Title: Some Modified Matrix Eigenvalue Problems

Abstract: 

This is the title of a well-known numerical linear algebra

survey article by Gene Golub published in 1973. The article

Title: Some Modified Matrix Eigenvalue Problems

Abstract: 

This is the title of a well-known numerical linear algebra

survey article by Gene Golub published in 1973. The article

Title: Generating Representative Samples: Neural Networks and More

Abstract: Approximating a probability distribution using a discrete set of points is a fundamental task in modern scientific computation, with applications in uncertainty quantification among other things. We discuss recent advances in this area, including the use of Stein discrepancies and various optimization techniques. In particular, we introduce Stein-Message-Passing Monte Carlo (Stein-MPMC), an extension of the original Message-Passing Monte Carlo model and the first machine-learning algorithm for generating low-discrepancy (space-filling) point sets. Additionally, we present a generalized Subset Selection algorithm, a simpler yet highly effective optimization method.

Title: Generating Representative Samples: Neural Networks and More

Abstract: Approximating a probability distribution using a discrete set of points is a fundamental task in modern scientific computation, with applications in uncertainty quantification among other things. We discuss recent advances in this area, including the use of Stein discrepancies and various optimization techniques. In particular, we introduce Stein-Message-Passing Monte Carlo (Stein-MPMC), an extension of the original Message-Passing Monte Carlo model and the first machine-learning algorithm for generating low-discrepancy (space-filling) point sets. Additionally, we present a generalized Subset Selection algorithm, a simpler yet highly effective optimization method.

Title: Log-Sum Regularized Kaczmarz Algorithms for High-Order Tensor Recovery

Abstract: Sparse and low rank tensor recovery has emerged as a significant area of research with applications in many fields such as computer vision. However, minimizing the $\ell_0$-norm of a vector or the rank of a matrix is NP-hard. Instead, their convex relaxed versions are typically adopted in practice due to the computational efficiency, e.g., log-sum penalty. In this presentation, we propose novel log-sum regularized Kaczmarz algorithms for recovering high-order tensors with either sparse or low-rank structures. We present block variants along with convergence analysis of the proposed algorithms. Numerical experiments on synthetic and real-world data sets demonstrate the effectiveness of the proposed methods.

Title: Log-Sum Regularized Kaczmarz Algorithms for High-Order Tensor Recovery

Abstract: Sparse and low rank tensor recovery has emerged as a significant area of research with applications in many fields such as computer vision. However, minimizing the $\ell_0$-norm of a vector or the rank of a matrix is NP-hard. Instead, their convex relaxed versions are typically adopted in practice due to the computational efficiency, e.g., log-sum penalty. In this presentation, we propose novel log-sum regularized Kaczmarz algorithms for recovering high-order tensors with either sparse or low-rank structures. We present block variants along with convergence analysis of the proposed algorithms. Numerical experiments on synthetic and real-world data sets demonstrate the effectiveness of the proposed methods.

Title: Understanding neutrophil dynamics during COVID-19 infection

Abstract: 

Infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) results in varied clinical outcomes, with virus-induced chronic inflammation and tissue injury being associated with enhanced disease pathogenesis. To determine the role of tissue damage on immune populations recruitment and function a mathematical model of innate immunity following SARS-CoV-2 infection has been proposed. The model was fitted to published longitudinal immune marker data from patients with mild and severe COVID-19 disease and key parameters were estimated for each clinical outcome. Analytical, bifurcation and numerical investigations were conducted to determine the effect of parameters and initial conditions on long-term dynamics. The results were used to suggest changes needed to achieve immune resolution.

Title: Understanding neutrophil dynamics during COVID-19 infection

Abstract: 

Infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) results in varied clinical outcomes, with virus-induced chronic inflammation and tissue injury being associated with enhanced disease pathogenesis. To determine the role of tissue damage on immune populations recruitment and function a mathematical model of innate immunity following SARS-CoV-2 infection has been proposed. The model was fitted to published longitudinal immune marker data from patients with mild and severe COVID-19 disease and key parameters were estimated for each clinical outcome. Analytical, bifurcation and numerical investigations were conducted to determine the effect of parameters and initial conditions on long-term dynamics. The results were used to suggest changes needed to achieve immune resolution.

Title: Forecasting patient-specific treatment response to neoadjuvant chemotherapy in triple-negative breast cancer via MRI-based digital twins

Abstract: Patients with locally advanced, triple-negative breast cancer (TNBC) typically receive neoadjuvant chemotherapy (NAT) to downstage the tumor and improve the outcome of subsequent breast conservation surgery. In this study, we integrated quantitative magnetic resonance imaging (MRI) data with biology-based mathematical modeling to address the currently unmet need for accurate prediction of TNBC response to NAT on an individual patient basis. Specifically, dynamic contrast-enhanced MRI and diffusion-weighted MRI was acquired in 56 patients before, after two, and after four cycles of Adriamycin/Cytoxan (A/C), and again after Taxol as part of the ARTEMIS (NCT02276443) trial. A biology-based mathematical model was established based on the reaction-diffusion equation to characterize the mobility of tumor cells, tumor proliferation, and treatment-induced cell death. Pre- and mid-treatment images were used for model calibration on a patient-specific basis. Two evaluation Frameworks were built: 1. using images acquired before and after two cycles of A/C for calibration and predicting tumor status after A/C, and 2. using images acquired before, after two cycles, and after four cycles of A/C for calibration and predicting response after NAT. For Framework 1, the Pearson correlation coefficients between the predicted and measured patient-specific, post-A/C changes in tumor cellularity and volume were 0.95 and 0.94, respectively. For Framework 2, the biologically-based model achieved an area under the receiver operator characteristic curve of 0.89 (sensitivity/specificity = 0.72/0.95) for differentiating pathological complete response (pCR) from non-pCR, which is statistically superior (P < 0.05) to the value of 0.78 (sensitivity/specificity = 0.72/0.79) achieved by the tumor volume measured after four cycles of A/C. Overall, our biology-based mathematical model successfully captured the patient-specific, spatiotemporal dynamics of TNBC response to NAT, providing highly accurate predictions of NAT response.  

Title: Is a Classification Procedure Good Enough?—A Goodness-of-Fit Assessment Tool for Classification Learning

Abstract: In recent years, many nontraditional classification methods, such as random forest, boosting, and neural network, have been widely used in applications. Their performance is typically measured in terms of classification accuracy. While the classification error rate and the like are important, they do not address a fundamental question: Is the classification method underfitted? To our best knowledge, there is no existing method that can assess the goodness of fit of a general classification procedure. Indeed, the lack of a parametric assumption makes it challenging to construct proper tests. To overcome this difficulty, we propose a methodology called BAGofT that splits the data into a training set and a validation set. First, the classification procedure to assess is applied to the training set, which is also used to adaptively find a data grouping that reveals the most severe regions of underfitting. Then, based on this grouping, we calculate a test statistic by comparing the estimated success probabilities and the actual observed responses from the validation set. The data splitting guarantees that the size of the test is controlled under the null hypothesis, and the power of the test goes to one as the sample size increases under the alternative hypothesis. For testing parametric classification models, the BAGofT has a broader scope than the existing methods since it is not restricted to specific parametric models (e.g., logistic regression). Extensive simulation studies show the utility of the BAGofT when assessing general classification procedures and its strengths over some existing methods when testing parametric classification models.