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Abstract

We consider the periodic homogenization of second order differential equations of non-divergence form. It is known that for the "majority" of coefficient matrices, the optimal rate of convergence in the homogenization is of order O(\epsilon). In this talk we will characterize coefficient matrices that give a homogenization rate O(\epsilon^2). As a consequence. we confirm a conjecture that all 2X2 diagonal positive-definite matrices with constant trace are of this type. Joint work with Timo Sprekeler (National University of Singapore) and Hung V. Tran (University of Wisconsin at Madison).

Speaker Bio

Xiaoqin Guo received his Ph.D. degree at University of Minnesota in 2012. After postdoctoral research experiences at various institutions, he became a tenure track assistant professor in Department of Mathematical Sciences at University of Cincinnati. His research interests are in the interface between probability and partial differential equations. In particular, he is currently focused on random motions in random media.

Thursday, February 24, 2022 (12:30 - 1:30PM)
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Abstract

There has been an explosion of interest in symmetry breaking and disorder in packings of spherical micelles in polymer chemistry. These include quasi-perioidic Frank Kasper phases and the C14, C15, and even C36 Laves phases. These are modeled by complex systems reduced from a molecular based Langevin formulation to a continuum formulation via self-consistent mean field theory. Frank Bates proposed a much simpler energy imposed upon a Voronoi tessellation that engenders similar structure. We propose a further reduction for a packing of soft spheres in a periodic rectangle in the plane. This simple energy possesses an astonishingly rich family of equilibrium with a structure evocative of point and continuum spectrum of a linear operator on an unbounded domain. This structure depends sensitively upon the number theoretic properties of N the number of soft balls in the packing. The energy admits a surprising reduction which leads to a seemingly new quantity, the max center of a convex polyhedron and an associated ``inverse'' isoperimetric inequality. The disordered equilibrium tessellations all possess defects - non-six-sided regions - and the extrapolation of equilibrium energy from some to zero defects has little correlation to the actual zero-defect equilibria energy. We examine the large N limit and identify a system frustration - the expected equilibrium energy above the zero-defect energy, which tends to a limiting value.

Speaker Bio

Keith Promislow received his PhD in 1991 from Indiana University and was an NSF Postdoctoral fellow at Penn �һ�������Ƶ. He has been a faculty member at Simon Fraser and Michigan �һ�������Ƶ Universities where he is currently the department chair. He represented the AMS at the Coalition for National Science Funding on Capital Hill, was a Kloosterman Professor at the University of Leiden, and a plenary speaker at the SIAM AGM and Nonlinear waves meetings. He is a past or current member of the editorial boards of SIADS, SIMA, and Physica D. His research blends PDE analysis with dynamical systems techniques and finds applications to the interaction of entropy and geometry in polymer chemistry, biology, and anywhere else that ionic solutions play a role.

Tuesday, February 8, 2022 (12:30 - 1:30PM)
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Abstract

With leaps and bounds progress in computing technology, it is increasingly feasible to simulate the physical process of protein-protein and protein-small molecule binding, and predicting native structures of the biomolecular complexes. Careful scrutiny of the simulation-generated structural models in light of existing biochemical, biophysical, and cellular data, can largely eliminate the false models. As examples, we will discuss our modeling of the full-length structures of JAK2 kinase and the structure of Ras-Raf signalosome, both are important in cancer biology and drug discovery. We will also show that such an approach can correctly recapitulate protein-small molecule binding in drug discovery settings. Recognizing that we are investigating complex systems and that the simulations are limited both in scale and in accuracy, we emphasize a soft qualitative approach that integrates substantial domain expertise, as opposed to a hard automatic protocol-based approach.

Speaker Bio

Dr Yibing Shan has spent more than two decades in the field of biomolecular simulations, computational structural biology, and computer-aided drug discovery. He was a founding member of D. E. Shaw Research and played a key role in the conceptualization and design of Anton Specialized Supercomputer for molecular dynamics simulations. He was twice a co-recipient of the Association of Computing Machinery (ACM) Gordon Bell Prize of Supercomputing. Yibing initiated the "swimming" simulations of biomolecular association and developed it into a tool that has become integral to a number of state-of-the-art drug discovery platforms. His team also pioneered simulation-based structural elucidation of large functional assemblies of proteins. In his recent pivot to philanthropy and entrepreneurship, Yibing founded Antidote Health Foundation for Cure of Cancer and AB Magnitude Ventures Group with a focus on facilitating computer-aided cancer drug discovery. He earned his A.B. in engineering physics from Shanghai University of Science and Technology, a M.S. in computer science and a PhD in physics from Drexel University.

Thursday, January 27, 2022 (12:30 - 1:30 PM)
1st Floor Room 1202 (Auditorium) E&CS Building

Abstract

Given the Hamiltonian, the evaluation of unitary operators, which correspond to solutions of the time-dependent Schrödinger equation, has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The concentration bound also provides a gate complexity estimate. The talk is also accessible to those with no prior knowledge of quantum computing.

Speaker Bio

Xiantao Li is a professor at Penn �һ�������Ƶ University. His primary research is in stochastic models, electron structure calculations, quantum transport, and model reduction.

Thursday, January 20, 2022 (12:30 - 1:30PM)
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Abstract

Many applications of applied sciences lead to differential equations with various types of singularities, including singularities of the geometry of the underlying space and singularities of the coefficients of the differential equations. The aim of this talk is to introduce the concept of singular manifolds, which can describe various kinds of singularities in a unified way. Then some recent work on the partial differential equation theory over singular manifolds will be presented. Based on this theory, I will investigate several linear and nonlinear parabolic equations arising from applied and theoretic mathematics.

Speaker Bio

Yuanzhen Shao is an assistant professor at Department of Mathematics at The University of Alabama (UA).

He earned his PhD in mathematics at Vanderbilt University in 2015 and was a postdoc fellow at Purdue University from 2015 to 2017. Before joining UA, Shao was an assistant professor at Georgia Southern University during 2017-2019. His research mainly focuses on geometric analysis, partial differential equations, mathematical biology, and general relativity.

Friday, December 10, 2021 (3:30 - 4:30PM)
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Abstract

In this talk, we present a systematic framework for deriving variational numerical schemes for generalized diffusions and gradient flows. The proposed numerical framework is based on the energy-dissipation law, which describes all the physics and the assumptions in each system and can combine various types of spatial discretizations including Eulerian, Lagrangian, and particle approaches. The resulting semi-discrete equation inherits the variational structures from the continuous energy-dissipation law. As examples, we apply such an approach to construct variational Lagrangian schemes to porous medium type generalized diffusions and Allen-Cahn type phase-field models, and particle methods for variational inference. Numerical examples demonstrate the advantages of our numerical approach.

Speaker Bio

I am currently a postdoctoral researcher at the Illinois Institute of Technology, working with Prof. Chun Liu. I obtained my Ph.D. at Peking University, advised by Prof. Pingwen Zhang, in 2018. Prior to that, I received a B.S. in Zhejiang University. My research centers around mathematical modeling, scientific computing, and machine learning, with applications in physics, biology, material science, and data science.

Tuesday, November 30, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Nonlocal models are becoming commonplace across application. Typically, these models are formulated using integral operators and integral equations, in lieu of the commonly used differential operators and differential equations. In this introductory lecture, I will present main properties of these nonlocal operators and equations. Tools of analyzing them such as nonlocal calculus, methods of showing well posedness of models, as well as corresponding solution/function spaces will be discussed. Behavior of solutions of nonlocal equations as a function of a measure of nonlocality will be analyzed. For example, with proper scaling, we will confirm consistency of models by showing that in the event of vanishing nonlocality, a limit of such solutions solves a differential equation. We will demonstrate all these in two model examples: a nonlocal model of heat conduction and a system of coupled equations from nonlocal mechanics.

Speaker Bio

Tadele Mengesha is an associate professor at the Department of Mathematics at the University of Tennessee, Knoxville (UTK). He received his doctorate in 2007 from Temple University under the supervision of Prof. Yury Grabovsky. Before joining UTK in 2014, he was an assistant professor at Coastal Carolina University, and later held postdoctoral positions at LSU and at Penn �һ�������Ƶ.

Dr. Mengesha's research areas include partial differential equations (PDE), the calculus of variations, nonlocal integral-differential equations, and the applications of these topics in materials science. His recent focus is on developing the basic analytical underpinnings of the peridynamic model, an integration-based model for the deformation of solids. He serves in the editorial board of the Journal of Peridynamics and Nonlocal Modeling (since 2019) and the SIAM Journal of Mathematical Analysis (since 2020).

Tuesday, November 23, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Motivated by the non-uniqueness results for multi-dimensional Euler equations and numerical results, Fjordholm, Lanthaler and Mishra (2017) suggested statistical solutions as a notion of solutions for multidimensional systems of nonlinear conservation laws. Statistical solutions are time-parameterized probability measures on spaces of integrable functions. We present a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of 'weak statistical scaling', which is inspired by Kolmogorov's 1941 turbulence theory, the approximations converge in an appropriate topology to statistical solutions. We will show numerical experiments which indicate that the assumption might hold true.

Speaker Bio

Franziska Weber got her PhD from University of Oslo in 2015 (Nils Henrik Risebro was her PhD advisor). Then she went to ETH Zürich for a postdoc (Sept 2015-Dec2016) and from there to University of Maryland for another postdoc (Dec 2016-Aug 2018) and in Sept 2018 she started as an assistant professor at CMU where she been since then. She mainly works on numerical methods and analysis for nonlinear time-dependent PDEs, mostly related to fluid dynamics, multiphase flows etc.

Thursday, November 18, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Many networks in modern infrastructure have large amount of agents correlated with each other and the size of data set on the network required to process is also huge. It is often impractical or even impossible to have a central server to collect and process the whole data set. Hence it is of great importance to design distributed and decentralized algorithms to solve the global optimization problems, that is, the storage and processing of data need to be distributed among agents and the local peer-to-peer communication should be employed to handle the coordination of results from each agent instead of a central server. In this work, we consider networks with topologies described by some connected undirected graph G = (V,E) and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem $\min_{x} \{ F(x) = \sum_{i\in V} f_i(x) \}$ with local objective functions $f_i$ depending only on neighboring variables of the vertex $i \in V $. We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide- and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. Our numerical demonstrations also indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods do for the least squares problem with/without l^1 penalty.

Speaker Bio

Nazar Emirov is currently a Postdoctoral Research Fellow at the Department of Computer Science, Boston College. He obtained his PhD from University of Central Florida very recently. His main areas of research are mathematical foundation of graph signal processing and distributed optimization over networks.

Thursday, November 11, 2021 (12:30 - 1:30PM)
Zoom

Abstract

In this talk we first introduce a scheme to construct a class of structured sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are discussed by leveraging their structures. We then present algorithms for nonconvex optimization problems regularized by two special structured sparsity promoting functions, namely the minimax concave penalty and log-sum penalty functions. Numerical experiments for image restoration will be presented as well.

Speaker Bio

Lixin Shen is currently a full professor in the Department of Mathematics at Syracuse University. He received the B.S. and M.S. degrees from Beijing University in 1987 and 1990, respectively, and the Ph.D. degree from Sun Ya-sen University in 1996, all in mathematics. His research focuses on applied computational harmonic analysis, optimization, and their applications in image processing. His research has been supported by NSF and AFRL.

Thursday, November 4, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Smectic liquid crystals are formed by elongated molecules that are aligned and arranged in fluid-like layers. They are remarkable examples of a geometrically frustrated, multi-layer, soft-matter system. Ground states of smectic liquid crystals are characterized by flat, equally spaced, parallel layers. Due to spontaneously broken translational and rotational symmetry, singularities form in regions where the smectic order breaks down. When defects are present, the layers must bend and the resulting curvature is, in general, incompatible with equal spacing between them. The subtle interplay between the geometry of the layers and equal spacing imposes theoretical complications, and understanding the layer structure of a smectic liquid crystal is a challenging task. Mathematically, this can be imposed as a singularly perturbed variational problem and the smectic state is described by the minimum configuration of the limiting energy as the penetration length parameter goes to zero. In this talk, I will discuss some recent progress on sharp lower bound and compactness for a nonlinear model of smectic A liquid crystals. This is based on joint work with Michael Novack.

Speaker Bio

Xiaodong Yan got her PhD from Minnesota, then worked as Courant Instructor at NYU before joining Michigan �һ�������Ƶ as assistant professor. She moved to UConn in 2007 and stayed ever since. Currently she is a professor at UConn. Her research area is Nonlinear PDE and Calculus of Variations. Her recent works are mainly on PDEs from materials science.

Thursday, October 28, 2021 (12:30 - 1:30PM)
Zoom

Abstract

In this talk we describe how the Landau - de Gennes energy and the constrained Ball-Majumdar energy are used to study nematic liquid crystal configurations. We illustrate how their minimizers capture patterns in liquid crystal materials for several two and three dimensional settings. Enforcing boundary conditions on a liquid crystal can introduce topological obstructions dictating that its texture contains defects. We determine the defect structures and estimates on the energies of minimizers, highlighting how these features depend on material parameters and the domain geometry.

Speaker Bio

Daniel Phillips received his Ph.D. degree from the University of Minnesota in 1981. He then joined Purdue University, retiring as a full professor in 2020. In addition he has held a number of visiting research positions at institutions including Northwestern University, Penn �һ�������Ƶ University, the I.M.A and the Newton Institute. Phillips' research centers on variational problems from nonlinear pde modelling soft material systems.

Thursday, October 21, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Active fluids consume fuel at the microscopic scale, converting this energy into forces that can drive macroscopic motions over scales far larger than their microscopic constituents. In some cases, the mechanisms that give rise to this phenomenon have been well characterized and can explain experimentally observed behaviors in both bulk fluids and those confined in simple stationary geometries. More recently, active fluids have been encapsulated in viscous drops or elastic shells to interact with an outer environment or a deformable boundary. Such systems are not as well understood. In this work, we examine the behavior of droplets of an active nematic fluid. We study their linear stability about the isotropic equilibrium over a wide range of parameters, identifying regions in which different modes of instability dominate. Simulations of their full dynamics are used to identify their nonlinear behavior within each region. When a single mode dominates, the droplets behave simply: as rotors, swimmers, or extensors. When parameters are tuned so that multiple modes have nearly the same growth rate, a pantheon of modes appears, including zigzaggers, washing machines, wanderers, and pulsators. This is a collaboration with David Stein and Mike Shelley.

Speaker Bio

Yuan-Nan Young is a professor of applied math at New Jersey Institute of Technology. Trained as a fluid dynamicist in astrophysical and geophysical flows at the University of Chicago, he worked on instability and turbulence in stratified fluids for his PhD in astrophysics. After graduation in 2000, he focused on nonlinear pattern formation in fluid dynamics and has worked on interfacial flow and fluid-structure interactions at low Reynolds number in the biological context since 2004, when he joined NJIT as faculty in applied math. He has recently worked on complex flows that involve vesicles, surfactant, fluid-elastic material interactions, electrokinetics and electrohydrodynamics. He also works on novel numerical schemes to simulate fluid-structure interactions. Motivated by complicated processes such as mechanosensing, he uses simple examples to illustrate the possible roles of hydrodynamics in the context of biophysics. Currently he has been working on flows in a deformable poroelastic medium, and active fluids enclosed by a deformable interface.

Thursday, October 14, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Simultaneous estimation problems have a long history in statistics and have become especially common and important in genomics research: modern technologies can simultaneously assay tens of thousands to even millions of genomic features that can each introduce an unknown parameter of interest. These applications reveal some conceptual and methodological gaps in the standard empirical Bayes approach to simultaneous estimation. This talk summarizes standard approaches, illustrates some difficulties, and introduces an alternative approach based on regression modeling, and illustrates some new estimators that can be applied to gene expression denoising, coexpression network reconstruction, and large-scale gene expression imputation.

Speaker Bio

Dr. Zhao received his bachelor's in Chemistry and Physics, master's in Statistics, and Ph.D. in Biostatistics from Harvard University. He then spent two years as a postdoc in biostatistics and statistics at the University of Pennsylvania before starting in the Department of Statistics at the University of Illinois at Urbana-Champaign, where he is now an associate professor. His research interests include Compound decision theory and empirical Bayes, Integrative genomics, and Mediation analysis. He has also managed to name one of his R packages after one of his cats, though sometimes he worries that the other cat is a little jealous.

Thursday, October 7, 2021 (12:30 - 1:30PM)
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Abstract

The semi-geostrophic equations, first studied by Eliasson and Hoskins, are fluid equations which describe the generation and subsequent dynamics of temperature fronts in the troposphere. They are used by the Met Office to diagnose flow data generated by the Unified Model, and they are also of interest to pure mathematicians owing to their connection with the theory of optimal transport. In this talk, following a brief introduction to the theory of optimal transport, I'll present mathematical results on the existence of dynamics of the semi-geostrophic equations, as well as results from numerical simulations on the dynamics which employ Laguerre diagrams (a weighted version of the famed Voronoi diagrams).

Speaker Bio

I did my BSc in mathematics at Strathclyde University, Glasgow, following which I did my graduate studies (Part III) at Trinity College, Cambridge. I studied for my PhD under Sir John Ball and Arghir Zarnescu at the University of Oxford. After graduating, I spent time as a Fondation Sciences Mathématiques de Paris (FSMP) Postdoctoral Fellow at the Ecole Normale Supérieure de Paris working with Laure Saint-Raymond. I was also a Courant Instructor at the Courant Institute, NYU, as well as a Research Fellow at Heriot-Watt University, Edinburgh, before becoming a Senior Lecturer (equivalent to Associate Professor) at Nottingham Trent University in Nottingham, England.

Thursday, September 30, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Modern data-acquisition techniques produce a wealth of data about the world we live in. Extracting the information from the data leads to machine learning/statistics tasks such as clustering, classification, regression, dimensionality reduction, and others. Many of these tasks seek to minimize a functional, defined on the available random sample, which specifies the desired properties of the object sought.

I will present a mathematical framework suitable for studies of asymptotic properties of such, variational, problems posed on random samples and related random geometries (e.g. proximity graphs). In particular we will discuss the passage from discrete variational problems on random samples to their continuum limits. Furthermore, we will discuss how tools of applies analysis help shed light on algorithms of machine learning.

Speaker Bio

Dejan received his Ph.D. degree at University of Texas Austin in 2002. After his postdoc positions at University of Toronto and UCLA, he joined the Department of Mathematical Sciences at Carnegie Mellon University as a tenure track assistant professor in 2006. He became a full professor in 2017 and he is currently serving as the associate director of Center for Nonlinear Analysis. His main research areas are in calculus of variations and PDE. In particular, he is interested in techniques of applied analysis in machine learning/data analysis, as well as nonlocal interaction equations and their applications.

Thursday, September 23, 2021 (12:30 - 1:30PM)
Zoom

Abstract

In the talk, a family of patch-like solutions of generalized surface quasi-geostrophic (GSQG) equation will be introduced. The equations of the contour dynamics under different geometrical situations will be formulated and the global well-posedness of the solutions will be proved under certain assumptions.

Speaker Bio

Qingtian Zhang is an assistant professor in West Virginia University. He graduated from Penn �һ�������Ƶ University in 2016 with a Phd degree. His research interests are the analysis of partial differential equations, mainly on the hyperbolic equations, dispersive equations, and conservation laws.

Thursday, September 9, 2021 (12:30 - 1:30PM)
Zoom

Abstract

Sparse signal recovery remains an important challenge in large scale data analysis and global-local (G-L) shrinkage priors have undergone an explosive development in the last decade in both theory and methodology. These developments have established the G-L priors as the state-of-the-art Bayesian tool for sparse signal recovery as well as default non-linear problems. While there is a huge literature proposing elaborate shrinkage and sparsity priors for high-dimensional real-valued parameters, there has been limited consideration of discrete data structures. In the first half of this talk, I will survey the recent advances in G-L shrinkage priors, focusing on optimality of these priors for both continuous as well as quasi-sparse count data. In the second half, I will discuss extension to discrete data structures including sparse compositional data, routinely observed in microbiomics. I will discuss the methodological challenges with the Dirichlet distribution as a shrinkage prior for high-dimensional probabilities for its inability to adapt to an arbitrary level of sparsity, and propose to address this gap by using a new prior distribution, specially designed to enable scaling to data with many categories. I will provide some theoretical support for the proposed methods and show improved performance in several simulation settings and application to microbiome data.

Speaker Bio

Jyotishka Datta is an assistant professor in the Department of Statistics at Virginia Tech. Prior to this, he was an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas Fayetteville from 2016-2020. Jyotishka received his PhD in Statistics from Purdue University in 2014 and worked as an NSF postdoctoral fellow at Duke University and SAMSI (Statistics and Applied Mathematical Sciences Institute) from 2014-16. His research interest spans developing new methodology and theory for high-dimensional or infinite-dimensional objects with low-dimensional structures such as sparsity. He has contributed to the area of multiple testing, sparse signal recovery, network analysis, changepoint detection as well as applications spanning cancer genomics, neuroscience, geotechnology and crime forecasting.

Thursday, April 29, 2021 (12:30 - 1:30PM)
Zoom

Abstract

I start with an overview of nematic liquid crystals (LCs) and their applications, including how they are modeled, such as Oseen-Frank, Landau-de Gennes, and the Ericksen model. For the rest of the talk, I will focus on Landau-de Gennes (LdG) and Ericksen.

Next, I describe some of the analytical difficulties of these models. For example, the Ericksen model exhibits a non-smooth constraint (for the PDE solution), and the LdG model with uniaxiality enforced as a hard constraint is also non-smooth. I will then discuss related numerical analysis issues that arise and how we handle these difficulties with a structure-preserving finite element method (FEM) for computing energy minimizers. We prove stability and consistency of the method without regularization, and $\Gamma$-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. Numerical simulations will be presented in two and three dimensions, some of which include non-orientable line fields, using a provably robust minimization scheme. Finally, I will conclude with some current problems and an outlook to future directions.

Speaker Bio

Shawn Walker received his Ph.D. degree in aerospace engineering at University of Maryland in 2007, and worked as a postdoc in Courant Institute, New York University from 2007-2010. He joined the math department of Louisiana �һ�������Ƶ University as a tenure track assistant professor in 2010, and became an associate professor in 2016. His research interests are mainly in numerical analysis and methods for partial differential related to fluids, free boundaries and geometric evolution problems, liquid crystals, and shape optimization.

Thursday, April 22, 2021 (12:30 - 1:30PM)
Zoom

Abstract

What does mathematics, materials science, biology and quantum information science have in common? It turns out, there are many connections worth exploring. I this talk, I will focus on graphs and random walks, starting from the classical mathematical constructs and moving on to quantum descriptions and applications. We will see how the notions of graph entropy and KL divergence appear in the context of characterizing polycrystalline material microstructures and predicting their performance under mechanical deformation, while also allowing to measure adaptation in cancer networks and entanglement of quantum states. We will discover unified conditions under which master equations for classical random walks exhibit nonlocal and non-diffusive behavior and see how quantum walks allow to realize the coveted exponential speedup in quantum Hamiltonian simulations. Recent classical and quantum breakthroughs and open questions will be discussed.

Speaker Bio

Maria Emelianenko is a Professor of Mathematics and an Associate Director of the Quantum Science and Engineering Center at George Mason University. She graduated with a PhD in Mathematics from Pennsylvania �һ�������Ƶ University and held a postdoctoral research associate position at the Center for Nonlinear Analysis of Carnegie Mellon University before joining Mason faculty. Her work lies at the interface between mathematics and other areas of science and engineering, such as materials science, chemistry and biology. Emelianenko's work has been supported by a number of National Science Foundation (NSF) grants, including the 2011 NSF CAREER award. She is a recipient of the 2009 ORAU Ralph E. Powe Junior Faculty Enhancement Award and 2008 MAA Project NExT Fellowship. She is serving as a Director of the Industrial Immersion Program and has co-directed several outreach and undergraduate research programs, including the first NSF-funded REU SITE at Mason. She is currently a Vice Chair of the SIAM Activity Group in Materials science and a SIAM representative to the US National Committee for Theoretical and Applied Mechanics.

Thursday, April 15, 2021 (12:30 - 1:30PM)
Zoom

Abstract

In this talk I will focus on the deterministic and stochastic Zakharov-Kuznetsov (ZK) equation with multiplicative noise in a bounded domain in space dimensions two and three. ZK equation is a long-wave small-amplitude limit of the Euler-Poisson system of the cold plasma uniformly magnetized along one space direction. It is also a multi-dimensional extension of the Korteweg-de Vries (KdV) equation. The talk will focus on the well-posedness and regularity of the deterministic and stochastic ZK equation. In the deterministic case, the global existence of strong solutions is established in space dimension three. For the stochastic ZK equation driven by a white noise, the existence of martingale solution in 3D, and the uniqueness and existence of the pathwise solution in 2D are established. The major challenge is to handle the mixed features of the equation, including the partial hyperbolicity, nonlinearity, anisotropy, and stochasticity. The main idea is to split up the dissipative and dispersion effect of the equation and use them for different purposes.

Speaker Bio

Chuntian Wang is an assistant professor at UA Department of Mathematics at The University of Alabama. She earned her PhD in mathematics at Indiana University Bloomington in 2015, and she was a residence Postdoc Fellow at Mathematical Sciences Research Institute in fall 2015, and a Hedrick Assistant Professor at the University of California, Los Angeles, from 2015-2018. Her main research interests are in nonlinear partial differential equations, mathematical modeling, computation and applied analysis.

Thursday, April 8, 2021 (12:30 - 1:30PM)
Zoom

Abstract

We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter epsilon entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as epsilon tends to zero. We demonstrate existence of local energy minimizers classified by their overall twist, find the Gamma-limit of these energies and show that it consists of twist and jump terms. This is a joint work with Michael Novack and Peter Sternberg.

Speaker Bio

Dmitry Golovaty is a professor at the University of Akron, specializing in mathematical problems that arise in materials science. His recent interest is primarily in variational models of nematic liquid crystals.

Thursday, April 1, 2021 (12:30 - 1:30PM)
Zoom

Abstract

It is believed that high-order methods have remarkable advantages in simulating wave propagations. In this talk, we shall discuss efficient techniques that can integrate with spectral and spectral-element solvers for time-harmonic wave scattering problems. One important building block is to introduce a truly exact perfect absorbing layer (PAL) for domain truncation of the scattering problem in an unbounded domain with a bounded scatterer. This technique is based on a complex compression coordinate transformation in radial direction, and a suitable substitution of the unknown field in the artificial layer. Compared with the widely-used perfectly matched layer (PML) methods, the distinctive features of PAL lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain reduced by the truncation layer; (ii) with the substitution, the PAL-equation is free of singular coefficients and the substituted unknown field is essentially non-oscillatory in the layer; and (iii) the construction is valid for general star-shaped domain truncation. By formulating the variational formulation in Cartesian coordinates, the implementation of this technique using standard spectral-element or finite-element methods can be made easy as a usual coding practice. We provide ample numerical examples to demonstrate that this method is highly accurate and robust for very high wavenumbers and thin layers.

Speaker Bio

Dr. Li-Lian Wang is currently an Associate Professor of Nanyang Technological University (NTU) in Singapore. Before he joined NTU as an Assistant Professor in 2006, he worked as a Visiting Assistant Professor and Postdoc in Purdue University since 2002. His main research interest resides in the spectral/high-order methods and computational acoustics & electromagnetics. He has published around 100 papers in respectful journals and one co-authored book on Spectral Methods (Springer, 2011). His research is well funded by funding agencies in Singapore.

Thursday, March 25, 2021 (10:30 - 11:30AM)
Zoom

Abstract

This presentation deals with the study of DNA in in static and time-dependent configurations. The term chromonic, that characterizes a class of water-based liquid crystals, emerged in connection with the crystalline phases that DNA forms in free solutions as well as in packed configurations of viruses. We first construct an energy that describes equilibrium packing configurations of DNA (and synthetic chromonic) in free solution, showing that energy minimizing configurations form faceted tori.

The next part of the presentation deals with the study of energy minimizing configurations of the DNA of a bacteriophage virus in a protein capsid, ichosahedral domain. The energy that we propose consists of the Oseen-Frank free energy of nematic liquid crystals that penalizes bending of the columnar DNA directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer.

In the last part of the talk, we turn into the time-dependent problem modeling the packing of the viral DNA inside the capsid by means of a motor.

Speaker Bio

Carme Calderer received her Ph.D. degree at Heriot-Watt University, Scotland in 1980. After working as faculty members in several universities including University of Maryland, University of Delaware, George Mason University, she became a full professor at Pennsylvania �һ�������Ƶ University in 1993. Then she moved to University of Minnesota in 2001. Her main research interests are in in modeling and mathematical analysis of problems of soft matter physics, arising in the fields of materials sciences and biophysics.

Thursday, March 18, 2021 (12:30 - 1:30 PM)
Zoom

Abstract

The lecture will review results on controllability of viscoelastic flows obtained over the last decade. For linear viscoelastic flows with any finite number of relaxation modes, approximate controllability has been shown. The result can be extended to an infinite number of relaxation modes, provided the relaxation times satisfy a gap condition. If there is more than one relaxation mode, exact null controllability does not hold. Approximate controllability holds within a subspace of the state space, which is invariant under the evolution of the system. This invariance is destroyed by nonlinear terms, as well as by variable coefficients. This leads to challenging and large unresolved problems.

Speaker Bio

Michael Renardy obtained his doctorate in mathematics from the University of Stuttgart in 1980. After spending a few years at the Universities of Wisconsin and Minnesota, he joined Virginia Tech in 1986. He retired from the position of Class of 1950 Professor in 2019. His principal field of research has been the analysis of partial differential equations arising in viscoelastic fluid dynamics. He has been a recipient of a Presidential Young Investigator Award and of the Virginia Tech Alumni Award for Research Excellence, and he is a Fellow of the American Mathematical �һ�������Ƶ.

Thursday, March 11, 2021 (12:30 - 1:30 PM)
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Abstract

In this talk we simulate the hydrodynamics of small unilamellar vesicles (sUVs) using a hybrid approach that is shown to capture the formation of sUVs in a solvent (SIAM J Multiscale Model. Simul., vol 18, pp. 79-103). In this hybrid formulation, the non-local interactions between the coarse-grained lipid molecules are described by a hydrophobicity functional, giving rise to forces and torques (between lipid particles) that dictate the motion of both particles and the fluid flow in the viscous solvent. Both the hydrophobic and hydrodynamic interactions between the coarse-grained amphiphilic particles are formulated into integral equations, which allow for accurate and efficient numerical simulations in both two- and three-dimensions. We validate our hybrid coarse-grained model by reproducing various physical properties of a lipid bilayer membrane, and we use this simulation tool to examine how a small unilamellar vesicle behaves under a planar shear flow and investigate the collective dynamics of sUVs under a shear flow. Finally. we also examine the possibility of membrane rupture by extreme flowing conditions.

Speaker Bio

Rolf Ryham is an Associate Professor and Associate Chair in the Department of Mathematics at Fordham University. He received his PhD from The Pennsylvania �һ�������Ƶ University under the joint supervision of Chun Liu and Ludmil Zikatanov, and held a postdoctoral appointment at Rice University working with Robert Hardt. He specializes in applied mathematics and membrane fluid mechanics.

Tuesday, March 2, 2021 (12:30 - 1:30 PM)
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Abstract

In this talk, we will discuss design of structure-preserving central-upwind finite volume methods for shallow water models. Shallow water models are widely used in many scientific and engineering applications related to modeling of water flows in rivers, lakes, and coastal areas. Shallow water equations are examples of hyperbolic systems of balance laws and such mathematical models can present a significant challenge for the construction of accurate and efficient numerical algorithms.

We will show that the developed structure-preserving central-upwind schemes for shallow water equations deliver high-resolution, can handle complicated geometry, and satisfy necessary stability conditions. We will illustrate the performance of the designed methods on a number of challenging numerical tests. Current and future research will be discussed as well.

Speaker Bio

Yekaterina Epshteyn received bachelor's degree in Applied Mathematics and Physics in 2000 from Moscow Institute of Physics and Technology (MIPT), Russia and a Ph.D. in Mathematics from the University of Pittsburgh in 2007. She completed a 3-year NSF-RTG postdoctoral position at the Department of Mathematical Sciences and Center for Nonlinear Analysis at Carnegie Mellon University. Since 2010, she has been a faculty member at the Department of Mathematics at the University of Utah. Yekaterina Epshteyn's research interests are in Numerical Analysis, Scientific Computing and Mathematical Modeling with applications to problems from Biology, Materials Science and Fluid Dynamics

Thursday, February 25, 2021 (12:30 - 1:30 PM)
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Abstract

The ability of tumors to metastasize is manifested by morphological instabilities such as chains or fingers that invade the host environment. In this talk, we develop a computational method for computing the nonlinear dynamics of a tumor-host interface within the sharp interface framework. We are interested in solid tumor growth with chemotaxis and cell-to-cell adhesion, together with the effect of the tumor microenvironment by the variability in spatial diffusion gradients, the uptake rate of nutrients inside/outside the tumor and the heterogeneous distribution of vasculature modeled using complex far-field geometries. We solve the nutrient field (modified Helmholtz equation) and the Stokes/Darcy flow field using a spectrally accurate boundary integral method, and update the interface using a non-stiff semi-implicit approach. Numerical results highlight the complexity of the problem, e.g. development of spreading branching-patterns and encapsulated morphologies in a long period of time.

Speaker Bio

Shuwang got his PhD from U. of Minnesota in 2005, then he did a three-year postdoc at U. of California. In 2008, he joined Illinois Institute of Technology as a faculty member. He is now a full professor at IIT. His research interests include modeling and computation of moving interface problems, and its applications in physical/biological systems, such complex fluids, materials science and tumor growth. His research is funded mainly by NSF.

Thursday, February 18, 2021 (12:30 - 1:30 PM)
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Abstract

I will discuss the effect of irregular transport on mixing properties in incompressible fluids, in particular describing measures of mixing and examples of optimal mixers. I will also discuss how mixing and transport can suppress singularity formation or lead to complete loss of regularity.

Speaker Bio

Anna Mazzucato received her bachelor and master's degrees from Milan University, and PhD degree from University of North Carolina at Chapel Hill in 2000. She then worked as Gibbs Instructor at Yale University between 2000-2003. Afterwards she joined Pennsylvania �һ�������Ƶ University as a tenure track assistant professor in 2003, where she became a full professor in 2013. Her main area is in partial differential equations that arise from problems in fluid mechanics, elasticity, and inverse problems.

Thursday, February 11, 2021 (12:30 - 1:30 PM)
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Abstract

In this talk we visit various models for traffic flow. These include local and nonlocal models, using both ODEs and PDEs. In particular, we discuss results and aspects on traveling waves, nonlocal to local convergence, uniform bound on total variation, and entropy admissibility.

Speaker Bio

Dr Wen Shen completed her education in Shanghai Jiao Tong University in China and University of Oslo in Norway, where she obtained her PhD in 1998. After research fellow positions at NTNU in Trondheim Norway and at SISSA in Italy, she joined the faculty of Penn �һ�������Ƶ University in 2003. She is currently a professor in the Mathematics Department, with research interests in the area of nonlinear partial differential equations and their applications.

Tuesday, February 2, 2021 (12:30 - 1:30 PM)
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Abstract

For any bounded smooth domain in dimension two, we established the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flows to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero for both uniaxial and biaxial cases. This is based on the compensated compactness property of the Ericksen stress tensors, which has been obtained by applying the Pohozaev type argument to the Ginzburg-Landau type nematic liquid crystal flows. As a byproduct, we obtain similar weak compactness results for the full Ericksen-Leslie system.

Speaker Bio

Tao Huang is Assistant Professor of Mathematics at Wayne �һ�������Ƶ University. He received his PhD at University of Kentucky in 2013, and then worked as Research Associate at Penn �һ�������Ƶ between 2013-2015 and as Visiting Assistant Professor at NYU Shanghai between 2015-2018. His main research interests lie in nonlinear partial differential equations from complex fluids and geometry analysis.

Thursday, January 21, 2021 (12:30 - 1:30 PM)
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Abstract

Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay in the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions.
I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model non-linear equations.

Speaker Bio

Gautam Iyer got his Ph.D. 2006 from the University of Chicago under the supervision of Peter Constantin. He was then a Szegö assistant professor at Stanford for 3 years. He has been at Carnegie Mellon university since 2009. His research mainly studies problems in PDE and probability motivated by applied mathematics.

Thursday, December 10, 2020 (12:30 - 1:30 PM)
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Abstract

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. We present in this talk numerical methods for nonlocal models characterized by a length parameter which measures the range of nonlocal interactions. Several numerical methods will be discussed with a focus on asymptotic compatibility of the schemes that are robust under the change of the nonlocal length parameter.

Speaker Bio

Dr. Tian is an assistant professor in Department of Mathematics at UC San Diego. Before arriving at UCSD, she was a Bing Instructor in the math department at UT Austin. She completed her PhD in 2017 at Columbia University Department of Applied Physics and Mathematics. Her research interests include numerical analysis, applied PDEs, nonlocal models, fractional PDEs and multiscale modeling.

Thursday, December 3, 2020 (12:30 - 1:30 PM)
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Abstract

The interaction between the deformation and fluid flow in a fluid-saturated porous medium is the object of study in poroelasticity theory. In this work, we consider a popular mixed finite-element (P1-RT0-P0) discretization of the three-field formulation of Biot's consolidation problem, which describes linear elastic, homogeneous porous media that is saturated by an incompressible Newtonian fluid.

Since this finite-element formulation does not satisfy an inf-sup condition uniformly with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. Thus, we proposed a stabilization technique that enriches the piecewise linear finite-element space of the displacement with the span of edge/face bubble functions. We have shown that for Biot's model this does give rise to discretizations that are uniformly stable with respect to the physical parameters. We also propose a perturbation of the bilinear form, which allows for local elimination of the bubble functions and provides a uniformly stable scheme with the same number of degrees of freedom as the classical P1-RT0-P0 approach. Additionally, we discuss robust linear solvers for this stabilized discretization of the poroelastic equations. Since the discretization is well-posed with respect to the physical and discretization parameters, it provides a framework to develop block preconditioners that are robust with respect to such parameters as well. We construct these preconditioners for the stabilized discretization and the perturbation of the stabilized discretization that leads to a smaller overall problem. Numerical results confirm the robustness of the block preconditioners.

Time permitting, we also discuss a monolithic geometric multigrid method for solving the stabilized discretization and compare its performance with the block preconditioners. This is joint work with Francisco Gaspar, Xiaozhe Hu, Peter Ohm, Carmen Rodrigo, and Ludmil Zikatanov.

Speaker Bio

James Adler is an Associate Professor in the Department of Mathematics at Tufts University. He is also currently the Director of Graduate Studies for the department. He received his PhD in Applied Mathematics in 2009 from the University of Colorado at Boulder and was a Research Assistant Professor at the Pennsylvania �һ�������Ƶ University from 2009-2011. His research is in scientific computing and numerical partial differential equations, studying efficient computational methods for applications in complex fluids, electromagnetism, and elastic materials.

Thursday, November 12, 2020 (12:30 - 1:30 PM)
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Abstract

Nematic liquid crystals are the most well studied and most technologically used liquid crystals. Their study involves a mixture of topology, analysis, algebra and gives raise to fascinating interplay between these areas. This talk surveys some of the work that I did over the past decade on the qualitative study of nematic liquid crystal defects, focusing on the most mysterious of them, the half-integer defects.

Speaker Bio

Prof. Arghir Zarnescu received his PhD degree at University of Chicago in 2006 under the supervision of Peter Constantine and Lenya Ryzhik. Then he worked as a research fellow at University of Oxford, and afterwards a lecturer in University of Sussex between 2011 and 2016. He started to work as Ikerbasque Research Professor at Basque Center for Applied Mathematics, Spain, as well as a Senior Researcher at ``Simion Stoilow" Institute, Romania in 2016. He studies analytical questions related to the study of partial differential equations and their applications to physically relevant models. Most of recent research concerns mathematical models of liquid crystals.

Thursday, November 5, 2020 (12:30 - 1:30 PM)
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Abstract

Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions that lead to the emergence of global behaviors: aggregation and flocking.

In particular, I will focus on the Euler-alignment system, and present some recent progress on the global wellposeness theory, large time behaviors, as well as interesting connections to classical equations in fluid mechanics.

Speaker Bio

Dr. Tan received his Ph.D. in the Department of Mathematics at the University of Maryland, College Park, in 2014. His advisor was Professor Eitan Tadmor. He then joined Rice University as a Lovett instructor. Currently, he is an assistant professor at the University of South Carolina.

Dr. Tan's research interests lie in the analysis and computation of nonlinear partial differential equations, particularly nonlocal models in fluid dynamics, collective motions, and hyperbolic conservation laws.

Wednesday, October 28, 2020 (3:00 - 4:00 PM)
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Abstract

New mesh algorithms were developed for numerical approximations of elliptic equations with singularities. These algorithms are simple, intuitive, and impose fewer geometric constraints on the domain. The resulting mesh is generally anisotropic and may not possess the maximum angle condition. In this talk, we present new regularity results on polyhedral domains and propose optimal anisotropic finite element algorithms approximating the 3D singular solution.

Speaker Bio

Hengguang Li received his Batchelor's degree at Peking University in 2002, and Doctor's degree at Pennsylvania �һ�������Ƶ University in 2008. After taking postdoctoral position at Syracuse University and Institute of Mathematics and its Applications at University of Minnesota, he joined Wayne �һ�������Ƶ University as a tenure track assistant professor in 2011. He became the department chair in 2017 and a full professor in 2018.

His main research interests are in numerical analysis, partial differential equations, and scientific computing.

His research has been highly recognized with various honors and awards.

Thursday, October 22, 2020 (12:30 - 1:30 PM)
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Abstract

The Patlak-Keller-Segel equation is a fundamental model for chemotaxis in the cell population. It is also used in astrophysics to describe gravitationally interacting massive particles. In this talk, we discuss global dynamics of 2D Patlak-Keller-Segel equation in subcritical regime with no additional assumptions. For the higher dimensional case in which the spatial dimension is greater than 2, we also obtain a conditional long-time asymptotics of global mild solutions to Patlak-Keller-Segel equation. This is a joint work with Chia-Yu Hsieh at The Chinese University of Hong Kong.

Speaker Bio

Prof. Yu obtained his bachelor and master degree from Fudan university. In 2004, he went to Courant institute and began to study PDE with professor Fang-hua Lin. During this period, he began to have interests in liquid crystal theories, Ginzburg-Landau theories and many other problems related to geometric flow and geometric analysis. After graduating from

Courant Insititute in 2009, he moved to university of Iowa as a visiting assistant professor. In 2012, he joined the Chinese University of Hong Kong, and now he is an associate professor at CU. His major interests are on hydrodynamical equations arising in liquid crystal theories and disclination problems in material sciences.

Thursday, October 15, 2020 (10:30 - 11:30 AM)
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Abstract

The mathematical analysis of liquid crystal models poses many challenging questions, as can be seen by their close relationship to the study of singularities for harmonic maps. In this talk, I will discuss the structure of defects in the context of different models of nematic liquid crystals, and their connection to classical results on harmonic maps into the sphere. To illustrate, I will present the physically fundamental problem of defects created by a colloid particle immersed in a nematic, and present recent results using the Landau-de Gennes energy. We find that the Landau-de Gennes model allows for a greater variety of types of singularity than the (harmonic map-based) Oseen-Frank energy, including line singularities such as the "Saturn Ring" defect.

Speaker Bio

Bronsard is originally from Québec. She did her undergraduate studies at the Université de Montréal, graduating in 1983, and earned her PhD in 1988 from New York University under the supervision of Robert V. Kohn. After short-term positions at Brown University, the Institute for Advanced Study, and Carnegie Mellon University, she moved to McMaster in 1992. She was president of the Canadian Mathematical �һ�������Ƶ for 2014-2016.

Bronsard was a plenary speaker at the Annual SIAM meeting in Boston in 2016, at the Mathematical Congress of the Americas in Montreal in 2017 and at the CMS Summer meeting in Charlottetown in 2018. She is on the editorial board of Nonlinear Analysis, Mathematics in Science and Industry, and the Canadian Applied Math Quarterly.

Her research has concentrated around singularly perturbed variational problems, including interfaces in reaction-diffusion systems, grain boundaries, superconducting vortices, and liquid crystal defects. Bronsard was the 2010 winner of the Krieger-Nelson Prize. In 2018 the Canadian Mathematical �һ�������Ƶ listed her in their inaugural class of fellows.

Thursday, October 8, 2020 (12:30 - 1:30 PM)
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Abstract

We present recent results on time-scales separation in fluid mechanics. The fundamental mechanism to detect in a precise quantitative manner is commonly referred to as fluid mixing. Its interaction with advection, diffusion and nonlocal effects produces a variety of time-scales which explain many experimental and numerical results related to hydrodynamic stability and turbulence theory.

Speaker Bio

Michele Zelati received his PhD at Indiana University in 2014, under the guidance of Roger Temam. He then moved to the University of Maryland as a Brin fellow for three years, and then in 2017 at Imperial College as a Chapman fellow. Since 2019, he is a Senior Lecturer and a Royal �һ�������Ƶ university research fellow.

His main research interest is in analysis of (stochastic) partial differential equations arising from mathematical physics and fluid mechanics.

Tuesday, September 29, 2020 (12:30 - 1:30 PM)
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Abstract

Fundamental tasks in data science, such as clustering, classification, and regression, involve solving large scale optimization problems. Theoretical properties of these optimization problems, and why their solutions tend to behave well, is often only partially understood. In this talk I will discuss how a growing body of work over the last twenty years has used continuum variational problems to analyze large data limits of statistical problems. In particular, I will discuss: 1) How Dirichlet energies from mathematical physics are linked with regression and statistical geometry problems, and 2) How classical isoperimetric problems are linked with clustering and classification problems. This talk will be designed to be broadly accessible to those with mathematical background, and in particular no specific knowledge of statistics or variational problems will be assumed.

Speaker Bio

Ryan Murray is an assistant professor of mathematics at the North Carolina �һ�������Ƶ University. He earned his PhD in mathematics at Carnegie Mellon University in 2016, and was a Chowla Research Assistant Professor at the Pennsylvania �һ�������Ƶ University from 2016-2019.

Tuesday, September 15, 2020 (12:30 - 1:30 PM)
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Abstract

For hyperbolic systems of conservation laws in one space dimension, a general existence-uniqueness theory is now available, for entropy weak solutions with bounded variation. In several space dimensions, however, it seems unlikely that a similar theory can be achieved.

For the 2-D Euler equations, in this talk I shall discuss ``simplest possible" examples of Cauchy problems admitting multiple solutions. Several numerical simulations will be presented, for incompressible as well as compressible flow, indicating the existence of two distinct solutions for the same initial data. Typically, one of the solutions contains a single spiraling vortex, while the other solution contains two vortices.

Some theoretical work, aimed at validating the numerical results, will also be discussed.

Speaker Bio

Prof. Bressan received his PhD degree in University of Colorado, Boulder in 1982. After working in various places in the U.S. and Italy, he joined Penn �һ�������Ƶ as a full professor in 2003, and became the Eberly Family Chair Professor of Mathematics later. His main research interests lie in Hyperbolic conservation laws, nonlinear PDEs, control theory, and differential games.

Prof. Bressan has received many prestigious honors and awards due to his contributions in PDEs, including the plenary lectureship at ICM 2002, the AMS Bocher prize in 2008, the SIAM Analysis of PDE prize in 2007, etc.

Thursday, September 10, 2020 (12:30 - 1:30 PM)
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Abstract

Heterogeneous problems with high contrast, multiscale and possibly also random coefficients arise frequently in practice, e.g., reservoir simulation and material sciences. However, due to the disparity of scales, their efficient and accurate simulation is notorious challenging. First, I will describe some important applications, and review several state-of-the-art multiscale model reduction algorithms, especially the Generalized Multiscale Finite Element Method (GMsFEM). Then I will describe recent efforts on developing a mathematical theory for GMsFEM, and ongoing works on algorithmic developments and novel applications.

Speaker Bio

Guanglian Li received her PhD degree from Texas A&M University in 2015, after postdoc experiences in University of Bonn, Germany and Imperial College London, United Kindom, she became an assistant professor at the University of Hong Kong. Her main research interests lie in

1. Homogenization, Multiscale Finite Element Methods (MsFEM), Generalized Multiscale Finite Element Methods (GMsFEM)
2. Model Order Reduction (MOR)
3. High Dimensional Approximation; Uncertainty Quantification

Tuesday, September 1, 2020 (10:30 - 11:30 AM)
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Abstract

Adaptive algorithms are convenient because they automatically determine the computational effort required to satisfy the error criterion. The function data acquired for constructing the approximate solution are used to compute a data-based error bound for the approximate solution. Computation proceeds until this error bound becomes small enough.
Unfortunately, most data-based error bounds are heuristic. We describe a framework for theoretically justified, adaptive function approximation algorithms. A key ingredient is identifying a suitable non-convex candidate cone of functions for which the data-based error bound succeeds. Our framework is illustrated via recent results for univariate functions obtained by the speaker and collaborators, which include proofs of asymptotic optimality. We also present work in progress for multivariate function approximation using reproducing kernels.

Speaker Bio

Fred J. Hickernell is Professor of Applied Mathematics and Vice Provost for Research at Illinois Institute of Technology, located in Chicago. He received his BA in mathematics and physics from Pomona College and his PhD in applied mathematics from Massachusetts Institute of Technology. Previously, Fred has served as a faculty member at the University of Southern California and at Hong Kong Baptist University. He enjoys solving problems using the tools of both computational mathematics and statistics. Fred is a fellow of the Institute of Mathematical Statistics and received the 2016 Joseph F. Traub Prize for Achievement in Information-Based Complexity. He has served on the editorial boards of the Journal of Complexity, Mathematics of Computation, and the SIAM Journal on Numerical Analysis. His recent interests are in merging theory with practice for integration and function approximation problems.

Thursday, March 5, 2020 (12:30 - 1:30 PM)
2nd Floor Room 2120 E&CS Building

Abstract

The concept of Pareto optimality has been utilized in the fields such as engineering, economics, and machine learning to understand fluid dynamics, consumer behavior, and identifying parameters that best optimize a set of m criteria (multi-objective optimization). During the process of model selection statisticians are often concerned with the model which has the single most optimal criterion (eg. AIC, R2 ) before continuing to check several other diagnostics. This strategy is multi-objective in nature but single-objective in its numeric execution. This talk will first introduce the general framework of Pareto optimality and a feasible solutions algorithm that can be used to estimate the Pareto boundaries for regression models. Then an overview how the algorithm can be applied to multi-objective problems in subset selection that will result in an ensemble of weak (regression-based) learners. Finally, an application of the method on a real dataset will be used to compare PIEs to other common machine learning techniques.

Speaker Bio

Dr. Lambert is Assistant Professor at the University of Cincinnati since 2018. He is the lead biostatistician in the College of Nursing at University of Cincinnati where he teaches and mentors PhD students, conducts statistical consultations, and works on his own methodology. He has been developing algorithms to identify statistical interactions in large datasets. His algorithm, the feasible solution algorithm(FSA) is now implemented in an R package, rFSA and is outlined in a publication in the R journal, and was featured in a recent books such as the 2020 Kuhn and Johnson: "Feature Engineering and Selection: A Practical Approach for Predictive Models".

Thursday, February 27 (12:30 - 1:30 PM)
1st Floor Room 1202 (Auditorium) E&CS Building

Abstract

Gamow's liquid drop model, initially developed to predict the shape of atomic nuclei, has recently resurfaced within the framework of the modern calculus of variations. The physical model includes two competing forces: an attractive surface energy associated with a depletion of nucleon density near the nucleus boundary, and repulsive Coulombic interactions due to the presence of positively charged protons. From the point of view of modern analysis, it can be considered as a phenomenological model of energy-driven pattern formation. This type of variational models appears in many different systems at all length scales.

In this talk, I will introduce the liquid drop model and review the state of the art for the global minimizers. I will then address certain anisotropic variants of this geometric variational problem and present on the recent results obtained in collaborations with Rustum Choksi (McGill), Alex Misiats (VCU), and Robin Neumayer (Northwestern). These extensions introduce surprising effects on the shape of local and global minimizers due to the additional competition between isotropic and anisotropic terms. In particular, regularity properties of the surface energy introduce some rigidity to the problem which affects the geometry of the ground states.

Speaker Bio

Ihsan Topaloglu received his PhD from Indiana University in 2012 under the direction of Peter Sternberg. He then worked as a postdoctoral fellow at McGill University in Montreal, Canada. Upon receiving the Fields-Ontario Postdoctoral Fellowship he spent one semester at the Fields Institute in Toronto, and three semesters at McMaster University. Since 2016, he has been an assistant professor at Virginia Commonwealth University in Richmond. Using tools of variational analysis, Ihsan works on problems stemming from mathematical physics, materials science and biological systems.

Friday, February 14 (12:30 - 1:30 PM)
1st Floor Room 1202 (Auditorium) E&CS Building

Abstract

The accuracy of machine learning (ML) based classification algorithms is greatly dependent on the availability of large amounts of representative data to train networks. However, in militarily-relevant applications, necessary training sets are expensive and often impossible to collect with appropriate sensing technology. This is especially true in synthetic aperture radar (SAR) imaging, where measured returns depend heavily on operating conditions such as target configurations, radar parameters, and environmental settings. Thus, to perform SAR automatic target recognition (ATR) using ML, we must augment the available measured data with synthetic (computer generated) data. However, due to model errors and assumptions made for feasible computational times, synthetic data often misrepresent the matching measured counterpart. To combat this domain discrepancy issue we developed Matching Component Analysis (MCA) for transfer learning. MCA is a Procrustes-type algorithm that estimates appropriate projections such that when applied to mismatched data, data are transferred into a common domain, more suitable for ATR tasks. We present theoretical results that describe the sample complexity of MCA and demonstrate, through numerical experiments, the performance of MCA on the problem of SAR measured/synthetic data domain adaptation.

Speaker Bio

Dr. Theresa Scarnati currently serves as a Research Mathematician for the Air Force Research Laboratory within the Multi-Domain Sensing & Autonomy Division. Her research interests include the implementation and analysis of regularization techniques for exploiting sparsity and prior knowledge in inverse problems, specifically for the applications of denoising synthetic aperture radar (SAR) images, SAR automatic target recognition via machine learning, three-dimensional image reconstruction and multi-sensor data fusion.

Thursday, February 6, 2020 (12:30 - 1:30 PM)
2nd Floor Room 2120 E&CS Building