My research interests lie in the intersection of several areas -
applied and computational stochastic processes, fluid dynamics,
homogenization for partial differential equations,
numerical methods for time-dependent PDEs, mathematical biology, and data science.
My projects
typically require a combination of various tools from data science, applied probability, and
computational mathematics. In particular, I am interested in aplying various ideas from data
science to computational fluid dynamics.
Other research directions include
mathematical biology, multi-agent systems, derivation of effective coarse equations for
collective behavior, kinetic theory,
underground pollution transport.
PDF
slides with Some Examples, Discussion, and Derivations on Stochastic Mode
Reduction
There has been a recent surge of interest in using Neural Networks (NNs) to analyze various datasets. I am primarily interested in understanding how NNs can be used in time-series analysis and model reduction.
NNs for Time-series analysis. My group is working on developing practical Neural Networks models for predicting observed (e.g. stock prices) and computer-generated time-series. I am interested in applications where computer-generated time-series are the result of numerical integration of partial differential equations representing complex spatio-temporal phenomena. In this application we use NNs as predictive models for future times. Currently, we're trying to understand which types of networks are suitable for predicting time-series and what are the main limiting factors for the accuracy of predictions.
NNs for Inverse problems. We would like to utilize machine learning and deep learning tools in combination with computational applied mathematics to address important problems related to inverse problems in partial differential equations. In particular, I'm currently working on developing modern computational tools to recover bottom topography from observations of the surface waves (water height) and velocity. Currently, we're investigating the perfomance of neural networks in recovering the leading-order approximation of the bottom topography in fluid flows. This is a very important practical problem related to recovering the bottom shape of riverbeds from suraface observations.
NNs for model reduction. One of my primarily interests in reducing the computational complexity in numerical simulations of partial differential equations (PDEs). Direct numerical simulations of PDEs often invovle a very large number of degrees of freedom which makes such simulations very computationally intensive. Therefore, in many practical situations it is desireable to obtain a coarse (or reduced) model which approximates the numerical behavior of the original PDE in some suitable metric. To this end, we use neural Networks to represent unresolved degrees of freedom in such reduced models. This approach allows to reduce the dimensionality of the discretized problem and perform much faster numerical integration of the model.
Currently, my group is applying various ideas from Deep Learning for several time-dependent problems, including turbulent fluid dynamics and reactive transport. In a recent work we use the generative adversarial network (GAN) to represent the subgrid-scale processes in a prototype model of fluid turbulence.
Agent (or particle) based modeling has been used in many areas of science and engineering. Currently, I am working on two projects in this area.
Mathemaical Biology. First project concerns modeling, simulation, and analysis of multi-agent systems representing collective motion of myxobacteria. Myxobacteria exhibits a complex combination of interaction rules, including slime following, regime switching, aligning, etc. One of the difficulties of this project is that the regime switching exhibits waiting times which do not follow exponential distribution. Thus, an extension of kinetic theory for such processes is necessary.
Information flow in parallel computing. Second project addresses developing a model for information flow during parallelcomputing and and subsequent analysis of this model. Continuous-time markov chains can be used as modeling framework for this problem. Next, we would like to derive an effective mean-field model for information propagation during different stages.
Projects in this area consist of two stages. In the first stage, it is necessary to develop an appropriate microscopic description of the model, including the mathematical description of interactions between agents. The second stage involves mathematical analysis of the microscopic model and derivation of a coarse description on a higher level in terms of, for instance, probability density. Numerical simulations are used to verify the underlying assumptions and demonstrate agreement between the microscopic and coarse descriptions.
These are typical "applied-math" projects which involve a combination of modeling and numerical and analytical tools from stochastic processes and kinetic theory. In addition, there are also questions from numerical analysis related to developing discretizations of PDEs for the evolution of the density. Such equations are, typically, conditionally hyperbolic conservation laws with a subtle balance between the advective and diffusive terms. Therefore, modern flux-correction techniques are required to develop accurate and efficient discretizations of these equations.
I am interested in developing efficient stochastic closures for sub-grid scale variables in finite-volume discretization of partial differential equations. Currently, I am working on applying this methodology to the Shallow-Water equations and advection-diffusion-reaction equations modeling reactive transport of pollution in the sub-surface.
The project is motivated by the necessity to simulate time-dependent partial differential equations on very large domains and for an extended period of time. Thefore, to achieve this goal, we consider how a fine-mesh discretization can be coarsened.
Typically, the mesh-size becomes large and the coarse discretization does not represent small (subgrid-) scales sufficiently accurately. Therefore, we need to derive additional terms which represent the effect of sub-grid scales in coarse models. These terms can be deterministic or stochastic.
I employ techniques from multiscale analysis, homogenization, empirical daa-driven approaches such as parameter estimation and deep learning.