Past Events

Summer 2020 - Fall 2021

The detailed information is in the following website:

https://sites.google.com/view/iit-multiscale-seminar/information


Fall 2022

Speaker: Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Topics of Energetic Variational Approaches 1

Abstract: We will discuss the introduction of the energetic variational approaches and their applications. I will start with basic mechanics and chemical reaction dynamics.


Speaker: Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Topics of Energetic Variational Approaches 2

Abstract: We will discuss the introduction of the energetic variational approaches and their applications. I will start with basic mechanics and chemical reaction dynamics.


Speaker: Prof. Gavish Nir, Department of Mathematics, Technion - Israel Institute of Tehnology

Title: Retrospective model-informed analysis of the Israeli booster campaign to curtail COVID-19 resurgence

Abstract: Israel was one of the first countries to administer mass vaccination. Consequently, it was among the first countries to experience substantial breakthrough infections due to the waning of vaccine-induced immunity, which led to a resurgence of the epidemic. In response, Israel launched a booster campaign to mitigate the outbreak, and was the first country to do so. Israel’s success in curtailing the Delta resurgence while imposing only mild non-pharmaceutical interventions influenced the decision of many countries to initiate a booster campaign. 

In this work, by constructing a detailed mathematical model and calibrating it to the Israeli data, we extend the understanding of the impact of the booster campaign from the individual to the population level. We used the calibrated model to explore counterfactual scenarios in which the booster vaccination campaign is altered by changing the eligibility criteria or the start time of the campaign and to assess the direct and indirect effects in the different scenarios. The results point to the vast benefits of vaccinating younger age groups that are not at a high risk of developing severe disease but play an important role in transmission. We further show that when the epidemic is exponentially growing the success of the booster campaign is highly sensitive to the timing of its initiation. Hence a rapid response is an important factor in reducing disease burden using booster vaccination.


Speaker: Prof. Ming Zhong, Department of Applied Mathematics, Illinois Tech

Title: Learning Self Organization from Observation 1

Abstract: Self-Organization (aka collective behavior) can be used to explain crystal formation, aggregation of cells, social behaviors of insects, synchronization of heart beats, etc. It is a challenging task to understand these types of phenomena from the mathematical point of view. We offer a statistical/machine learning approach to understand these behaviors from observation; moreover, our learning approach can aid in validating and improving the modeling of Self-Organization.

In the first part of the talk, we will focus on the forward modeling and backward learning of self organization.  We will review several important models which produce clustering, flocking, milling, and synchronization.  Then we will derive the learning method for inferring the interaction kernel from observation data and discuss its convergent properties.

In the second part of the talk, we will discuss how to expand the learning method to include more complicated models, complex geometries, missing feature variables, and how to handle real world data and observation noise.  We will also show a demo of the software suite for modeling and learning of self organization.


Speaker: Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Topics of Energetic Variational Approaches 3

Abstract: We will discuss the introduction of the energetic variational approaches and their applications. I will start with basic mechanics and chemical reaction dynamics.


Speaker: Prof. Shen Jie, Department of Mathematics, Purdue University

Title: Efficient structure preserving  schemes for complex nonlinear systems

Abstract: Solutions for a large class of partial differential equations (PDEs) arising from sciences and engineering applications  are required to be positive to be positive or within a specified bound, and/or energy dissipative.

It is of critical importance that their numerical approximations preserve these structures at the discrete level, as violation of these structures  may render the  discrete problems ill posed or inaccurate.

I will  review the existing approaches for constructing positivity/bound preserving  schemes, and then present  several efficient and accurate approaches: (i) through reformulation as Wasserstein gradient flows; (ii) through a suitable functional transform; and (iii) through a  Lagrange multiplier. These approaches have different advantages and limitations, are all relatively easy to implement and can be combined with most spatial discretizations.


Speaker: Prof. Ming Zhong, Department of Applied Mathematics, Illinois Tech

Title: How scientific machine learning can be used for knowledge discovery?

Abstract: Identifying the driving force for certain motion (i.e. planetary motion) or leading cause for certain disease (John Snow’s Cholera experiment in 1800 London) from data has been a crucial part of scientific development of human knowledge.  As observation/sensing techniques has boomed in the recently years, how to make informed guesses from large dataset within a certain time frame has become a great challenge.


Speaker:  Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Topics of Energetic Variational Approaches 4

Abstract: We will discuss the introduction of the energetic variational approaches and their applications. I will start with basic mechanics and chemical reaction dynamics.


Speaker: Dr. Yubin Lu, Department of Applied Mathematics, Illinois Tech

Title: Data-Driven Non-Gaussian Stochastic Dynamics: System Learning and Probability Distribution Estimation 

Abstract:  In this talk, we will introduce two aspects of data-driven analysis of non-Gaussian stochastic dynamics, i.e., extracting stochastic governing laws and estimating transition probability density. To be specific, on one hand, we will show how to learn a stochastic differential equation with L\'evy noise from data. On the other hand, we will show how to estimate the evolution of the transition probability density from sample path data. Some numerical results will also be introduced.


Speaker:  Prof. Shuwang Li, Department of Applied Mathematics, Illinois Tech

Title: Phase field modeling and computation of vesicle growth and shrinkage

Abstract:  We study a phase field model for vesicle growth or shrinkage based on osmotic pressure that arises due to a chemical potential gradient. The model consists of an Allen-Cahn-like equation, which describes the phase field evolution, a Cahn-Hilliard-like equation, which simulates the fluid concentration, and a Stokes-like equation, which models the fluid flow. It is mass conserved and surface area constrained during the membrane deformation. Conditions for vesicle growth or shrinkage are analyzed via the common tangent construction. The numerical computing is in two-dimensional space using a nonlinear multigrid method consisting of a FAS method for the PDE system. Convergence test suggests that the global error is of first order in time and of second order in space.  Numerical results are demonstrated under no flux boundary conditions and with boundary-driven shear flow respectively.  


Speaker:  Prof. Rolf Ryham, Department of Mathematics, Fordham University

Title: Hydrodynamics of Janus Particles Self-Assembled as Vesicles

Abstract:  Janus particles are widely used for self-assembly of mesoscopic structures with specific functions. We have constructed a model for self-assembly of Janus particles to form bilayer membranes under a hydrophobic potential (SIAM J. Multiscale Modeling, 2020). In our latest work (J. Fluid Mech., 2022), we illustrate the hydrodynamics of a vesicle made of such bilayer membranes. We use boundary integral equations to examine the hydrodynamics under various conditions: a quiescent flow, a planar shear flow, a linear elongation flow, and a Poiseuille flow. The simulation results show strong similarities to the vesicle hydrodynamics of a permeable lipid bilayer membrane and yield flowing conditions such as tank-treading motion, an asymmetric slipper, and membrane rupture. Moreover, the Janus-particle bilayers exhibit intermonolayer slip similar to that for two lipid monolayers and we calculate the friction coefficients.  Additionally, the talk will go into the physical considerations that led to the development of the hydrophobic attraction potential model, some mathematical results dealing with numerical issues, and proposes potential directions of research in mathematical analysis. 


Speaker:  Prof. Ming Zhong, Department of Applied Mathematics, Illinois Tech

Title: Learning Self Organization from Observation 2

Abstract:  Self-Organization (aka collective behavior) can be used to explain crystal formation, aggregation of cells, social behaviors of insects, synchronization of heart beats, etc. It is a challenging task to understand these types of phenomena from the mathematical point of view. We offer a statistical/machine learning approach to understand these behaviors from observation; moreover, our learning approach can aid in validating and improving the modeling of Self-Organization.

In the first part of the talk, we will focus on the forward modeling and backward learning of self organization.  We will review several important models which produce clustering, flocking, milling, and synchronization.  Then we will derive the learning method for inferring the interaction kernel from observation data and discuss its convergent properties.

In the second part of the talk, we will discuss how to expand the learning method to include more complicated models, complex geometries, missing feature variables, and how to handle real world data and observation noise.  We will also show a demo of the software suite for modeling and learning of self organization.


Speaker:  Prof. Ming Zhong, Department of Applied Mathematics, Illinois Tech

Title: Learning Self Organization from Observation 2

Abstract:  Self-Organization (aka collective behavior) can be used to explain crystal formation, aggregation of cells, social behaviors of insects, synchronization of heart beats, etc. It is a challenging task to understand these types of phenomena from the mathematical point of view. We offer a statistical/machine learning approach to understand these behaviors from observation; moreover, our learning approach can aid in validating and improving the modeling of Self-Organization.

In the first part of the talk, we will focus on the forward modeling and backward learning of self organization.  We will review several important models which produce clustering, flocking, milling, and synchronization.  Then we will derive the learning method for inferring the interaction kernel from observation data and discuss its convergent properties.

In the second part of the talk, we will discuss how to expand the learning method to include more complicated models, complex geometries, missing feature variables, and how to handle real world data and observation noise.  We will also show a demo of the software suite for modeling and learning of self organization.


Speaker:  Prof. Arkadz Kirshtein, Department of Mathematics, Tufts University

Title: Modeling and computations of multi-component fluid flow.

Abstract: In this talk I will introduce the systematic energetic variational approach for dissipative systems applied to multi-component fluid flows. These variational approaches are motivated by the seminal works of Rayleigh and Onsager. The advantage of this approach is that we have to postulate only energy law and some kinematic relations based on fundamental physical principles. The method gives a clear, quick and consistent way to derive the PDE system. I will discuss different approaches to multi-component flows using diffusive interface method. The diffusive interface method is an approach for modeling interactions among complex substances. The main idea behind this method is to introduce phase field labeling functions in order to model the contact line by smooth change from one type of material to another. Further I will introduce an energy stable numerical method for the proposed system system and discuss it's implementation, efficiency and possible further improvements.  


Speaker:  Dr. Senbao Jiang, Department of Mathematics, Illinois Tech

Title: Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations

Abstract: In this talk, we firstly propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals in n dimensions. The admissible class requires the singular part of the integrand in the weakly singular integral satisfies two simple hypotheses and is large enough to contain many fractional type singular kernels. The modified trapezoidal rule is the singularity-punctured trapezoidal rule plus correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule to exactly evaluate some monomials and solving corresponding linear systems. A long-standing difficulty of these types of methods is establishing the non-singularity of the linear system, despite strong numerical evidence. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence. 

Using the modified trapezoidal rule, we propose trapz-PiNN, a physics-informed neural network for solving the space-fractional Fokker-Planck equations in 2D and 3D. We demonstrate trapz-PiNNs have high expressive power through predicting solutions with low L2 relative error on a variety of numerical examples. Applications to backward problems are also presented.


Speaker: Trent Gerew, Department of Mathematics, Illinois Tech

Title: Swarmalators that Sync and Flock

Abstract: We present two new models, Swarmalator-Vicsek and Swarmalator-Cucker-Smale, which produce the synchronization, clustering, and flocking at the same time.  It is an generalization of the framework presented in the ``Oscillators that Sync and Swarm'' by Strogatz et al in 2017.  We present extensive numerical insights into how the synchronization of phases can affect the spatial pattern and correspondingly the flocking behavior.  We also discuss some major differences between the two new models and future directions.


Spring 2023

Speaker: Prof. Yuhan Ding, Department of Applied Mathematics, Illinois Tech

Title: Introduction to Monte Carlo and Quasi-Monte Carlo Methods 1

Abstract: This talk will introduce the basics of Monte Carlo methods and how to assess the performance of Monte Carlo methods and improve their effectiveness. The speaker will present the variance reduction methods, including control variates, importance sampling, antithetic variates, and stratified sampling. In addition, Quasi-Monte Carlo methods and their error estimations will be illustrated.


Speaker: Prof. Matthew A. Shapiro, Department of Political Science, Illinois Tech

Title: Crucial Problems in the Social Sciences: Opinion Formation, Governance/Institutions, and International Coordination

Abstract: This presentation will provide an overview of some of the most unresolved areas - theoretically and empirically - in the social sciences, broadly defined. Professor Matthew Shapiro (Social Sciences, IIT) will provide a sketch of these issues, make connections to his own past and present research, and offer avenues for fostering interdisciplinary research and curricula across the university.  


Speaker: Prof. Yuhan Ding, Department of Applied Mathematics, Illinois Tech

Title: Introduction to Monte Carlo and Quasi-Monte Carlo Methods 2

Abstract: This talk will introduce the basics of Monte Carlo methods and how to assess the performance of Monte Carlo methods and improve their effectiveness. The speaker will present the variance reduction methods, including control variates, importance sampling, antithetic variates, and stratified sampling. In addition, Quasi-Monte Carlo methods and their error estimations will be illustrated.


Speaker: Prof. Yuhan Ding, Department of Applied Mathematics, Illinois Tech

Title: Introduction to Monte Carlo and Quasi-Monte Carlo Methods 3

Abstract: This talk will introduce the basics of Monte Carlo methods and how to assess the performance of Monte Carlo methods and improve their effectiveness. The speaker will present the variance reduction methods, including control variates, importance sampling, antithetic variates, and stratified sampling. In addition, Quasi-Monte Carlo methods and their error estimations will be illustrated.


Speaker: Prof. Steven Wise, Department of Applied Mathematics, University of Tennessee

Title: Doubly Degenerate Cahn-Hilliard Models of Surface Diffusion

Abstract: Motion by surface diffusion is a type of surface-area-diminishing motion such that the enclosed volume is preserved and is important is many physical applications, including solid state de-wetting. In this talk I will describe a relatively recent diffuse interface model for surface diffusion, wherein the sharp-interface surface description is replaced by a diffuse interface, or boundary layer, with respect to some order parameter. One of the nice features of the new doubly degenerate Cahn-Hilliard (DDCH) model is that it permits a hyperbolic tangent description of the diffuse interfaces, in an asymptotic sense, but, at the same time, supports a maximum principle, meaning that the order parameter stays between two predetermined values. Furthermore, numerics show that convergence to the sharp interface solutions for the DDCH model is faster than that of the standard regular Cahn-Hilliard (rCH) model. The down side is that the new DDCH model is singular and much more nonlinear than the rCH model, which makes numerical solution difficult, and it is still only first order accurate asymptotically. We will describe positivity-preserving numerical methods for the new model and review some existing numerics. We will also describe very recent results on the rigorous Gamma convergence of the underlying diffuse interface energy.


Speaker: Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Free interface motion and phase field methods

Abstract:  I will present a class of free interface motions with application in geometry and physics. In particular, I will discuss the phase field methods and energetic variational approaches for these problems.


Speaker: Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Free interface motion and phase field methods 2

Abstract:  I will present a class of free interface motions with application in geometry and physics. In particular, I will discuss the phase field methods and energetic variational approaches for these problems.


Speaker: Prof. Kai Zhang, Department of Statistics and Operations Research, UNC

Title: BET and BELIEF

Abstract:  We study the problem of distribution-free dependence detection and modeling through the new framework of binary expansion statistics (BEStat). The binary expansion testing (BET) avoids the problem of non-uniform consistency and improves upon a wide class of commonly used methods (a) by achieving the minimax rate in sample size requirement for reliable power and (b) by providing clear interpretations of global relationships upon rejection of independence. The binary expansion approach also connects the symmetry statistics with the current computing system to facilitate efficient bitwise implementation. Modeling with the binary expansion linear effect (BELIEF) is motivated by the fact that wo linearly uncorrelated binary variables must be also independent. Inferences from BELIEF are easily interpretable because they describe the association of binary variables in the language of linear models, yielding convenient theoretical insight and striking parallels with the Gaussian world. With BELIEF, one may study generalized linear models (GLM) through transparent linear models, providing insight into how modeling is affected by the choice of link. We explore these phenomena and provide a host of related theoretical results. This is joint work with Benjamin Brown and Xiao-Li Meng.


Speaker: Dr. Tran Hoang, Oak Ridge National Lab

Title: High-Dimensional Optimization with a Novel Nonlocal Gradient

Abstract:  The problem of minimizing multi-modal loss functions with a large number of local optima frequently arises in machine learning and model calibration problems. Since the local gradient points to the direction of the steepest slope in an in infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea of DGS is to conducts 1D long-range exploration with a large smoothing radius along d orthogonal directions in Rd, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. The d directional derivatives are then assembled to form the nonlocal gradient. We use the Gauss -Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We provide a convergence theory in the scenario where the objective function is composed of a convex function perturbed by a highly oscillating, deterministic noise. We prove that our method exponentially converge to a tightened neighborhood of the solution, whose size is characterized by the noise wavelength. The superior performance of our method is demonstrated in several high-dimensional benchmark tests, machine learning and model calibration problems.


Speaker: Prof. Xiang Wan, Department of Mathematics and Statistics, Loyola University Chicago

Title: Poisson Equations in Two-dimensional Domains with Line Fracture: from Qualitative to Quantitative Analysis

Abstract: In this talk, we investigate the 2-d Poisson equation with the forcing term being a Dirac delta function on a line segment, modeling a singular line fracture. Numerically, such a fracture imposes additional treatment of the meshing while constructing the triangular Finite Element space. Inspired by the 1-d case, we can see that a graded meshing is naturally called for, where the level of grading depends on the distance to the fracture.

In order to tune the numerical analysis of this system to the optimal convergence rate, one has to look closer into the regularity of the solution in weighted Sobolev spaces - in contrast to the regularity results in standard Sobolev spaces from the classic Elliptic theory. Such examination reveals deeper connections between the qualitative regularity and quantitative behavior of the system.  Last but not least, we will present how the characteristics, and lack thereof, of different geometries of domains plays a role via numerical demonstrations.


Speaker: Prof. Robert Pego, Department of Mathematical Sciences, CMU

Title: Breaking glass optimally and Minkowski's problem for polytopes

Abstract: Motivated by a study of least-action incompressible flows, we study all the ways that a given convex body in Euclidean space can break into countably many pieces that move away from each other rigidly at constant velocity, following geodesic motions in the sense of optimal transport theory. These we classify in terms of a countable version of Minkowski's geometric problem of determining convex polytopes by their face areas and normals. Illustrations involve a number of curious examples both fractal and paradoxical, including Apollonian packings and other types of full packings by smooth balls.

Fall 2023

Speaker:  Prof. Takashi Kagaya, Muroran Institute of Technology

Title: Asymptotic behavior of geometric flows with contact angle conditions

Abstract: Several geometric flows were derived from interface phenomena. In this talk, contact angle conditions for the geometric flows are dealt with, motivated by surface tension problems. The asymptotic behavior of the geometric flows depends on the contact angle conditions. In particular, traveling waves have the asymptotic stability if we assume specific contact angle conditions. The uniqueness and shape of the traveling wave also depend on the geometric flow equations. I will introduce my results related to the asymptotic behavior and properties of the traveling wave. This talk contains joint works with Prof. Shimojo (Tokyo metropolitan university) and Prof. Kohsaka (Kobe university).


Speaker:   Prof. Masashi Mizuno, Department of Mathematics, Nihon University

Title: The Simon-Lojasiewicz gradient inequality for a PDE related to grain boundary motion

Abstract: The Simon-Lojasiewicz gradient inequality is widely applied to study long-time asymptotics for time-dependent PDEs, especially gradient descent flows. Here, I present a PDE related to grain boundary motion. Next, we derive the Simon-Lojasiewicz gradient inequality related to the PDE. We will mainly focus on how to set the function space. This talk is based on the joint work with Ayumi Sakiyama (Nissay Information Technology Co, Ltd.) and Keisuke Takasao (Kyoto University).


Speaker:  Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Introduction to energetic variational approaches

Abstract: This is a series of lectures that will introduce the general framework of energetic variational approaches, with application  in biology, engineering and data sciences. The first lecture will be accessible to first year graduate students to energetic variational approaches.


Speaker:  Prof. Trevor Leslie, Department of Applied Mathematics, Illinois Tech

Title: Questions and Philosophy of the Mathematical Theory of Collective Behavior

Abstract: I will describe in broad strokes some goals in the mathematical study of collective behavior and give examples of several relevant models.  We will discuss the multi-scale framework, inspired by the kinetic theory of gas dynamics, which is used to study some of these models.  Finally, we will discuss several possible communication rules for the Cucker--Smale system of ODE's, and how these communication rules tend to affect the dynamics.  This will be the first talk in a several-part series.  Depending on the interests of the audience, future talks may concern the mean-field (discrete-to-kinetic) and/or kinetic-to-hydrodynamic limits, the Eulerian and Lagrangian perspectives, regularity theory for the PDE's involved, and the prediction of limiting states based on the initial data and communication rules.  These topics will be presented in the context of the Cucker--Smale and Euler alignment systems, but the focus will be on the tools involved, which are relevant to more general equations.


Speaker:  Prof. Chun Liu, Department of Applied Mathematics, Illinois Tech

Title: Lecture 2: Introduction to Energetic Variational Approaches: Calculus of Variations.

Abstract: In this talk, I will go over some basic concepts and calculations in calculus of variations. Will also introduce the concepts of various weak/variational solutions, Pokhozhaev's identity and Hamilton's principle.


Speaker:  Prof. Trevor Leslie, Department of Applied Mathematics, Illinois Tech

Title: Euler-alignment system: the Eulerian and Lagrangian formulations

Abstract: We continue our discussion of the Cucker--Smale system and its kinetic and hydrodynamic analogues, focusing primarily on the hydrodynamic Euler-alignment system.  For bounded interaction rules, we write the system in both Eulerian and Lagrangian coordinates and discuss the existence (or nonexistence) of global-in-time classical solutions.


Speaker:  Prof. Trevor Leslie, Department of Applied Mathematics, Illinois Tech

Title: More basics of the Euler Alignment System: 1D classical solutions and the heavy-tail condition for flocking

Abstract: Last time, starting from a kinetic version of the Cucker--Smale ODE's, we derived the Eulerian and Lagrangian formulations of the pressureless Euler Alignment system.  This time, we show how both of these formulations simplify drastically in 1 spatial dimension, due to the appearance of additional conserved quantities.  These quantities furthermore lead naturally to an essentially sharp criterion for the existence of classical solutions [Carrillo-Choi-Tadmor-Tan 2016, Leslie 2020].  Next, we describe the now well-known "heavy-tail" criterion which guarantees (in any spatial dimension) exponentially fast convergence to a flock.  If time allows, we will give the proof of this statement, which boils down to finding an appropriate Lyapunov function.  The argument we present is essentially due to [Ha-Liu 2009] but has been optimized by other authors. 


Speaker:  Prof. Chiun-Chang Lee, National Tsing Hua University

Title: An Introduction to Schrödinger equations and semiclassical limits 1

Abstract: Nonlinear Schrödinger equations are a research topic with a wide range of applications, including problems in nonlinear optics and models of Bose–Einstein condensates. The semiclassical limit serves as a bridge between quantum mechanics and classical mechanics, enabling us to understand how classical physics emerges from quantum mechanical principles as Planck's constant tends to zero.

In this talk, I will present a method for studying the semiclassical limits of Schrödinger equations. Using Madelung's transformation, I will formally derive its hydrodynamic equations and the corresponding limiting equations, specifically the compressible Euler equations.


Speaker:  Prof. Wei Cai, Department of Mathematics, Southern Methodist University

Title: DeepMartNet - A Martingale based Deep Neural Network Algorithm for Eigenvalue/BVP Problems of PDEs and Optimal Stochastic Controls

Abstract: In this talk, we will present a deep neural network (DNN) learning algorithm for solving high dimensional Eigenvalue (EV) and boundary value problems (BVPs)  for elliptic operators and initial BVPs (IBVPs) of quasi-linear parabolic equations as well as optimal stochastic controls. The method is based on the Martingale property in the stochastic representation for the eigenvalue/BVP/IBVP problems and martingale principle for optimal stochastic controls. A loss function based on the Martingale property can be used for an efficient optimization by sampling the stochastic processes associated with the elliptic operators or value process  for stochastic controls. The Martingale property conforms naturally with the stochastic gradient descent process  for the DNN optimization. The proposed algorithm can be used for eigenvalue problems and BVPs and  IBVPs with Dirichlet, Neumann, and Robin boundaries in bounded or unbounded domains and some feedback stochastic control problems. Numerical results for BVP and EV problems in high dimensions will be presented.


Speaker:  Prof. Alexey Cheskidov, Department of Mathematics, University of Illinois at Chicago

Title: Dissipation anomaly for long time averages

Abstract: In turbulent flows, the energy injected at forced low modes (large scales) cascades to small scales through the inertial range where viscous effects are negligible, and only dissipates above Kolmogorov’s dissipation wavenumber. The persistence of the energy flux through the inertial range is what constitutes dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. We first analyze these intrinsically linked phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where anomalous dissipation occurs without dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp.


Speaker:  Dr. Pawan Negi, Department of Applied Mathematics, Illinois Tech

Title: Weakly compressible smoothed particle hydrodynamics method

Abstract: In my talk, I will introduce the smoothed particle hydrodynamics (SPH) followed by the weakly compressible SPH method to solve time accurate incompressible fluid flow problems. If time permits I will talk about the application of the method to various other fields.


Speaker:  Prof. Jay Schieber, Department of Chemical Engineering, Illinois Tech

Title: Multiscale Modeling Beyond Equilibrium: Viscoelasticity

Abstract: Multiscale modeling is used to predict properties of systems that have microstructure (Physics Today, March 2020, 10.1063/PT.3.4430). Such predictions require a conversation between two different length scales. In our focus, we have a macroscopic flow or deformation that influences the evolution of the microstructure, and the microstructure provides the stresses that determine the flow. Hence, we simultaneously solve two coupled dynamical equations. I will discuss the restrictions provided by the first and second laws of thermodynamics on such models. We show a general approach, but apply them specifically to polymer rheology. We need little mathematics beyond the sort of partial differential equations and tensorial mathematics necessary in transport phenomena. Nonetheless, we discover that several recent and popular models violate these fundamental laws of physics. All results are published in detail (Physics of Fluids, 2021, 33, 083103).


Speaker:  Prof. Chiun-Chang Lee, National Tsing Hua University

Title: An Introduction to Schrödinger equations and semiclassical limits 2

Abstract: In continuation of the last discussion, I will present a case study on the semiclassical analysis of nonlinear Schrödinger equations. The focus will be on introducing a modulated energy functional to govern mass density and linear momentum, particularly in relation to the compressible Euler equations. Additionally, I will introduce a nonlocal Schrödinger equation, drawing connections to a nonlocal Euler equation through Madelung's transformation. It is important to note that rigorous analysis regarding the convergence and existence of solutions for this nonlocal Euler equation remains an open challenge in my current research. I welcome discussions on this unresolved problem.

Spring 2024

Speaker: Prof. Ming Zhong

Title: Numerical Study of the Bertozzi Model for Modeling Anti-social Activities with PyTorch Application

Abstract: We present the ODE and PDE models from the Bertozzi's model for understanding some anti-social activities.  We compare such model to Gray Scott and Keller Segel, and present some ongoing results on how to build a data-driven model from observation using PyTorch.  Code demo via PyTorch is also given.


Speaker: Prof. Ming Zhong

Title: Lecture II - Numerical Study of the Bertozzi Model for Modeling Anti-social Activities with PyTorch Application

Abstract: We present the ODE and PDE models from the Bertozzi's model for understanding some anti-social activities.  We compare such model to Gray Scott and Keller Segel, and present some ongoing results on how to build a data-driven model from observation using PyTorch.  Code demo via PyTorch is also given.


Speaker: Jhih-Hong Lyu (National Taiwan University)

Title: PB-Steric Equation: A General Model of Poisson-Boltzmann Equations

Abstract: Poisson-Boltzmann equation is a fundamental model for describing the distribution of ions. When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson-Boltzmann equation, which treats ions as a point without volume, should be modified. Several modified PB equation were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter Λ, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effect and chemical potentials. As Λ=0, the PB-steric equation becomes the conventional PB equation but as Λ>0, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as Λ goes to infinity. Our theoretical results show that the PB-steric equations (for 0≤Λ≤∞) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations.


Speaker: Dr. Yubin Lu (IIT)

Title: Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems

Abstract: The prediction of stochastic dynamical systems and the capture of dynamical behaviors are profound problems. In this talk, I will introduce a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue, which mimics error modeling to improve traditional Reservoir Computing performance and integrates the virtues of both approaches. With few assumptions about the underlying stochastic dynamical systems, this model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. We verify the effectiveness of the proposed framework in several experiments, including the stochastic Van der Pal oscillator, El Niño-Southern Oscillation simplified model, and stochastic Lorenz system. These experiments consist of Markov/non-Markov and stationary/non-stationary stochastic processes which are defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. Additionally, we explore the noise-induced tipping phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and replication of the strange attractor.


Speaker: Prof. Jiahong Wu (University of Notre Dame)

Title: Stabilizing phenomenon for incompressible fluids

Abstract: This talk presents recent stability results on several PDE systems modeling fluid flows.  These results reflect a seemingly universal stabilizing phenomenon exhibited in quite different fluids. The 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor in the Oldroyd-B model, small smooth data always lead to global and stable solutions. Solutions of the 2D Navier-Stokes in R^2  with dissipation in only one direction are not known to be stable, but the Boussinesq system involving this Navier-Stokes is always stable near the hydrostatic equilibrium. The buoyancy forcing helps stabilize the fluid. The 3D incompressible Navier-Stokes equation with dissipation in only one direction is not known to always have global solutions even when the initial data are small. However, when this Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamic system, solutions near a background magnetic field are shown to be always global in time. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect.


Speaker: Prof. Gavish Nir (Israel Institute of Technology)

Title: Epidemiology at Elevated Reproduction Numbers 

Abstract: During the COVID-19 pandemic, the emergence of the Omicron variant created an unprecedented scenario of an epidemic driven by a highly transmissible variant. Our data-driven research during that time showed that a variant with a basic reproduction number as high as 10 can defy conventional theory in certain circumstances.  Motivated by this experience, I will present two works focusing on the epidemic theory of highly transmissible variants:

1) Optimizing vaccine allocation is crucial for effective vaccination campaigns against epidemics. Contrary to intuition and classic vaccination theory, we show that for leaky vaccines and high basic reproduction numbers, the optimal allocation strategy for minimizing infections prioritizes those least likely to be infected. These findings have important implications for managing vaccination campaigns against highly transmissible infections.  

2) The competitive exclusion principle in epidemiology implies that when competing strains of a pathogen provide complete protection for each other, the strain with the largest reproduction number outcompetes the other strains and drives them to extinction. Various trade-off mechanisms facilitate the coexistence of competing strains in epidemic systems, particularly when their respective basic reproduction numbers are close so that the competition between the strains is weak. One may expect that a substantial competitive advantage of one of the strains will eventually outbalance mechanisms that facilitate coexistence, aligning with the principles of competitive exclusion. Yet, the literature lacks a rigorous validation of this statement.

In this work, we challenge the validity of the exclusion principle at an ultimate limit in which one strain has a vast competitive advantage over the other strains.  Our results show that the competitive exclusion principle does not, unconditionally, hold beyond the established case of complete immunity.

Joint work with Guy Katriel.



Speaker: Prof. Yuanzhen Shao (University of Alabama)

Title: Thermodynamically consistent models for non-isothermal magnetoviscoelastic fluids

Abstract: In this talk, we consider the motion of a magnetoviscoelastic fluid in a non-isothermal environment. When the deformation tensor field is governed by a regularized transport equation, the motion of the fluid can be described by a quasilinear parabolic system. We will establish the local existence and uniqueness of a strong solution. Then it will be shown that a solution initially close to a constant equilibrium exists globally and converges to a (possibly different) constant equilibrium. Further, we will show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria. If time permits, we will discuss some recent advancements regarding the scenario where the deformation tensor is modeled by a transport equation. In particular, we will discuss the local existence and uniqueness of a strong solution as well as global existence for small initial data.


Speaker: Prof. Cheng Wang (Umass, Dartmouth)

Title: Global in time energy estimate for the exponential time differencing Runge-Kutta (ETDRK) numerical scheme for the phase field crystal equation

Abstract: The global in time energy estimate is derived for the ETDRK2 numerical scheme for the phase field crystal (PFC) equation, a sixth order parabolic equation to model crystal evolution. The energy stability is available for the exponential time differencing Runge-Kutta (ETDRK) numerical scheme to the gradient flow equation, under an assumption of global Lipschitz constant. To recover the stabilization constant value, some local-in-time convergence analysis has been reported, so that the energy stability becomes available over a fixed final time. In this work, we develop a global in time energy estimate for the ETDRK2 numerical scheme to the PFC equation, so that the energy dissipation property is valid for any final time. An a-priori assumption at the previous time step, combined with a single-step H^2 estimate of the numerical solution, turns out to be the key point in the analysis. Such an H^2 estimate recovers the maximum norm bound of the numerical solution at the next time step, so that the stabilization parameter value could be theoretically justified. This justification in turn ensures the energy dissipation at the next time step, so that the mathematical induction could be effectively applied, and the global-in-time energy estimate is accomplished. This technique is expected to be available for many other Runge-Kutta numerical schemes.


Speaker: Aleksei Sorokin (IIT)

Title: Walsh Functions and Spaces

Abstract: The Fourier series expands a periodic function into a sum of trigonometric functions. For smooth periodic functions, the Fourier coefficients decay rapidly with some regularity. In this talk, we explore the Walsh functions which form a complete orthonormal system in L2([0,1)) without requiring periodicity in the function. Like the Fourier coefficients, when a function is known to have certain smoothness we can say something about the decay of its Walsh coefficients. Spaces of smooth periodic functions derived from decays of Fourier coefficients and spaces containing smooth functions derived from the decay of the Walsh coefficients are explored. The speaker makes connections to his research at the intersection of kernel methods and Quasi-Monte Carlo.

This is a joint seminar between the Computational Math and Stats group and the Multiscale group.