In statistics and machine learning, logistic regression is a widely-used supervised learning technique primarily employed for binary classification tasks. When the number of observations greatly exceeds the number of predictor variables, we present a simple, randomized sampling-based algorithm for logistic regression problem that guarantees high-quality approximations to both the estimated probabilities and the overall discrepancy of the model. Our analysis builds upon two simple structural conditions that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized numerical linear algebra. We analyze the properties of estimated probabilities of logistic regression when leverage scores are used to sample observations, and prove that accurate approximations can be achieved with a sample whose size is much smaller than the total number of observations. To further validate our theoretical findings, we conduct comprehensive empirical evaluations. Overall, our work sheds light on the potential of using randomized sampling approaches to efficiently approximate the estimated probabilities in logistic regression, offering a practical and computationally efficient solution for large-scale datasets.

High-fidelity modeling is an essential design tool for many engineering applications. However, for complex systems, computational cost can be a limiting factor. Analyzing parameter sensitivity, uncertainty quantification, and design optimization require many model evaluations. Surrogate models are often used to develop the relationship between model parameters and quantities of interest. However, in the case of complex systems, surrogate models require several degrees of freedom and, thus, a large number of data points to determine the correct dependencies. For many applications, this may be prohibitively expensive. The reduction of computational requirements can be achieved by leveraging low-fidelity models. Low-fidelity models represent the system at a coarser resolution with the advantage of computational efficiency. Therefore, a bi-fidelity modeling paradigm, which augments the accuracy of a low-fidelity model in a computationally efficient manner by invoking limited runs of a high-fidelity model, can be leveraged to sufficiently balance the accuracy and computational requirements. In this work, a bi-fidelity weighted transfer learning method using neural networks was applied to a computational fluid dynamics heat transfer modeling problem. The transfer learning advantage was investigated as a function of hyperparameters. Our main finding is that the use of a bi-fidelity modeling paradigm achieves accuracy close to that of a high-fidelity Gaussian process model while significantly reducing computational cost. The bi-fidelity model achieves comparable performance with 90 high-fidelity samples-that is, 60% less than the samples needed to achieve similar accuracy without the use of bi-fidelity modeling,

Blind restoration of single-input/multiple-output images finds application in diverse fields including multi-sensor remote sensing, non-destructive testing and medical imaging. In this paper, an iterative forward propagation dynamic programming based approach is proposed for restoring images in a single-input/multiple-output multichannel framework. This algorithm exploits cross channel correlation for restoring images, leading to improved performance as compared to single channel restoration. The algorithm is applied to both synthetic as well as experimental data and initial results indicate the feasibility of the proposed algorithm.

The solution of partial differential equations (PDE) arises in a wide variety of engineering problems. Solutions to most practical problems use numerical analysis techniques such as finite-element or finite-difference methods. The drawbacks of these approaches include computational costs associated with the modeling of complex geometries. This paper proposes a finite-element neural network (FENN) obtained by embedding a finite-element model in a neural network architecture that enables fast and accurate solution of the forward problem. Results of applying the FENN to several simple electromagnetic forward and inverse problems are presented. Initial results indicate that the FENN performance as a forward model is comparable to that of the conventional finite-element method (FEM). The FENN can also be used in an iterative approach to solve inverse problems associated with the PDE. Results showing the ability of the FENN to solve the inverse problem given the measured signal are also presented. The parallel nature of the FENN also makes it an attractive solution for parallel implementation in hardware and software.

Neural networks, particularly the multilayer perceptron, have been used extensively in automated signal classification systems with classification accuracy as the figure of merit. Three important issues that can enhance the utility of these systems are (i) incremental learning, (ii) confidence or reliability measures and (iii) performance improvement through continual learning. This paper investigates these issues using a fuzzy ARTMAP network. A hypothesis testing based algorithm is developed for computing reliability measures, which are fed back to the network for retraining and performance improvement. Implementation results on ultrasonic data are presented.

Neural network based signal classification systems are being applied increasingly in nondestructive evaluation to solve the inverse problem. In general, two issues not usually addressed are (i) estimation of reliability measures of the network decision and (ii) ability of the network to learn and improve its performance with time. This paper presents a signal classification system using the fuzzy ARTMAP network. Fuzzy logic based reliability measures are developed for the fuzzy ARTMAP network and used as a feedback for retraining the network to improve its performance. The performance of the algorithm is demonstrated using ultrasonic data obtained from the inspection of welds in nuclear flow plant piping.