Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten p-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions. * This article is partially based on preliminary results published in the proceeding of the 32nd International Conference on Machine Learning (ICML 2015).
This article presents the application of three black-box modeling methods to two industrial polymerization processes to predict the melt index, which is considered an important quality variable determining product specifications. The modeling methods covered in this study are support vector machines (SVMs; known as state-of-the-art modeling methods), partial least squares (PLS), and artificial neural networks (ANNs); the processes are styrene-acrylonitrile (SAN) and polypropylene (PP) polymerizations currently operated for commercial purposes in Korea. Brief outlines of the modeling procedure are presented for each method, followed by the procedures for training and validating the models. The SVM models yield the best prediction performances for both the SAN and PP polymerization processes. However, the ANN models fail to accurately predict the melt index when sufficient data are not available for model training in the PP polymerization process. The PLS models are not effective either when applied to the SAN polymerization process, for which the melt index has strong nonlinear functionality with the process variables. The good prediction performance that the SVM models show despite the insufficient data or strong process nonlinearity suggests that SVMs can be effectively used as alternative to PLS or ANNs for modeling the melt indices in other polymerization processes as well.
A new methodology is proposed to design a soft sensor for a polypropylene (PP) process with
grade changeover operation. In contrast to the general polyolefin process, the PP process usually
produces more than 100 different grades of products. Its reaction mechanism, based on seven
catalysts, is so complex that neither mechanistic nor empirical models have been successful in
describing full-scale industrial applications. The proposed methodology was developed based
on the hybrid modeling of novel clustering and black-box and mechanistic models. Clustering
based on critical to quality enables the soft sensor to handle the complexity of many different
grades. Hybrid modeling offers good predictive power for transient behaviors as well as normal
behaviors. The methodology also allows us to reduce the cost of building and updating the model.
The developed soft sensor was successfully applied to a real industrial process. The accurate
and reliable monitoring of the melt index in the PP process helped to significantly reduce the
amount of off-specification product generation.
A systematic procedure is presented for the optimal curing of rubber compounds showing reversion type cure behavior. First, a cure kinetic model is proposed that can explain the reversion and the induction period commonly found in the vulcanization of rubber compounds. The state of cure behavior is analyzed as a function of cure temperature and time on the basis of the derived kinetic model. Then, the problem of determining optimal cure temperature profile for a rubber slab in a simple cure press is addressed and formulated into a constrained dynamic optimization problem. Finally, numerical algorithms and simulation results are presented to demonstrate the proposed cure optimization procedure.
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