Abstract: The lowest-order nonstandard finite element schemes namely Morley, discontinuous Galerkin, and 𝐶0-interior penalty methods are presented for numerical discretization of the biharmonic wave equation with clamped boundary conditions in a unified framework. The regularity results we presented ensure the boundedness of various norms for the different derivatives of the continuous solution required in the error analysis. Stability results are obtained for both semidiscrete and fully discrete (implicit and explicit) schemes. A novel Ritz projection operator is used to obtain the error estimates. The expected convergence rates are confirmed by the numerical results, highlighting the versatility of the methods.

Abstract:  The limiting spectral distribution (LSD) of high-dimensional Kendall's correlation matrix and tests of independence based on this matrix have been studied in the literature when observations are absolutely continuous with respect to the Lebesgue measure, and are independent and identically distributed. We investigate the LSD of this matrix under much weaker assumptions which accommodate discrete and/or non-identical observations, and also identify the limit distribution by using free probability. We also propose a graphical test of independence under these assumptions.

Abstract: Statistical test for high-dimensional means under missing observations seems to be very rare in the literature. We propose a new two-sample test for high-dimensional means based on independent observations with missing values. The critical region of the proposed test is based on a bootstrap estimate of the sample quantiles of the proposed test statistic. Unlike the existing tests, this test does not require any distributional assumptions or any particular correlation structure of the covariance matrices.  We establish the Gaussian approximation result for the proposed test statistic which is a non-trivial extension of the two-sample Gaussian approximation result with no missing values.  The rate of accuracy of the bootstrap approximation of the sample quantile of the proposed test statistic is also derived. This Gaussian approximation result and the accuracy of the bootstrap estimators together provide the theoretical guarantees on the size and power of the proposed test.

Annual Progress seminar

Wed, 28 Feb 11:00 am -12:00 pm

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Venue: Ramanujan Hall

Host: Suresh Kumar

Speaker: Vikrant Desai (174093013)

Title: Mean field games with degenerate diffusions

Abstract: We study mean field games with degenerate diffusions. We prove

the existence of a mean field equilibrium for class of mean field games with state dynamics given by controlled degenerate diffusions.

Abstract: In this seminar, we discuss three types of Kirchhoff-type PDE with memory involving usual time derivative, fractional time derivative, and fractional Laplacian. In the first half, we focus on well-posedness and regularity results. Then, we derive a semi-discrete error analysis of Galerkin FEM by defining a modified Ritz-Volterra projection  operator.  Finally, we develop fully discrete formulations and implement some examples to validate the proven theoretical results.

Abstract: In the first half of the talk, I will introduce key ideas and methodologies of causal inference in the scientific foundations of evidence-based medicine. I will trace the history (20th century) of modern drug development through the evolution of the law known as The Food, Drugs and Cosmetic Act. The law was instituted and further evolved in response to some major global health events/crises. The US Congress gave sweeping powers to the FDA to interpret and execute the law to develop a drug approval process based on empirical data and science. We will also trace the law’s impact on global drug policies. In that context, we will discuss ‘adequate and well-controlled studies and the role of statistics in the process. We will discuss some of the important examples on the way. 

Bio of the speaker:  As a statistician, I worked in academia, the federal government, and the Biotechnology/Pharma industry. I was in leadership and supervisory positions in the government (Department of Biostatistics at Center for Drug Development and Research, Food and Drug Administration), and in the pharma industry at Johnson & Johnson  (large pharma), Amgen (mid-size pharma when I began there) and Esperion (small biotech). The early years of my career (about 10 years) were in academic positions starting as a lecturer at Bombay University followed by several years at the University of California, Santa Barbara with a brief sojourn to IIT, Mumbai. I continued methodological research and publishing during my time in the government and industry as well. Currently, I offer statistical and regulatory consulting services to pharmaceutical companies specializing in strategic and secondary review of submission modules, DMC statistical membership, strategic reviews of protocols, and statistical analysis plans mostly in dermatology, immunology, cardiovascular, and antiviral therapeutic areas.  I am an elected Fellow of the American Statistical Association.

Abstract: The classical theorem of Beckman and Quarles states the following: A function from the Euclidean plane to itself that preserves unit distances must preserve all distances. We will discuss some key steps of the proof. We will also discuss our recent result generalizing this theorem. This is joint work with Ben Schmidt. We will try to make the talk accessible to a wide audience.

Abstract: We will discuss a variety of topics in Algebraic groups through reading some of the papers that have become classics in the subject. The first few seminars will be on the paper by Robert Steinberg, Regular elements of semi-simple algebraic groups Publications mathématiques de l'I.H.É.S., tome 25 (1965), p. 49-80

 Abstract: In this talk, I will consider the question of compactness of (approximate) solutions to some Yamabe-type problems. We will see that by appropriately blowing up the singularities a finer analysis often reveals some analytical or geometric rigidity, thereby ruling out explosions. This establishes uniform bounds which implies compactness (in the appropriate topology) and gives existence and non-existence results.