IITG Mathematics Seminar Series

Lecture Number: | 296 |

Title: | On short recurrence Krylov type methods for Fourier Galerkin Based Homogenization of Periodic Media |

Speaker: | Dr. Nachiketa Mishra, Airbus prized postdoc researcher |

Affiliation: | ICTS-TIFR Bangalore |

Date: | 7th March, 2017 (Tuesday) |

Time: | 4:30 PM |

Abstract: The first FFT-ased algorithm for numerical homogenization from high resolution images was proposed by Moulinec and Suquet in 1994 as an alternative to finite elements and twenty years later, it is still widely used in computational micromechanics of materials. The method is based on an iterative solution to an integral equation of the Lippmann-Schwinger type, whose kernel can be explicitly expressed in the Fourier domain. Only recently, it has been recognized that the algorithm has a variational structure arising from a Fourier Galerkin method. In this talk, I will show how this insight can be used to significantly improve the performance of the original Moulinec-Suquet solver. In particular, I will focus on (i) influence of Krylov subspace methods used to solve non-symmetric rank-deficient linear systems (ii) effects of numerical integration of the Galerkin weak form, and (iii) convergence of an aposteriori bound on thee solution during iterations.

Lecture Number: | 295 |

Title: | Multi-scale Classification using Localized Spatial Depth |

Speaker: | Dr. Anil K. Ghosh |

Affiliation: | Associate Professor, Theoretical Statistics and Mathematics Unit, ISI, Kolkata |

Date: | 24th February, 2017 (Friday) |

Time: | 4:00 PM |

Abstract: In this article, we develop and investigate a new classifier based on features extracted using spatial depth. Our construction is based on fitting a generalized additive model to posterior probabilities of different competing classes. To cope with possible multi-modal as well as non-elliptic nature of the population distribution, we also develop a localized version of spatial depth and use that with varying degrees of localization to build the classifier. Final classification is done by aggregating several posterior probability estimates, each of which is obtained using this localized spatial depth with a fixed scale of localization. The proposed classifier can be conveniently used even when the dimension of the data is larger than the sample size, and its good discriminatory power for such data has been established using theoretical as well as numerical results.

Lecture Number: | 294 |

Title: | Computational Mathematics-Some Challenges |

Speaker: | Prof. M. K. Kadalbajoo, Distinguished Professor, LNMIIT Jaipur (Retd. Professor, Department of Mathematics, IIT Kanpur) |

Affiliation: | LNMIIT Jaipur |

Date: | 20th February, 2017 (Monday) |

Time: | 4:00 PM |

Abstract: We shall describe and discuss but with elementary and rudimentary details the two important subfields of Computational Mathematics, namely, the "Numerical Linear Algebra" and the "Computational Fluid Dynamics" and also state several challenging problems one faces in these two fundamental and important areas.

Lecture Number: | 293 |

Title: | Hyperbolic Geometry and Chaos in the Complex Plane |

Speaker: | Prof. Mahan Mj |

Affiliation: | School of Mathematical Sciences, TIFR Mumbai |

Date: | 17th February, 2017 (Friday) at 4:15 PM |

Venue: | Lecture Hall 1 |

Abstract: Instances of hyperbolic geometry come up in nature whenever a system starts developing fast interconnections. Examples include trees, the human brain and the internet. A tell-tale signature is the existence of a fractal in one dimension less, e.g. the surfaces of trees and brains in the above examples.

After dealing with the above examples, we shall discuss a special case where the fractals emerge in the complex plane as a result of symmetries of hyperbolic 3-space. These symmetries act on the complex plane as well; however the dynamics being chaotic, it is hard to get a hold on them directly. Instead we go to hyperbolic geometry in 3 dimensions, set up a dictionary between the two and finally get a hold on the fractals in the complex plane through our study of hyperbolic geometry in 3 dimensions.

Lecture Number: | 292 |

Title: | Absence of the Riemann mapping theorem in higher dimensions |

Speaker: | Dr. Diganta Borah |

Affiliation: | IISER Pune |

Date: | 14th February, 2017 (Tuesday) |

Time: | 4:15 PM |

Abstract: The Riemann mapping theorem states that any proper simply connected domain in the complex plane is holomorphically equivalent to the unit disc. H. Poincare discovered that this theorem fails spectacularly in higher dimensions. We will discuss a proof of this surprising phenomenon. This will be an elementary talk and should be accessible to anyone with some familiarity with basic complex analysis in one variable.

Lecture Number: | 291 |

Title: | Studies of human microbiome and the role of statistics |

Speaker: | Dr. Siddhartha Mandal |

Affiliation: | Research Scientist, Public Health Foundation of India |

Date: | 25th January, 2017 (Wednesday) |

Time: | 4:00 PM |

Abstract: Human microbiome plays a crucial role in health and diseases with recent research unravelling the myriad ways in which these associations are manifested. While traditional microbiology focused on microbial pathogens and few beneficial bacteria, advances in high-throughput technologies have presented evidence of a broader spectrum of functions in relation to chronic diseases, brain function and neurodevelopmental outcomes. Understanding factors regulating our microbiota and the impact of microbiota on health requires appropriate statistical methodology. Microbial communities associated with the human body sites are complex communities, with unknown interactions and functions. In this talk, we shall discuss some of the statistical designs and methodologies that are useful to study questions related to microbiome. Few major questions of interest in studies of microbiome may be "How do microbial communities cluster between different groups ?" or "Which microbial taxa are differentially abundant between these groups?". Data generated from microbial surveys are relative abundances of microbial taxa (which are themselves members of a phylogenetic tree), while the actual abundances are unobservable quantities. This results in compositional data, where observations on each subject are multivariate vectors belonging to a simplex.

To answer the first question, we shall explore the Unifrac distance (Lozupone et al. 2005), a phylogenetic distance based metric that is used to separate microbial communities according to groups. Existing approaches to detect differentially abundant microbes either discount the underlying compositional structure in the microbiome data or use inappropriate probability distributions including the multinomial and Dirichlet-Multinomial that may potentially increase false discovery rate. For the second question, we introduced a novel statistical framework called Analysis of Composition of Microbiomes (ANCOM, Mandal et al. 2015) that accounts for the underlying compositional structure in the data, and, unlike existing approaches, can compare the composition of microbiomes across populations. ANCOM makes no distributional assumptions, and is sufficiently general to enable easy adjustment for covariates. ANCOM also scales well to compare samples involving thousands of taxa. We shall illustrate these methodologies using publicly available microbial datasets in the human gut. In addition, we shall explore some of the recent developments in microbiome research and how these may be relevant in the case of Indian populations and related health problems.

Lecture Number: | 290 |

Title: | Control of compressible Navier-Stokes system |

Speaker: | Dr. Debanjana Mitra |

Affiliation: | Post Doctoral Fellow, Department of Mathematics, Virginia Tech, Blacksburg, USA |

Date: | 23rd January, 2017 (Monday) |

Time: | 4:00 PM |

Abstract: We consider the one dimensional compressible Navier-Stokes system near a constant steady state with the periodic boundary conditions. The linearized system around the constant steady state is a hyperbolic-parabolic coupled system. We discuss some of the properties of the linearized system and its spectrum. Next we study some controllability results of the system.

Lecture Number: | 289 |

Title: | Numerical Solution for a Transient Incompressible Viscous Fluid Flow in a Cavity |

Speaker: | Prof. Dambaru Bhatta |

Affiliation: | Professor of Mathematics, School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas |

Date: | 4th Jauary, 2017 (Wednesday) |

Time: | 11:30 AM |

Abstract: We consider time dependent flow for a Newtonian, viscous and incompressible fluid in a rectangular cavity. The governing system consists of the conservation of mass and momentum equations. The momentum equations considered here are linearization of the Navier-Stokes Equations. We derive the weak formulation for the governing system. Using the Galerkin method, we obtain matrix form at element level. The elements considered here are of Taylor-Hood type elements. Time dependency part is solved by using the Crank-Nicolson method. Numerical results for the velocity for a square cavity are presented.