Spring 2022

Speaker: Dr. Fei Lu from Johns Hopkins University

Date and Time: Friday, October 22, 2021, 11am-12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: A statistical learning perspective of model reduction

Abstract: Stochastic closure models aim to make timely predictions with uncertainty quantified. We discuss the statistical learning framework that achieves this goal by accounting for the effects of the unresolved scales. A fundamental idea is the approximation of the discrete-time flow map for the dynamics of the resolved variables. The flow map is an infinite-dimensional functional of the history of resolved scales, as suggested by the Mori-Zwanzig formalism. Thus its inference faces the curse of dimensionality. We investigate a semi-parametric approach that derives parametric models from numerical approximations of the full model. We show that this approach leads to effective reduced models for deterministic and stochastic PDEs, such as the Kuramoto-Sivashisky equation and the viscous stochastic Burgers equations. In particular, we highlight the shift from the classical numerical methods (such as the nonlinear Galerkin method) to statistical learning, and discuss space-time reduction.

Speaker: Dr. Xuerong Wen from Missouri University of Science and Technology

Date and Time: Friday, October 15, 2021, 11:15am-12:15pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Variable dependent partial dimension reduction

Abstract: Sufficient dimension reduction reduces the dimension of a regression model without loss of information by replacing the original predictor with its lower-dimensional linear combinations. Partial (sufficient) dimension reduction arises when the predictors naturally fall into two sets X and W, and pursues a partial dimension reduction of X. Though partial dimension reduction is a very general problem, only very few research results are available when W is continuous. To the best of our knowledge, these methods generally perform poorly when X and $\W$ are related, furthermore, none can deal with the situation where the reduced lower-dimensional subspace of $\X$ varies with W. To address such issue, we in this paper propose a novel variable dependent partial dimension reduction framework and adapt classical sufficient dimension reduction methods into this general paradigm. The asymptotic consistency of our method is investigated. Extensive numerical studies and real data analysis show that our Variable Dependent Partial Dimension Reduction method has superior performance comparing to the existing methods.

Speaker: Dr. Toan Nguyen from the Penn State University

Date and Time: Thursday, October 7, 2021, 11am-12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Landau damping in plasma physics

Abstract: After a quick overview on the classical notion of Landau damping discovered by Landau in 1946, the colloquium will highlight recent mathematical advances on understanding the damping and the large time behavior of a plasma modeled by Vlasov-Poisson and Vlasov-Poisson-Landau systems, including (1) an elementary proof of nonlinear Landau damping for analytic and Gevrey data (joint work with E. Grenier from ENS Lyon and I. Rodnianski from Princeton) and (2) nonlinear Landau damping in the weakly collisional regime for a threshold of initial data with Sobolev regularity (joint work with S. Chaturvedi and J. Luk from Stanford).

Speaker: Dr. Haiying Wang from the University of Connecticut

Date and Time: Friday, October 1, 2021, 11am-12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data

Abstract: We investigate the issue of parameter estimation with nonuniform negative sampling for imbalanced data. We first prove that, with imbalanced data, the available information about unknown parameters is only tied to the relatively small number of positive instances, which justifies the usage of negative sampling. However, if the negative instances are subsampled to the same level of the positive cases, there is information loss. To maintain more information, we derive the asymptotic distribution of a general inverse probability weighted (IPW) estimator and obtain the optimal sampling probability that minimizes its variance. To further improve the estimation efficiency over the IPW method, we propose a likelihood-based estimator by correcting log odds for the sampled data and prove that the improved estimator has the smallest asymptotic variance among a large class of estimators. It is also more robust to pilot misspecification. We validate our approach on simulated data as well as a real click-through rate dataset with more than 0.3 trillion instances, collected over a period of a month. Both theoretical and empirical results demonstrate the effectiveness of our method.

Speaker: Dr. Yanming Li from the University of Kansas Medical Center

Date and Time: Thursday, September 23, 2021, 2pm-3pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Disease Prediction by Detecting and Integrating Connectomic Networks and Marginally Weak Signals

Abstract: Many contemporary studies use individual genomic or imaging profiles for early prediction of cancer or neuropsychological outcomes, such as cancer subtypes and Alzheimer’s disease stages. Current approaches base prediction using only biomarkers that are strongly correlated with the disease outcome. However, the connection structures of the genome and the brain (e.g. gene pathways or brain networks) are ignored in such marginal approaches. Many genetic and imaging markers, despite having marginally weak effects, may exude strong predictive effects once considered together with their connected biomarkers. Weak signals are not detectable by themselves because of their small marginal effect sizes. To find weak signals, the inter-feature connection (or network) structure of the genome or brain (which is termed the genome or brain connectome) has to be explored first. However, given the ultrahigh-dimensional characteristic of genomic/neuroimaging profiles, identifying the whole genome/brain connectome is computational prohibitive. This is also an impediment for detecting weak signals. In this work, we hypothesize that a large portion of the predictiveness of diseases attributes to inter-marker connections as well as marginally weak signals. By detecting and integrating them, accuracy of prediction can be significantly improved. We develop novel statistical/machine-learning algorithms for detecting connectomic genetic or brain networks for cancer or AD related outcome prediction. The proposed methods can be extended to detecting connectomic profiles for numerous outcome types using pan-cancer, pan-omic and multi-modality neuroimaging data. The identified network or pathway signatures will also enhance our understanding about the underlying mechanisms of disease development and progression.

Speaker: Dr. Felix Xiaofeng Ye from SUNY Albany

Date and Time: Friday, September 3, 2021, 11am-12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Nonlinear model reduction for slow-fast stochastic systems near unknown invariant manifolds

Abstract: We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is ecient in that it exploits of the low dimension of the invariant manifold, and takes time steps of size dependent on the regularity of the selective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.

Speaker: Dr. Yimei Li from St. Jude Children’s Research Hospital

Date and Time: Friday, April 30, 2021, 11am – 12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Group sequential design for historical control trials using error spending functions

Abstract: Randomized clinical trials (RCTs) are considered the gold standard for clinical trials comparing treatment groups. However, historical control trials (HCTs) are an alternative to RCTs if randomization is not feasible because of ethical concerns, patient preference, limited patient populations, or regulatory acceptability. The major benefit of HCTs is that all patients can receive the new treatment with historical data providing the information for the control arm. Therefore, HCTs are useful for studies with limited patient populations. Group sequential designs using Lan-DeMets error spending functions are proposed for historical control trials with time-to-event endpoints. Both O’Brien–Fleming and Gamma family types of sequential decision boundaries are derived based on sequential log-rank tests, which follow a Brownian motion in a transformed information time. Simulation results show that the proposed group sequential designs using historical controls preserve the overall type I error and power. An example is provided to show how to use the method to design a Children Oncology Group High Grade Glioma trial.

Speaker: Dr. Andreas Artemiou from Cardiff University

Date and Time: Friday, April 16, 2021, 11am – 12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: SVM-based real time sufficient dimension reduction

Abstract: We discuss in this talk one of the first efforts for real-time sufficient dimension reduction. Support Vector Machine (SVM) based sufficient dimension reduction algorithms were proposed the last decade to provide a unified framework for linear and nonlinear sufficient dimension reduction. We present our idea of using a variant of the classic SVM algorithm known as Least Squares SVM (LSSVM) to achieve real time sufficient dimension reduction. We demonstrate the computational advantages as well as the computational efficiency of our algorithm through simulated and real data experiments. This is joint work with my collaborators Yuexiao Dong (Temple University) and Seung Jun Shin (Korea University).

Speaker: Dr. Runhuan Feng from the University of Illinois at Urbana-Champaign

Date and Time: Friday, April 9, 2021, 12pm – 1pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Modeling Financial Market Movement with Winning Streaks: Sticky Maximum Process

Abstract: Winning streaks appear frequently in all financial markets including equity, commodity, foreign exchange, real estate, etc. Most stochastic process models for financial market data in the current literature focus on stylized facts such as fat-tailedness relative to normality, volatility clustering, mean reversion. However, none of existing financial models captures the pervasive feature of persistent extremes: financial indices frequently report record highs or lows in concentrated periods of time. In this paper, we apply the technique of time change with local time to capture the market anomaly of persistent extremes. The new model which is driven by a sticky processes with moving boundaries { running maxima enables us to measure and assess the impact of persistent extremes on financial derivatives. Despite the time change construction, option prices are still solvable analytically. In addition, the model in this paper reveals a paradox that investors who bet on the growth of financial market may be worse off with pervasive winning streaks in the market.

Speaker: Dr. Yanzhi Zhang from Missouri University of Science and Technology

Date and Time: Friday, April 2, 2021, 11am – 12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Numerical Methods for Nonlocal Problems with the Fractional Laplacian

Abstract: Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, its nonlocality introduces considerable challenges in both mathematical analysis and numerical simulations. So far, numerical methods for the fractional Laplacian still remain limited. It is well known that the fractional Laplacian can be defined either in a pseudo-differential form via the Fourier transforms or in a hypersingular integral form. In this talk, I will discuss three different groups of numerical methods to discretize the fractional Laplacian. In the first group, we introduce the Fourier pseudospcetral methods based on the pseudo-differential form of the fractional Laplacian. The second group is operator factorization methods based on the hypersingular integral definition.  In the third group, we combine both pseudo-differential and hypersingular integral forms of the fractional Laplacian and introduce meshfree methods with radial basis functions. The properties of these methods will be discussed, and some applications of nonlocal problems with the fractional Laplacian will also be demonstrated.