Carmeliza Navasca

Philip K. Hopke, Maggie Leung, Na Li et al.

The ambient particulate chemical composition data with three particle diameter sizes (2.5mm<D< 1.15mm, 1.15mm<D<0.34mm and 0.34mm<D<0.1mm) collected at a major industrial center in Allen Park in Detroit, MI is examined. Standard multiway (tensor) methods like PARAFAC and Tucker tensor decompositions have been applied extensively to many chemical data. However, for multiple particle sizes, the source apportionment analysis calls for a novel multiway factor analysis. We apply the regularized block tensor decomposition to the collected air sample data. In particular, we use the Block Term Decomposition (BTD) in rank-(L;L;1) form to identify nine pollution sources (Fe+Zn, Sulfur with Dust, Road Dust, two types of Metal Works, Road Salt, Local Sulfate, and Homogeneous and Cloud Sulfate).

Na Li, Stefan Kindermann, Carmeliza Navasca

We study the convergence of the Regularized Alternating Least-Squares algorithm for tensor decompositions. As a main result, we have shown that given the existence of critical points of the Alternating Least-Squares method, the limit points of the converging subsequences of the RALS are the critical points of the least squares cost functional. Some numerical examples indicate a faster convergence rate for the RALS in comparison to the usual alternating least squares method.

Xiaofei Wang, Carmeliza Navasca

The goal of this paper is to find a low-rank approximation for a given tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP-hard problem. In this paper, we formulate a sparse optimization problem via an $l_1$-regularization to find a low-rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank-one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low-rank approximation.

Michael Brazell, Na Li, Carmeliza Navasca et al.

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format.

Stefan Kindermann, Carmeliza Navasca

We study the least-squares (LS) functional of the canonical polyadic (CP) tensor decomposition. Our approach is based on the elimination of one factor matrix which results in a reduced functional. The reduced functional is reformulated into a projection framework and into a Rayleigh quotient. An analysis of this functional leads to several conclusions: new sufficient conditions for the existence of minimizers of the LS functional, the existence of a critical point in the rank-one case, a heuristic explanation of "swamping" and computable bounds on the minimal value of the LS functional. The latter result leads to a simple algorithm -- the Centroid Projection algorithm -- to compute suboptimal solutions of tensor decompositions. These suboptimal solutions are applied to iterative CP algorithms as initial guesses, yielding a method called centroid projection for canonical polyadic (CPCP) decomposition which provides a significant speedup in our numerical experiments compared to the standard methods.

Dipak Dulal, Joseph J. Charney, Michael Gallagher et al.

This research project investigated the correlation between a 10 Hz time series of thermocouple temperatures and turbulent kinetic energy (TKE) computed from wind speeds collected from a small experimental prescribed burn at the Silas Little Experimental Forest in New Jersey, USA. The primary objective of this project was to explore the potential for using thermocouple temperatures as predictors for estimating the TKE produced by a wildland fire. Machine learning models, including Deep Neural Networks, Random Forest Regressor, Gradient Boosting, and Gaussian Process Regressor, are employed to assess the potential for thermocouple temperature perturbations to predict TKE values. Data visualization and correlation analyses reveal patterns and relationships between thermocouple temperatures and TKE, providing insight into the underlying dynamics. The project achieves high accuracy in predicting TKE by employing various machine learning models despite a weak correlation between the predictors and the target variable. The results demonstrate significant success, particularly from regression models, in accurately estimating the TKE. The research findings contribute to fire behavior and smoke modeling science, emphasizing the importance of incorporating machine learning approaches and identifying complex relationships between fine-scale fire behavior and turbulence. Accurate TKE estimation using thermocouple temperatures allows for the refinement of models that can inform decision-making in fire management strategies, facilitate effective risk mitigation, and optimize fire management efforts. This project highlights the valuable role of machine learning techniques in analyzing wildland fire data, showcasing their potential to advance fire research and management practices.

Jiahua Jiang, Carmeliza Navasca

We address the optimal control problems arising from partial differential equations with large discrete dimensional control systems. To obtain reduced order models, we find basis elements from the canonical polyadic (CP) decomposition. Tensor datasets are from snapshots of the large models. Our method to reduce the control system is to use dimensionality reduction approaches through sparse optimization and flexible hybrid methods is to obtain low rank CP tensor basis elements. The reduced optimal control problem leads to reduced state-dependent Riccati Equations which can be solved efficiently.

Dipak Dulal, Joseph J. Charney, Michael R. Gallagher et al.

This study explores the potential for predicting turbulent kinetic energy (TKE) from more readily acquired temperature data using temperature profiles and turbulence data collected concurrently at 10 Hz during a small experimental prescribed burn in the New Jersey Pine Barrens. Machine learning models, including Deep Neural Networks, Random Forest Regressor, Gradient Boosting, and Gaussian Process Regressor, were employed to assess the potential to predict TKE from temperature perturbations and explore temporal and spatial dynamics of correlations. Data visualization and correlation analyses revealed patterns and relationships between thermocouple temperatures and TKE, providing insight into the underlying dynamics. More accurate predictions of TKE were achieved by employing various machine learning models despite a weak correlation between the predictors and the target variable. The results demonstrate significant success, particularly from regression models, in accurately predicting the TKE. The findings of this study demonstrate a novel numerical approach to identifying new relationships between temperature and airflow processes in and around the fire environment. These relationships can help refine our understanding of combustion environment processes and the coupling and decoupling of fire environment processes necessary for improving fire operations strategy and fire and smoke model predictions. The findings of this study additionally highlight the valuable role of machine learning techniques in analyzing the complex large datasets of the fire environments, showcasing their potential to advance fire research and management practices.

Ramin Goudarzi Karim, Carmeliza Navasca, Da Yan

We propose a block coordinate descent type algorithm for estimating the rank of a given tensor. In addition, the algorithm provides the canonical polyadic decomposition of a tensor. In order to estimate the tensor rank we use sparse optimization method using $\ell_1$ norm. The algorithm is implemented on single moving object videos and color images for approximating the rank.

Xiaofei Wang, Carmeliza Navasca, Stefan Kindermann

In this paper, we discuss the acceleration of the regularized alternating least square (RALS) algorithm for tensor approximation. We propose a fast iterative method using a Aitken-Stefensen like updates for the regularized algorithm. Through numerical experiments, the fast algorithm demonstrate a faster convergence rate for the accelerated version in comparison to both the standard and regularized alternating least squares algorithms. In addition, we analyze the global convergence based on the Kurdyka- Lojasiewicz inequality as well as show that the RALS algorithm has a linear local convergence rate.