We consider the problem of predicting the nonzero structure of a product of two or more matrices. Prior knowledge of the nonzero structure can be applied to optimize memory allocation and to determine the optimal multiplication order for a chain product of sparse matrices. We adapt a recent algorithm by the author and show that the essence of the nonzero structure and hence, a near-optimal order of multiplications, can be determined in near-linear time in the number of nonzero entries, which is much smaller than the time required for the multiplications. An experimental evaluation of the algorithm demonstrates that it is practical for matrices of order 103 with 104 nonzeros (or larger). A relatively small pre-computation results in a large time saved in the computation-intensive multiplication.

Numerische Mathematik 1

On Optimizing Multiplications of Sparse Matrices

In this paper, we propose an OpenMP implementation of a recursive algorithm for parallel fast Fourier transform (FFT) on shared-memory parallel computers. A recursive three-step FFT algorithm improves performance by effectively utilizing the cache memory. Performance results of one-dimensional FFTs on the DELL PowerEdge 7150 and the hp workstation zx6000 are reported. We successfully achieved performance of about 757MFLOPS on the DELL PowerEdge 7150 (Itanium 800MHz, 4CPUs) and about 871MFLOPS on the hp workstation zx6000 (Itanium2 1GHz, 2CPUs) for 224-point FFT.

An OpenMP implementation of parallel FFT and its performance on IA-64 processors

This thesis presents new methods for classification and thematic grouping of billions of web pages, at scales previously not achievable This process is also known as document clustering, where similar documents are automatically associated with clusters that represent various distinct topic These automatically discovered topics are in turn used to improve search engine performance by only searching the topics that are deemed relevant to particular user queries

Document clustering algorithms, representations and evaluation for information retrieval

Substructure coupling techniques allow one to predict the response of an assembly from dynamic models for each subcomponent. Linear substructures are routinely used in analysis (e.g. the Craig-Bampton method) to reduce the computational cost of vibration, noise and load predictions for structures. They also provide a designer with insight into the influence that each subcomponent has on the assembled system’s natural frequencies, mode shapes and damping. These concepts are currently limited to assemblies of linear substructures. This work explores substructuring with nonlinear subcomponents (substructures), using the nonlinear normal modes of each substructure to seek to understand how those contribute to the nonlinear modes of the assembly. The goal is to extend the insights that have been developed over the past several years for linear substructures, to nonlinear ones. A specific type of describing function model, which captures the variation of the structure’s natural frequencies and mode shapes with energy is introduced, which is here dubbed a Representative Linear Modal Model (RLMM). The results show that this type of model can be used with linear substructuring techniques to effectively predict the dynamics of a nonlinear assembly, suggesting that linear substructuring concepts may be applied advantageously to nonlinear assemblies.

Substructuring with Nonlinear Subcomponents: A Nonlinear Normal Mode Perspective

1 Univariate Linear Regression Model.- 1.1 Probability and Data Generating Process.- 1.1.1 Random Variable and Probability Distribution.- 1.1.2 Example.- 1.1.3 Data Generating Process.- 1.1.4 Example.- 1.2 Estimators and Properties.- 1.2.1 Regression Parameters and their Estimation.- 1.2.2 Least Squares Method.- 1.2.3 Example.- 1.2.4 Goodness of Fit Measures.- 1.2.5 Example.- 1.2.6 Properties of the OLS Estimates of a, ss and ?2.- 1.2.7 Examples.- 1.3 Inference.- 1.3.1 Hypothesis Testing about ss.- 1.3.2 Example.- 1.3.3 Testing Hypothesis Based on the Regression Fit.- 1.3.4 Example.- 1.3.5 Hypothesis Testing about ?.- 1.3.6 Example.- 1.3.7 Hypotheses Testing about ?2.- 1.4 Forecasting.- 1.4.1 Confidence Interval for the Point Forecast.- 1.4.2 Example.- 1.4.3 Confidence Interval for the Mean Predictor.- 2 Multivariate Linear Regression Model.- 2.1 Introduction.- 2.2 Classical Assumptions of the MLRM.- 2.2.1 The Systematic Component Assumptions.- 2.2.2 The Random Component Assumptions.- 2.3 Estimation Procedures.- 2.3.1 The Least Squares Estimation.- 2.3.2 The Maximum Likelihood Estimation.- 2.3.3 Example.- 2.4 Properties of the Estimators.- 2.4.1 Finite Sample Properties of the OLS and ML Estimates ofss.- 2.4.2 Finite Sample Properties of the OLS and ML Estimates of ?2.- 2.4.3 Asymptotic Properties of the OLS and ML Estimators of ss.- 2.4.4 Asymptotic Properties of the OLS and ML Estimators of ?2.- 2.4.5 Example.- 2.5 Interval Estimation.- 2.5.1 Interval Estimation of the Coefficients of the MLRM..- 2.5.2 Interval Estimation of ?2.- 2.5.3 Example.- 2.6 Goodness of Fit Measures.- 2.7 Linear Hypothesis Testing.- 2.7.1 Hypothesis Testing about the Coefficients.- 2.7.2 Hypothesis Testing about a Coefficient of the MLRM.- 2.7.3 Testing the Overall Significance of the Model.- 2.7.4 Testing Hypothesis about ?2.- 2.7.5 Example.- 2.8 Restricted and Unrestricted Regression.- 2.8.1 Restricted Least Squares and Restricted Maximum Likelihood Estimators.- 2.8.2 Finite Sample Properties of the Restricted Estimator Vector.- 2.8.3 Example.- 2.9 Three General Test Procedures.- 2.9.1 Likelihood Ratio Test (LR).- 2.9.2 The Wald Test (W).- 2.9.3 Lagrange Multiplier Test (LM).- 2.9.4 Relationships and Properties of the Three General Testing Procedures.- 2.9.5 The Three General Testing Procedures in the MLRM Context.- 2.9.6 Example.- 2.10 Dummy Variables.- 2.10.1 Models with Changes in the Intercept.- 2.10.2 Models with Changes in some Slope Parameters.- 2.10.3 Models with Changes in all the Coefficients.- 2.10.4 Example.- 2.11 Forecasting.- 2.11.1 Point Prediction.- 2.11.2 Interval Prediction.- 2.11.3 Measures of the Accuracy of Forecast.- 2.11.4 Example.- 3 Dimension Reduction and Its Applications.- 3.1 Introduction.- 3.1.1 Real Data Sets.- 3.1.2 Theoretical Consideration.- 3.2 Average Outer Product of Gradients and its Estimation.- 3.2.1 The Simple Case.- 3.2.2 The Varying-coefficient Model.- 3.3 A Unified Estimation Method.- 3.3.1 The Simple Case.- 3.3.2 The Varying-coefficient Model.- 3.4 Number of E.D.R. Directions.- 3.5 The Algorithm.- 3.6 Simulation Results.- 3.7 Applications.- 3.8 Conclusions and Further Discussion.- 3.9 Appendix. Assumptions and Remarks.- 4 Univariate Time Series Modelling.- 4.1 Introduction.- 4.2 Linear Stationary Models for Time Series.- 4.2.1 White Noise Process.- 4.2.2 Moving Average Model.- 4.2.3 Autoregressive Model.- 4.2.4 Autoregressive Moving Average Model.- 4.3 Nonstationary Models for Time Series.- 4.3.1 Nonstationary in the Variance.- 4.3.2 Nonstationarity in the Mean.- 4.3.3 Testing for Unit Roots and Stationarity.- 4.4 Forecasting with ARIMA Models.- 4.4.1 The Optimal Forecast.- 4.4.2 Computation of Forecasts.- 4.4.3 Eventual Forecast Functions.- 4.5 ARIMA Model Building.- 4.5.1 Inference for the Moments of Stationary Processes..- 4.5.2 Identification of ARIMA Models.- 4.5.3 Parameter Estimation.- 4.5.4 Diagnostic Checking.- 4.5.5 Model Selection Criteria.- 4.5.6 Example: European Union G.D.P.- 4.6 Regression Models for Time Series.- 4.6.1 Cointegration.- 4.6.2 Error Correction Models.- 5 Multiplicative SARIMA models.- 5.1 Introduction.- 5.2 Modeling Seasonal Time Series.- 5.2.1 Seasonal ARIMA Models.- 5.2.2 Multiplicative SARIMA Models.- 5.2.3 The Expanded Model.- 5.3 Identification of Multiplicative SARIMA Models.- 5.4 Estimation of Multiplicative SARIMA Models.- 5.4.1 Maximum Likelihood Estimation.- 5.4.2 Setting the Multiplicative SARIMA Model.- 5.4.3 Setting the Expanded Model.- 5.4.4 The Conditional Sum of Squares.- 5.4.5 The Extended ACF.- 5.4.6 The Exact Likelihood.- 6 Auto Regressive Conditional Heteroscedastic Models.- 6.1 Introduction.- 6.2 ARCH(1) Model.- 6.2.1 Conditional and Unconditional Moments of the ARCH(1).- 6.2.2 Estimation for ARCH(1) Process.- 6.3 ARCH(q) Model.- 6.4 Testing Heteroscedasticity and ARCH(1) Disturbances.- 6.4.1 The Breusch-Pagan Test.- 6.4.2 ARCH(1) Disturbance Test.- 6.5 ARCH(1) Regression Model.- 6.6 GARCH(p,q) Model.- 6.6.1 GARCH(1,1) Model.- 6.7 Extensions of ARCH Models.- 6.8 Two Examples of Spanish Financial Markets.- 6.8.1 Ibex35 Data.- 6.8.2 Exchange Rate US Dollar/Spanish Peseta Data (Continued).- 7 Numerical Optimization Methods in Econometrics.- 7.1 Introduction.- 7.2 Solving a Nonlinear Equation.- 7.2.1 Termination of Iterative Methods.- 7.2.2 Newton-Raphson Method.- 7.3 Solving a System of Nonlinear Equations.- 7.3.1 Newton-Raphson Method for Systems.- 7.3.2 Example.- 7.3.3 Modified Newton-Raphson Method for Systems.- 7.3.4 Example.- 7.4 Minimization of a Function: One-dimensional Case.- 7.4.1 Minimum Bracketing.- 7.4.2 Example.- 7.4.3 Parabolic Interpolation.- 7.4.4 Example.- 7.4.5 Golden Section Search.- 7.4.6 Example.- 7.4.7 Brent's Method.- 7.4.8 Example.- 7.4.9 Brent's Method Using First Derivative of a Function..- 7.4.10 Example.- 7.5 Minimization of a Function: Multidimensional Case.- 7.5.1 Neider and Mead's Downhill Simplex Method (Amoeba).- 7.5.2 Example.- 7.5.3 Conjugate Gradient Methods.- 7.5.4 Examples.- 7.5.5 Quasi-Newton Methods.- 7.5.6 Examples.- 7.5.7 Line Minimization.- 7.5.8 Examples.- 7.6 Auxiliary Routines for Numerical Optimization.- 7.6.1 Gradient.- 7.6.2 Examples.- 7.6.3 Jacobian.- 7.6.4 Examples.- 7.6.5 Hessian.- 7.6.6 Example.- 7.6.7 Restriction of a Function to a Line.- 7.6.8 Example.- 7.6.9 Derivative of a Restricted function.- 7.6.10 Example.

Numerische Mathematik 1

Citations

On Optimizing Multiplications of Sparse Matrices

An OpenMP implementation of parallel FFT and its performance on IA-64 processors

Document clustering algorithms, representations and evaluation for information retrieval

Substructuring with Nonlinear Subcomponents: A Nonlinear Normal Mode Perspective

Computer-Aided Introduction to Econometrics

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