Publications and Preprints

Current Projects

Fast Solvers for Stokes Flow past Axisymmetric Geometries

At very small length scales (e.g. for describing motions of bacteria), the motion of a fluid is approximated by Stokes equation. Using the Greens’ function for this elliptic partial differential equation, we can write the solution of a boundary value problem as a boundary integral equation (BIE). It is well established that using BIE formulation, we can construct a more accurate solution faster compared to traditional finite difference, finite volume or finite element schemes.

Nevertheless, due to the relatively poor quality of two-dimensional singular quadratures, we require a high level of discretization to achieve high accuracy for general surface boundaries. In this project, we assume that these surfaces are axisymmetric, and leverage this rotational symmetry to decouple 2D BIE into a series of 1D BIEs. Using good quality quadrature schemes available in 1D, we can easily achieve high accuracy. In addition, the decoupling improves the overall computational complexity of our solver.

Scalable Solvers for Frictional Rigid Body Contact

Granular media is simply a collection of rigid particles. It is one of the most common form of materials in several fields of applications (e.g. molecular dynamics, terramechanics, robotics). However, unlike other states of matter, there is no simple constitutive law governing granular flow. For this reason, the discrete element method (DEM), which tracks each particle individually, remains the most accurate way to simulate the these materials.

Naturally, since each particles needs to be tracked separately, the computational cost of running DEM simulations for large-scale problems is very high. In this project, we aim to

Reduced-Order Model Generation with Tensor Factorizations

Learning tasks (e.g., optimization, control, uncertainty quantification) require a large number of data points before they are able to produce reliable predictions. In this setup, it is practically impossible to utilize the highly accurate outputs from physics-based simulations for modeling the system under consideration. We therefore seek reduced-order models (ROM) that can faithfully approximate the full-fidelity model to used in these learning tasks.

High-fidelity models typically generate a huge amount of high-dimensional data during the course of the simulation, so much so that even storing all of it becomes challenging. We propose to utilize tensor decompositions (e.g., tensor-train decomposition) to compress the data for future use. We also propose to use tensor factorizations to construct surrogate models out of the simulation data.

Bayesian Low-Rank Factorization of Matrices and Tensors

Low rank matrix and tensor completion has many different practical applications in fields such as relational data analysis and image/video recovery. Over the last few years, many optimization based approaches have been proposed to solve this problem efficiently. Recently, Bayesian inference based approaches have also gained popularity, since they allow for robust uncertainty quantification.

A key challenge in these Bayesian approaches for matrix and tensor factorizations is that the factorizations are not unique, and this non-identifiability issues often lead to symmetries in the posteriors. These symmetries degrade the performances of Markov-chain Monte-Carlo (MCMC) sampling algorithms. In this project, our goal is to develop theoretical results that would break these symmetries, and lead to better performance in Bayesian inference.

Quantum Computing for Fast Optimizers

Quantum computing promises to be a revolutionary paradigm of scientific computing. Quantum algorithms for certain classes of tasks have already been designed that can outstrip their classical counterparts, at least theoretically. One such class of problems is combinatorial optimization.

In this project, we propose to extend this approach for continuous optimization, which is omnipresent in many computational tasks (e.g., in the complementarity approach to frictional contact described above). A general framework for quantum continuous optimization has the potential to accelerate many simulations far beyond the capabilities of classical computing.