PhD Candidate in Mathematics
My research focuses on Langevin-based sampling algorithms for complex probability distributions on both unconstrained and constrained domains, with applications in Bayesian statistics and machine learning. I combine theoretical convergence analysis with numerical experiments to understand and evaluate their performance.
Seeking postdoctoral opportunities starting Summer 2027.
I study how Langevin dynamics can be designed and analysed to efficiently sample from complex distributions. My current work falls under the first two themes below; the third is a direction I want to develop.
Current research
When does adding a nonreversible drift make a Langevin sampler converge faster, and how can the gain be measured?
Overdamped Langevin dynamics is the starting point for many sampling algorithms. Adding a skew-symmetric drift leaves the target distribution unchanged but changes how the process moves through it. I study when this helps, using large deviations and nonasymptotic convergence bounds.
Current research
How can we combine penalization and nonreversible Langevin dynamics to efficiently sample from probability distributions supported on constrained sets?
Many models in statistics and machine learning restrict parameters to compact convex sets. We develop Langevin sampling algorithms that combine squared-distance penalties with nonreversible dynamics. Our research establishes convergence and accuracy guarantees for full-gradient and stochastic-gradient methods, accounting for both numerical discretization and the approximation introduced by penalization. We also investigate when nonreversibility can accelerate convergence, combining theoretical analysis with numerical experiments on constrained sampling problems.
Future direction
How can we generate samples more accurately and at lower computational cost?
Building on my research in Langevin dynamics, I plan to investigate how numerical discretization and the choice of sampling dynamics affect convergence and approximation error in diffusion-based generative models. I aim to combine theoretical analysis with numerical experiments to develop sampling algorithms that achieve a desired accuracy with less computation.
Interactive demo
Both panels sample the same stretched Gaussian distribution, starting from the same point and using the same random noise. The right panel adds the drift −J∇U, which leaves the target unchanged but makes the points circulate around it, so they settle much sooner. Drag the slider to change the strength of the drift.
Standard Langevin J = 0
Nonreversible Langevin J ≠ 0
Distance to the target log scale
t = 0.0
Papers and preprints, newest first. Each entry is marked Published, Accepted, Preprint or In preparation. A full list is on Google Scholar.
arXiv:2610.11139
Nonreversible anchored Langevin dynamics for target densities that may be non-differentiable: NALD on the whole Euclidean space, where the target may also be heavy-tailed, and NRALD on bounded domains with normal reflection. A circulation drift built from a possibly state-dependent, divergence-free skew-symmetric matrix field and a stream potential preserves the target without using derivatives of its density. Nonasymptotic convergence bounds, a large deviations analysis and asymptotic variance reduction show that both can converge faster than their reversible counterparts, and numerical experiments demonstrate their efficiency.
arXiv:2609.25381
Sampling on a compact convex set by combining a squared-distance penalty with a skew-symmetric perturbation of Langevin dynamics. The preprint gives nonasymptotic bounds relative to the penalized target: in total variation with full gradients, including nonconvex potentials, and in 2-Wasserstein distance with stochastic gradients. For a quadratic model, tuning the perturbation improves the sufficient number of iterations from linear to logarithmic in the curvature ratio.
arXiv:2503.19066
Large deviation principles with explicit rate functions for mirror, high-order and Hessian-free high-resolution Langevin dynamics. Comparing rate functions shows when these dynamics, and underdamped Langevin, accelerate overdamped Langevin. Experiments cover Bayesian linear and logistic regression, including real data.
International Journal of Analysis and Applications 19(1), 29–46
Estimates for risk-neutral prices of temperature-based weather derivatives when temperature follows a G-stochastic differential equation driven by G-Brownian motion.
@article{hussain2021weather,
title = {On Modelling and Pricing Weather Derivatives Driven by Nonlinear {B}rownian Motion},
author = {Hussain, Javed and Ali, Pervez},
journal = {International Journal of Analysis and Applications},
volume = {19},
number = {1},
pages = {29--46},
year = {2021}
}I am a Fulbright Scholar and a PhD candidate in the Department of Mathematics at Florida State University, where I am advised by Dr. Lingjiong Zhu. I develop and analyze sampling algorithms based on Langevin dynamics. My work spans reversible and nonreversible dynamics, as well as unconstrained and constrained sampling.
I am also a Lecturer in Mathematics at Sukkur IBA University in Pakistan, where I taught from 2017 to 2022 and completed my MS in Mathematics. I have been on study leave from Sukkur IBA since July 2022. At Florida State, I have served as vice president of the SIAM Student Chapter.
Outside of research, I play tennis, pickleball, and cricket, and never pass up a chance to try a new sport.
PhD in Mathematics
Florida State University, Tallahassee, Florida
Expected June 2027 · Advisor: Dr. Lingjiong Zhu
Dissertation: Non-Reversible Langevin Dynamics for Unconstrained and Constrained Sampling
MS in Mathematics
Sukkur IBA University, Pakistan
Probability, stochastic processes and mathematical finance
BS in Mathematics
University of Sindh, Jamshoro, Pakistan
I have taught undergraduate mathematics since 2017: first as a lecturer in Pakistan, and since 2022 as an instructor and teaching assistant at Florida State University.
| Course | Role | Terms |
|---|---|---|
| Calculus I | Instructor | Spring 2025, Summer 2025 |
| Calculus II | Instructor | Fall 2025, Spring 2026 |
| Calculus III | Instructor | Summer 2026, Fall 2026 |
| Calculus, Trigonometry, College Algebra | Teaching assistant | 2024 |
| Course | Role | Terms |
|---|---|---|
| Linear Algebra, Scientific Computing, Probability, Statistics, Inferential Statistics | Lecturer | Aug 2017 – Jul 2022 |
Also taught Calculus, Ordinary Differential Equations, Business Mathematics, Business Statistics and Actuarial Mathematics.
My CV lists my education, publications, talks, teaching and awards.
A current CV is available on request. Please email me.
Email is the best way to reach me.
pa22g [at] fsu [dot] edu
Department of Mathematics