# Past Colloquia

# Spring 2017

*Approximation Algorithms for Big Data*

*Dongbin Xiu**Professor and Ohio Eminent Scholar**Department of Mathematics**The Ohio State University*

**Thursday, May 4, 2017****4:00-5:15 PM****Toomey Hall 295**

One of the central tasks in scientific computing is to accurately approximate unknown target functions. This is typically done with the help of data — samples of the unknown functions. In statistics this falls into the realm of regression and machine learning. In mathematics, it is the central theme of approximation theory. The emergence of Big Data presents both opportunities and challenges. On one hand, big data introduces more information about the unknowns and, in principle, allows us to create more accurate models. On the other hand, data storage and processing become highly challenging. Moreover, data often contain certain corruption errors, in addition to the standard noisy errors. In this talk, we present some new developments regarding certain aspects of big data approximation. More specifically, we present numerical algorithms that address two issues: (1) how to automatically eliminate corruption/biased errors in data; and (2) how to create accurate approximation models in very high dimensional spaces using stream/live data, without the need to store the entire data set. We present both the numerical algorithms, which are easy to implement, as well as rigorous analysis for their theoretical foundation.

*Dongbin Xiu received his Ph.D degree from the Division of Applied Mathematics of Brown University in 2004. He conducted postdoctoral studies in Los Alamos National Laboratory, Princeton University, and Brown University, before joining the Department of Mathematics of Purdue University as an Assistant Professor in the fall of 2005. He was promoted to the rank of Associate Professor in 2009 and to Full Professor in 2012. In 2013, he moved to the University of Utah as a Professor in the Department of Mathematics and Scientific Computing and Imaging (SCI) Institute. In 2016, He moved to Ohio State University as Professor of Mathematics and Ohio Eminent Scholar. He has received NSF CAREER award in 2007, as well as a couple of teaching awards at Purdue. He is on the editorial board of several journals, including SIAM Journal on Scientific Computing and Journal of Computational Physics. He is the founding Associate Editor-in-Chief of the International Journal for Uncertainty Quantification. His research focuses on developing efficient numerical algorithms for stochastic computations and uncertainty quantification.*

*Host: He*

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*On the Rate of Convergence for Online Principal Component Estimation and Tensor Decomposition*

*Junchi Li**Postdoctoral Reseach Associate**Department of Operations Research and Financial Engineering**Princeton* University

**Friday, April 21, 2017****4:00-5:15 PM****Rolla Building G5**

Principal component analysis (PCA) and tensor component analysis has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the component by processing streaming data are of tremendous practical and theoretical interests. In this talk, we cast these methods into stochastic nonconvex optimization problems, and we analyze the online algorithms as a stochastic approximation iteration. We will sketch the proof (for the first time) a nearly optimal convergence rate result for both online PCA algorithm and online tensor decomposition. We show that the finite-sample error closely matches the corresponding results of minimax information lower bound.

* Dr. Junchi Li obtained his B.S. in Mathematics and Applied Mathematics at Peking University in 2009, and his PhD in Mathematics at Duke University under Professor Rick Durrett in 2014. He is currently working as a postdoctoral research associate at Department of Operations Research and Financial Engineering, Princeton University, with Professor Han Liu and Professor Tong Zhang. His research interests include statistical machine learning and optimization, stochastic algorithms for big data analytics, and stochastic dynamics on graphs and complex networks.*

* Host: Hu*

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*Representation Theorems for Indefinite Quadratic Forms and Applications*

*Stephan Schmitz**Postdoctoral Fellow**Department of Mathematics*

University of Missouri-Columbia

**Monday, April 17, 2017****4:00-5:15 PM****Rolla Building G5**

We will discuss the connection between quadratic forms and operators. The main question is whether a symmetric sesquilinear form defines a self-adjoint operator and, if yes, whether knowing the operator allows to reconstruct the form again. For bounded or semibounded closed forms, classic results give an affirmative answer to these questions. In this talk, we address these questions for (possibly) non-semibounded forms. As an application, the Stokes block operator and indefinite differential operators of the type $\mathrm{div} H$grad in $L^2(\Omega)$, where $H$ is sign indefinite and $\Omega$ is a bounded domain in $\mathbb{R}^n$, will be discussed.

This talk is based on joint work with A. Hussein, V. Kostrykin, D. Krej¿i¿ík, K. A. Makarov, and K. Veseli¿.

*Stephan Schmitz earned his PhD in from the University of Mainz, Germany, in Spring 2014 under the supervision of Vadim Kostrykin. Since Fall 2015 he has been a Postdoctoral Fellow at MU Columbia. Dr. Schmitz's main research interest lie in the areas of Functional Analysis and Partial Differential Equations with primary focus on indefinite quadratic forms, diagonalization of operator matrices and their applications in Mathematical Physics.*

*Host: Clark*

*Generalized Semiparametric Varying-Coefficient Model for Longitudinal Data with Applications to Adaptive Treatment Randomizations*

*Yanqing Sun**Professor**Department of Mathematics and Statistics*

University of North Carolina at Charlotte

**Friday, April 14, 2017****4:00-5:15 PM****Rolla Building G5**

This paper investigates a generalized semiparametric varying-coefficient model for longitudinal data that can flexibly model three types of covariate effects: time-constant effects, time-varying effects, and covariate-varying effects. Different link functions can be selected to provide a rich family of models for longitudinal data. The model assumes that the time-varying effects are unspecified functions of time and the covariate-varying effects are parametric functions of an exposure variable specified up to a finite number of unknown parameters. The estimation procedure is developed using local linear smoothing and profile weighted least squares estimation techniques. Hypothesis testing procedures are developed to test the parametric functions of the covariate-varying effects. The asymptotic distributions of the proposed estimators are established. A working formula for bandwidth selection is discussed and examined through simulations. Our simulation study shows that the proposed methods have satisfactory finite sample performance. The proposed methods are appliedto the ACTG 244 clinical trial of HIV infected patients being treated with Zidovudine to examine the effects of antiretroviral treatment switching before and after HIV develops the T215Y/F drug resistance mutation. Our analysis shows benefits of treatment switching to the combination therapies as compared to continuing with ZDV monotherapy before and after developing the 215-mutation.

This is a joint work with Li Qi, Biostatistics and Programming, Sanofi and Peter B. Gilbert, Fred Hutchinson Cancer Research Center.

*Yanqing Sun received her Ph.D degree from Florida State University in 1992. She is a Full Professor of Statistics at the University of North Carolina at Charlotte. She is an Elected fellow of American Statistical Association and Elected member of International Statistical Institute. Her research interest includes developing semiparametric and nonparametric methods for univariate and multivariate failure time data, competing risks data, longitudinal data, and missing data. She has published 59 professional articles.Dr. Sun has been continuously funded by NSF for 15 years and by NIH for 10 years. She was appointed on the Statistics Panel of the NSF Division of Mathematical Sciences for statistics grant applications in 2013 and 2016. She served as an associate editor for International Journal of Biostatistics. Dr. Sun has supervised eight Ph.D graduates and is currently supervising six Ph.D graduate students.*

*Host: Adekpedjou*

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**Regularity of Solutions to the 3D Navier-Stokes Equations**

*Zachary Bradshaw**Visiting Scholar, Department of Mathematics **University of Virginia*

**Friday, February 10, 2017****4:00-5:15 PM****Rolla G-5**

In 1934, Jean Leray gave the first construction of a solution to the Navier-Stokes equations. 83 years later, the regularity, i.e.~smoothness and boundedness, of Leray's solutions remains an open question. Presently, only conditional regularity criteria are available. In this talk, we introduce the Ladyzhenskaya-Prodi-Serrin regularity criteria, a classical conditional regularity criteria for Leray's weak solutions to the Navier-Stokes equations. In our discussion, special attention is paid the roles of critical versus supercritical norms in regularity issues, and how these relate to the difficulty of solving the problem of global regularity for the Navier-Stokes equations (which is one of the Millennium prizes). We also present a recent refinement of the Ladyzhenskaya-Prodi-Serrin criteria highlighting which frequencies play an essential role in singularity formation.

*Dr. Zachary Bradshaw graduated from the University of Virginia in 2014. From July 2014 to July 2016, he was a postdoctoral research fellow at the University of British Columbia. He is currently a visiting scholar at the University of Virginia. His research is in PDEs, particularly the analysis of fluid models such as the Navier-Stokes equations. Major topics in his work are turbulence and regularity.*

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**On Contraction of Large Perturbations of Shock Waves**

*Moon-Jin Kang**R. H. Bing Instructor, Department of Mathematics **The University of Texas at Austin*

**Monday, February 6, 2017****4:00-5:15 PM****Rolla G-5**

Although mathematical understanding on hyperbolic conservation laws has made huge contributions across many fields of science, there remain many important unsolved questions. In particular, a global well-posedness of entropy solutions to the system of conservation laws in a class of large initial datas is completely open even in one space dimension. Recently, we have obtained a contraction (up to shift) of entropy shock waves to the hyperbolic systems in a class of large perturbations satisfying strong trace property. Moreover, concerning viscous systems, we have verified the contraction of large perturbations of viscous shock waves to the isentropic Navier-Stokes system with degenerate viscosity. Since the contraction of viscous shocks is uniformly in time and independent of viscosity coefficient, based on inviscid limit, we have the contraction (thus, uniqueness) of entropy shocks to the isentropic Euler in a class of large perturbation without any local regularity such as strong trace property. In this talk, I will present this kind of contraction property for entropy inviscid shocks and viscous shocks.

*Dr. Kang received his Ph.D. from the Department of Mathematical Science, Seoul National University, Korea in 2013, under the supervision of professor Seung-Yeal Ha. From August 2013 to July 2014, he was a visiting scholar at the Univ. Texas at Austin. He is currently R.H. Bing Instructor at the same university. During Spring semester 2015, he visited LJLL(Laboratoire Jacques-Louis Lions), Paris as a FSMP postdoctoral fellow. Dr. Kang's research interests lie in (partial) differential equations arising in the fluid dynamics, engineering, neuroscience, biology and social dynamics, etc.*

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**Diffusive Stability of Spatially Periodic Patterns**

*Alim Sukhtayev**Visiting Assistant Professor, Department of Mathematics **Indiana University*

**Friday, February 3, 2017****4:00-5:15 PM****Rolla G-5**

The topic of pattern formation has been the object of considerable attention since the fundamental observation of Alan Turing that reaction diffusion systems modeling biological/chemical processes can spontaneously develop patterns through destabilization of the homogeneous state. Going beyond the question of existence, an equally fundamental topic is stability of periodic patterns, and linear and nonlinear behavior under perturbation. Here, two particular landmarks are the formal small-amplitude theory of Eckhaus and the rigorous linear and nonlinear verification of this theory for the Swift-Hohenberg equation, a canonical model for hydrodynamic pattern formation. In this talk, I will present a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model, and for the model introduced by Cox-Matthews for pattern formation with a conservation law.

*Dr. Sukhtayev received his Ph.D. from the University of Missouri, Columbia in 2012, under the supervision of professor Yuri Latushkin. From July 2012 to July 2015, **he was a visiting assistant professor at Texas A&M University. He is currently a visiting assistant professor at Indiana University Bloomington. **Dr Sukhtayev's research interests lie in Applied Analysis, Partial Differential Equations and infinite dimensional Dynamical Systems with emphasis on problems related to stability theory and operator theory.*

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**On Two-Phase Flow in Karstic Geometry: Modeling, Analysis and Numerical Simulations**

*Daozhi Han**Zorn Postdoctoral Fellow, Department of Mathematics **Indiana University*

**Monday, January 30, 2017****4:00-5:15 PM****Rolla G-5**

Multiphase flow phenomena are ubiquitous. In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit and porous media are termed karstic geometry. In this talk, we derive a diffuse interface model for two-phase flow in karstic geometry utilizing Onsager's extremum principle. The model together with the interface boundary conditions satisfies a physically important energy law. We show that the model admits a global finite-energy weak solution which agrees with the strong solution provided the strong solution exists. Finally, we present a novel decoupled unconditionally energy-stable numerical scheme for solving this diffuse interface model.

*Dr. Han obtained his PhD in Applied and Computational Mathematics from the Florida State University in 2015, under the supervision of Prof. Xiaoming Wang. He then joins the Department of Mathematics at Indiana University as a Zorn postdoctoral fellow working with Prof. Roger Temam. Dr. Han's research is centered around applied analysis, numerical analysis and computation of partial differential equations from fluid dynamics. Dr. Han is currently working on the mathematical validity of Prandtl boundary layer theory; modeling, analysis and numerical simulations of multiphase flow phenomena; and flow instabilities.*

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A Multiscale Approach for Seafloor Identification in Sonar Imagery

*Christina Frederick**NSF IMPACT Postdoctoral Fellow, School of Mathematics **Georgia Institute of Technology *

**Friday, January 27, 2017****4:00-5:15 PM****Rolla G-5**

Modern day sonar systems are capable of probing the ocean floor and obtaining acoustic measurements with an unprecedented level of precision. Despite rapid advances in technology, only about .05% of the oceans are mapped to a resolution of a couple meters, needed for tasks such as finding plane wreckage. The main obstacle to rapid seafloor characterization is dealing with the complex scattering effects of structures on the ocean floor. In this talk, I'll describe a multiscale strategy for solving the inverse problem of recovering details of the ocean floor using sonar data. Forward solvers incorporate simulations of Helmholtz equations on a wide range of spatial scales, allowing for detailed recovery of seafloor parameters including the material type and roughness. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor configurations. The inversion is performed using efficient discrete optimization techniques.

*Dr. Frederick earned her Ph.D. in mathematics from the University of Texas at Austin in 2014 under the supervision of Björn Engquist. She has held a postdoctoral fellowship at the Mittag-Leffler Institute of the Royal Swedish Academy of Sciences in Stockholm (2014), and from 2015 until the present time, has been an NSF IMPACT postdoctoral fellow in the School Mathematics at the Georgia Institute of Technology and mentored by Haomin Zhao. Dr. Fredrick's research has focused on multiscale methods, a very active branch of computational and applied mathematics due to developments in computing technology and information science. Her work has encompassed multiscale computation and numerical homogenization for inverse problems based on elliptic PDE's that has application in porous media and medical imaging, as well as sampling strategies that exploit special microstructures of functions to reduce the computation cost, and retain theoretical optimality in terms of efficiency and stability. Recent work includes multiscale methods for sonar imaging, as well as robotics and stochastic differential equations. *

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From Buckling to Rigidity of Shells: Recent Mathematical Progress

*Davit Harutyunyan**Postdoctoral Associate, Department of Mathematics **Swiss Federal Institute of Technology, Lausanne (EPFL) *

**Monday, January 23, 2017****4:00-5:15 PM****Rolla G-5**

It is known that the rigidity of a shell (for instance under compression) is closely related to the optimal Korn's constant in the nonlinear Korn's first inequality (geometric rigidity estimate) for $H^1$ fields under the appropriate conditions (with no or with Dirichlet type boundary conditions arising from the nature of the compression). In their celebrated work, Frisecke, James and Mueller (2002, 2006) derived an asymptotically sharp nonlinear geometric rigidity estimate for plates, which gave rise to a derivation of a hierarchy of nonlinear plate theories for different scaling regimes of the elastic energy depending on the thickness $h$ of the plate (the optimal constant scales like $h^2$). Frisecke-James-Mueller type theories have been derived by Gamma-convergence and rely on $L^p$ compactness arguments and of course the underlying nonlinear Korn's inequality. While plate deformations have been understood almost completely, the rigidity, in particular the buckling of shells is less well understood. This is first of all due to the luck of sharp rigidity estimates for shells. In our recent work we derive linear sharp geometric estimates for shells by classifying them according to the Gaussian curvature. It turns out, that for zero Gaussian curvature (when one principal curvature is zero, the other one never vanishes) the amount of rigidity is $h^{3/2},$ for negative curvature it is $h^{4/3}$ and for positive curvature it is $h$. These results represent a breakthrough in both the shell buckling and nonlinear shell theories. All three estimates have completely new optimal constant scaling for any sharp geometric rigidity estimates to have appeared, and have a classical flavor. This is partially joint work with Yury Grabovsky (Temple University)

*Dr. Harutyunyan graduated from the Hausdorff Center for Mathematics (University of Bonn) in June, 2012, where he did his doctoral work in applied analysis under the supervision of prof. Stefan Mueller. He spent 2 years at Temple University (2011-2013) as a postdoctoral research assistant professor working with prof. Yury Grabovsky and 3 years at the University of Utah (2013-2016) as a research assistant professor working with distinguished prof. Graeme Walter Milton. He is now a scientific collaborator at Ecole Polytechnique Federale de Lausanne working with prof. Hoai-Minh Nguyen. Dr. Harutyunyan is working in various directions of analysis (both applied and pure) with broad research interests such as Partial Differential Equations, Calculus of Variations, Material Science, Composite Materials and Metamaterials, Continuum Mechanics, Homogenization, Stability Estimates and Micromagnetics.*

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**Long-time Behavior for Nonlinear Schrödinger Equations**

*Jason Murphy**NSF Postdoctoral Fellow, Department of Mathematics**University of California-Berkeley*

**Friday, January 20, 2017****4:00-5:15 PM****Rolla G-5**

We will discuss several results concerning the long-time behavior of solutions to nonlinear Schrödinger equations (NLS). The study of two special cases (the mass- and energy-critical problems) over the last 15-20 years led to the development of a powerful set of techniques, namely, the concentration compactness approach to induction on energy. These techniques have been further developed and refined to address a wide range of problems in the field of nonlinear dispersive equations. I will first discuss some results for pure power-type NLS at `non-conserved critical regularity'. I will also discuss some results for other models, including NLS in the presence of an external potential, as well as NLS with non-vanishing boundary conditions at spatial infinity.

*Dr. Murphy earned his PhD in Mathematics from UCLA in Spring 2014 under the supervision of Rowan Killip and Monica Visan. Since Fall 2014, he has been an NSF Postdoctoral Fellow at UC Berkeley with sponsoring scientist Daniel Tataru. Dr. Murphy's main research interests lie in the areas of harmonic analysis and nonlinear dispersive equations, with a primary focus on the asymptotic behavior of solutions to nonlinear Schrödinger equations.*

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# Fall 2016

**Accurate and Efficient Computation of Nonlocal Potentials Based on Gaussian-sum Approximation**

*Yong Zhang**Visiting Fellow, Courant Institute of Mathematical Sciences**New York University*

**Wednesday, November 16, 2016****4:00-5:15 PM****Rolla G-5**

We introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials, including the 3D/2D Coulomb, 2D Poisson and 3D dipole-dipole potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel combined with a Taylor expansion of the density. Starting from the convolution formulation of the nonlocal potential, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. The potential is separated into a regular integral and a near-field singular correction integral. The first is computed with the Fourier pseudospectral method, while the latter is well resolved utilizing a low-order Taylor expansion of the density. Both parts are accelerated by fast Fourier transforms (FFT). The method is accurate (14-16 digits), efficient ($O(N\log N)$ complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelizable.

*Dr. Zhang graduated from the Department of Mathematical Science, Tsinghua University, in 2012. From July 2012 to July 2015, he was a post doctoral fellow at the Wolfgang Pauli Institute, University of Vienna. From Sep 2015 till July 2016, he worked at IRMAR, University of Rennes 1 in France. Supported by an Erwin Schrödinger grant from the Austrian Science Fund (FWF) awarded last June, he is now visiting with Professor L. Greengard at the Courant Institute, New York University. Dr Zhang is a specialist in applied and computational mathematics whose research interests include: Bose-Einstein Condensates, analysis-based fast algorithms, artificial boundary conditions, and highly oscillatory problems.*

*Host: Zhang*

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**(Conditional) Positive Semidefiniteness in a Matrix-Valued Context**

*Michael Pang**Professor, Department of Mathematics**University of Missouri*

**Friday, October 28, 2016****4:00-5:15 PM****Rolla G-5**** **

We extend Schoenberg's classical theorem, which relates conditionally positive semidefinite functions $F \colon \mathbb{R}^n \to \mathbb{C}$ to their positive semidefinite exponentials $\exp(tF) \colon \mathbb{R}^n \to \mathbb{C}, \,\, t > 0,$ to matrix-valued conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}^{m \times m}, \,\, m \in \mathbb{N}.$ Moreover, we study the closely related property that $\exp(tF)(-i\nabla),\,\, t > 0,$ is positivity preserving and its failure to extend directly to the matrix-valued context. If time permits, we will discuss some of the main tools used in the proofs.

This is joint work with Fritz Gesztesy.

*Dr. Pang's research focuses on spectral properties of linear second order elliptic operators. These include heat kernel bounds, Lp properties of singular elliptic operators, eigenfunctions of the Dirichlet Laplacians defined on regions with fractal boundaries, elliptic operators defined on graphs, and the hot spots conjecture. Most recently he has been interested in problems related to perturbations of eigenspaces of Dirichlet Laplacians caused by perturbations of the regions.*

*Host: Clark*

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**Suppression of Chemotactic Explosion by Mixing**

*Xiaoqian Xu**Postdoctoral Associate, Department of Mathematics**Carnegie Mellon University*

**Friday, October 21, 2016****4:00-5:15 PM****Rolla G-5**

The Keller-Segel equation is one of the most studied PDE models of processes involving chemical attraction. However, a solution of the Keller-Segel equation can exhibit dramatic collapsing behavior where the population density of bacteria concentrates positive mass in a measure zero region. In other words, there exist initial data leading to finite time blow up. In this talk, we will discuss the possible effects resulting from interaction of chemotactic and fluid transport processes; namely, we will consider the Keller-Segel equation with additional advection term modeling ambient fluid flow. We will prove that the presence of fluid can prevent the singularity formation. We will discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers.

*Dr. Xu is currently a postdoctoral associate in the Mathematics Department of Carnegie Mellon University. Before joining CMU, he finished a PhD in Mathematics in 2016 at the University of Wisconsin-Madison under the direction of Professors Alexander Kiselev and Andrej Zlatos. Dr. Xu works in the area of partial differential equations with a focus on fluid dynamics, active scalar and mixing.*

*Host: Hu*

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**Padé Approximants and Difference Operators**

*Maksym Derevyagin**Visiting Assistant Professor, Department of Mathematics**University of Mississippi*

**Thursday, October 20, 2016****4:00-5:15 PM****Rolla G-5**

We will discuss Padé approximants, which serve as a very efficient tool for numerical analysis of objects that are described by analytic or even meromorphic functions. Actually, it will be shown how one can use operator theory to prove convergence results for this kind of approximation. In a word, Padé approximants arise as approximants to continued fractions of a special type and, in turn, continued fractions are intimately related to difference equations of the second order, which in fact give us the difference operators in question.

In order to demonstrate the method, several instances when it can be applied will be analyzed and, thus, a couple of convergence results will be presented. Also, we will consider what kind of obstacles may appear while applying the scheme.

*Dr. Derevyagin has been a Visiting Assistant Professor in the University of Mississippi Department of Mathematics since 2014. Before that, he has held post-doctoral positions in the Katholieke Universiteit Leuven, the Technische Universität Berlin, and Université Lille 1 : Sciences et Technologies. His research interests include operator theory, in particular, the spectral theory of differential and difference operators, approximation theory and special functions, orthogonal polynomials and random matrix theory*

*Host: Clark*