Theory and Algorithms Commons^™

A New Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

Logistic-T Multinomial Mixture Model For Clustering For Microbiome Data, Wenshu Dai, Yuan Fang, Sanjeena Subedi

Mathematics & Statistics Faculty Publications

The logistic-normal multinomial distribution has been used for modelling microbiome data obtained from high-throughput sequencing technologies, which are compositional in nature. A logistic-normal multinomial distribution is a hierarchical multinomial distribution that assumes the latent variable which are the additive log-ratio (ALR) transformed proportions in a multinomial distribution follows a Gaussian distribution. Model-based clustering algorithms have also been developed for clustering microbiome data based on the logistic-normal models. However, the Gaussian assumption may violated when the ALR transformed variable exhibit heavy-tailed distributions or has outliers. Our study introduces a novel mixture of logistic-t multinomial models that effectively address these challenges. Utilizing …

Utilizing Machine Learning Techniques For Computer-Aided Covid-19 Screening Based On Clinical Data, Honglun Xu, Andrews T. Anum, Michael Pokojovy, Sreenath Chalil Madathil, Yuxin Wen, Md Fashiar Rahman, Tzu-Liang (Bill) Tseng, Scott Moen, Eric Walser

Mathematics & Statistics Faculty Publications

The COVID-19 pandemic has highlighted the importance of rapid clinical decision-making to facilitate the efficient usage of healthcare resources. Over the past decade, machine learning (ML) has caused a tectonic shift in healthcare, empowering data-driven prediction and decision-making. Recent research demonstrates how ML was used to respond to the COVID-19 pandemic. This paper puts forth new computer-aided COVID-19 disease screening techniques using six classes of ML algorithms (including penalized logistic regression, random forest, artificial neural networks, and support vector machines) and evaluates their performance when applied to a real-world clinical dataset containing patients’ demographic information and vital indices (such as …

A Time-Domain Boundary Integral Equation For Moving Acoustic Sources In Uniform Flow And Its Solution By An Advanced Time Propagation Approach, Fang Q. Hu, Douglas M. Nark

Mathematics & Statistics Faculty Publications

This paper presents a time-domain boundary integral equation (TDBIE) formulation for predicting acoustic scattering from moving sources in a uniform mean flow. This work is motivated by the increasing need for accurate aeroacoustic modeling of modern aircraft configurations, including VTOL and eVTOL systems with rotating components. A key challenge in time-domain scattering simulations with moving sources is the determination of retarded time for a given observer time, which involves solving an implicit equation at each time step. This can be computationally costly, particularly for numerical solution of the TDBIE where every surface element on the scattering body acts as an …

Hyperparameter Estimation For Sparse Bayesian Learning Models, Feng Yu, Lixin Shen, Guohui Song

Mathematics & Statistics Faculty Publications

Sparse Bayesian learning (SBL) models are extensively used in signal processing and machine learning for promoting sparsity through hierarchical priors. The hyperparameters in SBL models are crucial for the model’s performance, but they are often difficult to estimate due to the nonconvexity and the high-dimensionality of the associated objective function. This paper presents a comprehensive framework for hyperparameter estimation in SBL models, encompassing well-known algorithms such as the expectation-maximization, MacKay, and convex bounding algorithms. These algorithms are cohesively interpreted within an alternating minimization and linearization (AML) paradigm, distinguished by their unique linearized surrogate functions. Additionally, a novel algorithm within the …

Machine-Learning-Enabled Diagnostics With Improved Visualization Of Disease Lesions In Chest X-Ray Images, Md. Fashiar Rahman, Tzu-Liang (Bill) Tseng, Michael Pokojovy, Peter Mccaffrey, Eric Walser, Scott Moen, Alex Vo, Johnny C. Ho

Mathematics & Statistics Faculty Publications

The class activation map (CAM) represents the neural-network-derived region of interest, which can help clarify the mechanism of the convolutional neural network’s determination of any class of interest. In medical imaging, it can help medical practitioners diagnose diseases like COVID-19 or pneumonia by highlighting the suspicious regions in Computational Tomography (CT) or chest X-ray (CXR) film. Many contemporary deep learning techniques only focus on COVID-19 classification tasks using CXRs, while few attempt to make it explainable with a saliency map. To fill this research gap, we first propose a VGG-16-architecture-based deep learning approach in combination with image enhancement, segmentation-based region …

Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan

Mathematics & Statistics Faculty Publications

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …

Generalized Sparse Bayesian Learning And Application To Image Reconstruction, Jan Glaubitz, Anne Gelb, Guohui Song

Mathematics & Statistics Faculty Publications

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are …

Fast Multiscale Functional Estimation In Optimal Emg Placement For Robotic Prosthesis Controllers, Jin Ren, Guohui Song, Lucia Tabacu, Yuesheng Xu

Mathematics & Statistics Faculty Publications

Electromyogram (EMG) signals play a significant role in decoding muscle contraction information for robotic hand prosthesis controllers. Widely applied decoders require a large amount of EMG signals sensors, resulting in complicated calculations and unsatisfactory predictions. By the biomechanical process of single degree-of-freedom human hand movements, only several EMG signals are essential for accurate predictions. Recently, a novel predictor of hand movements adopted a multistage sequential adaptive functional estimation (SAFE) method based on the historical functional linear model (FLM) to select important EMG signals and provide precise projections.

However, SAFE repeatedly performs matrix-vector multiplications with a dense representation matrix of the …

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

An Assessment Of Pet Dose Reduction With Penalized Likelihood Image Reconstruction Using A Computationally Efficient Model Observer, Howard C. Gifford, C. Ross Schmidtlein, Andrzej Krol, Yuesheng Xu

Mathematics & Statistics Faculty Publications

Developing PET reconstruction algorithms with improved low-count capabilities may provide a timely and cost- effective means of reducing radiation dose in promising clinical applications such as immuno-PET that require long-lived radiotracers. For many PET clinics, the reconstruction protocol consists of postsmoothed ordered-sets expectation-maximization (OSEM) reconstruction, but penalized likelihood methods based on total-variation (TV) regularization could substantially reduce dose. We performed a task-based comparison of postsmoothed OSEM and higher-order TV (HOTV) reconstructions using simulated images of a contrast-detail phantom. An anthropomorphic visual-search model observer read the images in a location-known receiver operating characteristic (ROC) format. Acquisition counts, target uptake, and target …

On Hybrid Temporal Basis Functions For Stable Numerical Solution Of Time Domain Boundary Integral Equations, Fang Q. Hu

Mathematics & Statistics Faculty Publications

Problems in unsteady aerodynamics and aeroacoustics can sometimes be formulated as integral equations, such as the boundary integral equations. Numerical discretization of integral equations in the time domain often leads to so-called March-On-in-Time (MOT) schemes. In the literature, the temporal basis functions used in MOT schemes have been largely limited to low-order shifted Lagrange basis functions. In order to evaluate the accuracy and effectiveness of the temporal basis functions, a Fourier analysis of the temporal interpolation schemes is carried out. Based on the Fourier analysis, the spectral resolutions of various temporal basis functions are quantified. It is argued that hybrid …