The Schedule

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 Shuwang Li

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 Rolf Ryham

Title: Learning Self Organization from Observation

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

Title: Learning Self Organization from Observation (III)

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

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:  Arkadz Kirshtein, Ph. D.

Norbert Wiener Assistant Professor

Department of Mathematics

Tufts University

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:  Senbao Jiang

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.

Speaker:  Trent Gerew

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

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 

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 Matthew Shapiro

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

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 

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

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 

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 Steven Matthew

Title: 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 

Title: Free interface motion and phase field methods II 

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 

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:  Prof  Zhang, Kai

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:  Dr. Tran, Hoang ( Oak Ridge National Lab)

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  Xiang Wan

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.

Speaker:  Prof  Robert Pego (CMU)

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. Takashi Kagaya (Muroran Institute of Technology)

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. Masashi Mizuno 

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. Chun Liu

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. Trevor Leslie 

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. Chun Liu