Abstract: In this seminar, I will be presenting high-order accurate entropy-stable ADER (Arbitrary high-order DERivative) predictor–corrector numerical schemes as an efficient numerical method for solving Hyperbolic Conservation laws. Traditional high-order methods such as WENO, TVD, and RKDG typically rely on multi-stage Runge-Kutta timestepping to maintain the temporal accuracy. However, these approaches become inefficient beyond the third order because of the Butcher barriers and the high memory traffic they impose on modern CPU and GPU hardware. ADER methods overcome these limitations using a predictor–corrector strategy that achieves arbitrarily high-order accuracy within a single-step update in both space and time, making them well-suited for parallel architectures.

Moreover, for a general nonlinear hyperbolic system, the solutions may break down in a finite amount of time, leading to infinitely many weak solutions. While conservation laws are necessary, they do not guarantee physical admissibility. Entropy stability is essential to enforce the second law of thermodynamics and select the physically admissible weak solutions at shocks. In this work, we construct high-order accurate, robust, and efficient entropy stable ADER schemes by combining entropy-conservative fluxes with suitable dissipation and limiter corrections inside the ADER framework. The designed numerical schemes are entropy stable and show high-order convergence, sharp shock capturing and robustness over various multidimensional test problems.

Abstract: In this talk, I will discuss the reduced order modeling for parametric PDE eigenvalue problems and briefly discuss our contributions in this direction. First, I will discuss the reduced order modeling for PDE eigenvalue problems in general setup. I will discuss how to choose the snapshots for finding eigenvalues and eigenvectors in reduced space. The success of the projection-based order modeling lies on the assumption that the problem is affine parameter dependent. For non-affine parameter dependent problem, we have proposed data-driven model. So, I will discuss the data-driven model for parametric eigenvalue problems using Gaussian Process regression (GPR). Then, I will talk about image processing problems such as image inpainting and segmentation. I will discuss an inpainting model and present some theoretical results related to this model. Some numerical results will be presented to demonstrate the improved performance of our model.

Abstract: We investigate the average number of lattice points within a ball where the lattice is chosen at random from the set of unit determinant ideal or modules lattices of some cyclotomic number field. The goal is to consider the space of such lattice as a probabilistic space and then study the distribution of lattice point counts. This is inspired by the connections of this problem to lattice-based cryptography and sphere packings in a high dimensional Euclidean space. Based on joint work with Vlad Serban, Maryna Viazovska, Ilaria Viglino.

Abstract: This talk focuses on my recent research on grouped multiple hypothesis testing inan online setting. Classical multiple testing procedures are offline in nature, meaning thatthe entire collection of hypotheses and corresponding test statistics is available before thetesting procedure begins. This setting allows for efficient use of available resources, such as the overall error budget and auxiliary structural information about the hypotheses, leading to procedures with high statistical power while maintaining control of a global error measure.

In contrast, online multiple testing procedures are a relatively recent development in theliterature. In the online framework, hypotheses arrive sequentially over time, and decisionsmust be made in real time based only on past information, before future test statistics areobserved. The lack of knowledge about future test statistics makes the task of controlling an overall error measure substantially more challenging than in the offline setting.

The talk introduces the ‘Grouped Online Testing Algorithm (GOTA)’ , which integrates ideasfrom both online and offline multiple testing to address settings in which hypotheses arrivein groups over a potentially infinite sequence. Unlike most existing multiple testingprocedures that rely on p-values, GOTA is built using the local false discovery rate as itsfundamental building block. I will discuss the theoretical properties of the algorithm,including its guarantees for controlling an overall error measure, as well as its practicalperformance. Simulation studies demonstrate that the proposed method achieves substantially higher power than a comparable p-value–based procedure.

Given the current lack of multiple testing methods tailored to such grouped online settings,this work aims to fill an important methodological gap. The talk will also briefly reviewfoundational concepts in multiple hypothesis testing and highlight my related recentresearch. No prior background in multiple testing is assumed, and the talk is intended to be accessible to everyone interested.

Abstract: The study of statistics of random permutations is arguably the earliest result in probability. These statistics bring out deep connections with fields like combinatorics, number theory, and representation theory. The probability that a uniform random permutation has $k$ orbits/cycles is log-concave (in $k$). In fact, it was observed by Levy that the number of orbits of a random permutation has the same distribution as the sum of independent Bernoullis. This allows one to deduce a central limit theorem for the number of orbits of a uniform random permutation. The situation is more delicate for a random pair of commuting permutations. Consider a pair of commuting permutations drawn uniformly at random from the set of all commuting pairs of permutations. It was conjectured by (Nekrasov--Okunkov) Heim-Neuhauser that the probability it has $k$ orbits is (unimodal) log-concave. This problem remains widely open. In this talk, we will discuss some recent partial progress on this problem. In particular, we prove a CLT for the number of orbits of a random pair of commuting permutations.

Abstract: We describe a framework for Bayesian analysis of vector-valued time series of counts. The approach consists of a flexible level correlated model (LCM) framework for building hierarchical models that incorporate correlated latent level effects and temporal effects to model the multivariate data. This allows for faster computation than using the multivariate Poisson distribution, whose likelihood calculation can be slow as the vector dimension increases. The LCM framework is versatile and allows us to model many types of multivariate time series such as counts, positive-valued observations, etc. For count time series, this framework allows us to combine univariate distributions for counts (Poisson, negative binomial, ZIP, etc.) for each component series, while accounting for association among the components via an unobserved (latent) Gaussian random vector. We also allow for univariate autoregression (AR) or vector autoregression (VAR) evolution of the latent states. We employ the integrated nested Laplace approximation (INLA) setup for fast approximate Bayesian modeling via the R-INLA package, building custom functions to handle the VAR evolution. We illustrate our approach using intra-day financial data streams. We show an application to analyzing financial data streams. This flexible framework can be easily extended to other scenarios such as modeling multivariate positive-valued time series, with application in several domains including ecology, marketing, and transportation safety.