modified by deleting or adding new observations or variables are found not to be very efficient, due to the necessity of removing the endogeneity of the new data or by re-estimating the disturbance covariance matrix. Three methods have been described for solving SEMs subject to separable linear equalities constraints. The first method considers the constraints as additional precise observations while the other two methods reparameterized the constraints to solve reduced unconstrained SEMs. Method for computing the main matrix factorizations illustrate the basic principles to be adopted for solving SEMs on serial or parallel computer

A DEIM Induced CUR Factorization

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a good approximation. Given a low-rank singular value decomposition $A \approx V S W^T$, the DEIM procedure uses $V$ and $W$ to select the columns and rows of $A$ that form $C$ and $R$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of $V$ and $W$. For large-scale problems, $V$ and $W$ can be approximated using an incremental QR algorithm that makes one pass through $A$. Numerical examples illustrate the favorable performance of the DEIM-CUR method, compared to CUR approximations based on leverage scores

A DEIM Induced CUR Factorization

We derive a CUR approximate matrix factorization based on the discrete empirical interpolation method (DEIM). For a given matrix ${\bf A}$, such a factorization provides a low-rank approximate decomposition of the form ${\bf A} \approx \bf C \bf U \bf R$, where ${\bf C}$ and ${\bf R}$ are subsets of the columns and rows of ${\bf A}$, and ${\bf U}$ is constructed to make $\bf C\bf U \bf R$ a good approximation. Given a low-rank singular value decomposition ${\bf A} \approx \bf V \bf S \bf W^T$, the DEIM procedure uses ${\bf V}$ and ${\bf W}$ to select the columns and rows of ${\bf A}$ that form ${\bf C}$ and ${\bf R}$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of ${\bf V}$ and ${\bf W}$. For very large problems, ${\bf V}$ and ${\bf W}$ can be approximated well using an incremental QR algorithm that makes only one pass through ${\bf A}$. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores

Regularized Decomposition of Stochastic Programs: Algorithmic Techniques and Numerical Results

A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported