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Ph.D., University of Texas at Austin
E-mail: slepcev AT math DOT cmu DOT edu
Office: Wean Hall 7123
Personal web site
My research is in applied analysis. It has two main themes. One of them is investigation of energy-driven systems --- that is systems whose dynamics is driven by dissipating an associated free energy. The dissipation mechanism endows the underlying configuration space with a geometric structure. Investigating the geometry of the energy landscape enables one to obtain important information about the behavior of the given system. With collaborators, I have studied dynamics of thin-liquid films, demixing of fluids, evolution of interfaces, systems with nonlocal interactions, and analyzed models of collective behavior in biological systems. The phenomena I have studied include dynamical scaling in coarsening processes, singularity formation, diffuse interface models and their sharp interface limits, evolution of grain-boundary networks and fundamental questions of existence, uniqueness, and asymptotic behavior in nonlocal interaction equations. Studies of these phenomena connect partial-differential equations, fluid mechanics, calculus of variations, optimal transport, and applied fields.
The other line of my research, which involves a broad collaboration, is on application of variational techniques to problems of analysis of large data sets. The task of data and image analysis is to enable scientists to analyze often high dimensional, datasets and interpret the information they contain. Many of these tasks like classification, clustering, data parameterization, and representation have a variational description, where the goal is to minimize an objective functional. Motivated by variational approaches to image analysis, I am studying total-variation based approaches to analysis of data clouds. As the natural data structure is often a graph, this involves building tools to analyze variational problems on graphs. In particular I have studied behavior of functionals and algorithms on graphs as the number of data points goes to infinity. I am also working on using optimal-transportation-based techniques for studies of medical images, for tasks such as cancer detection. Another research topic is the use of energy based approaches to find optimal ways to parameterize high-dimensional datasets by low dimensional objects. This research involves modeling, rigorous analysis of nonlocal energy functionals and design of efficient numerical algorithms.
Recent CNA Publications: