Past Events

• December 17, 11:30am-12:30pm (CST) online via zoom Transport information flows for Bayesian sampling problems

Wuchen Li, Department of Mathematics, University of South Carolina

Abstract: In AI and inverse problems, the Markov chain Monte Carlo (MCMC) method is a classical model-free method for sampling target distributions. A fact is that the optimal transport first-order method (gradient flow) forms the MCMC scheme, known as Langevin dynamics. Therefore, a natural question arises: Can we propose accelerated or high order optimization techniques for MCMC? We positively answer this question by applying optimization methods from optimal transport and information geometry, a.k.a. transport information geometry. E.g., we introduce a theoretical framework for accelerated gradient flow and Newton's flows in probability space w.r.t. the optimal transport type metric. Numerical examples are given to demonstrate the effectiveness of these optimization-based sampling methods.

• December 16, 11:30am-12:30pm (CST) online via zoom How Differential Equations and Random Graph Insights Benefit Deep Learning

Bao Wang, Scientific Computing and Imaging Institute, University of Utah

Abstract: We will present recent results on developing new deep learning algorithms leveraging differential equations and random graph insights.

First, we will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions. Third, if time permits, we will discuss building an efficient and reliable overlay network for decentralized federated learning based on the random graph theory.

• December 10, 11:30am-12:30pm (CST) online via zoom Flexoelectricity and three-dimensional solitons in nematic liquid crystals

Ashley Earls, Basque Center for Applied Mathematics

Abstract: This presentation deals with experimentally observed three dimensional solitons that develop in flexoelectric nematic liquid crystals, with negative dielectric and conduction anisotropies, when subject to an alternating electric field. The liquid crystal is confined in a thin region between two plates perpendicular to the applied field. The initial uniformly aligned director field is at equilibrium due to the negative anisotropy of the media. However, this state is unstable to perturbations that manifest themselves as confined, bullet-like, director distortions traveling up and down the sample at a speed of several hundred microns per second. We develop a variational model that couples the Ericksen-Leslie equation of director field distortion and the Poisson-Nernst-Planck equations governing the diffusion and transport of electric charge and the electrostatic potential. We perform a stability analysis of the equilibrium state to determine the threshold wave numbers and speed of the disturbance. We apply the Fourier transform, time and space averaging tools, and arguments based on asymptotic analysis to the linear system. Our predictions for the size, phase-shift, and speed of the solitons show good agreement with experiments.

• December 3, 11:30am-12:30pm (CST) online via zoom Supervised Optimal Transport

Yanxiang Zhao, Department of Mathematics, George Washington University

Abstract: Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan which traditional optimal transport cannot enforce. Here we introduce Supervised Optimal Transport (sOT) that formulates a constrained optimal transport problem where couplings between certain elements are prohibited according to specific applications. sOT is proved to be equivalent to an $l^1$ penalized optimization problem, from which efficient algorithms are designed to solve its entropy regularized formulation. We demonstrate the capability of sOT by comparing it to other variants and extensions of traditional OT in color transfer problems. We also study the barycenter problem in sOT formulation, where we discover and prove a unique reverse and portion selection (control) mechanism. Supervised optimal transport is broadly applicable to applications in which a constrained transport plan is involved and the original unit should be preserved by avoiding normalization.

• December 2, 11:30am-12:30pm (CST) online via zoom Variational approach to poroelasticity

Arkadz Kirshtein, Department of Mathematics, Tufts University

Abstract: In this talk I will discuss modeling fluid flow through a deformable porous medium. I will start from introducing a variational approach for fluids and elasticity in Lagrangian coordinates. Next I will discuss an existing approach based on Biot's consolidation model. Then I will introduce a system derived using energetic variational approach and discuss numerical methods that could be applied to it.

• November 5, 3:00pm-4:00pm in RE 119 The reference map technique for simulating complex materials and multi-body interactions

Prof. Chris Rycroft, John A. Paulson School of Engineering and Applied Sciences, Harvard University

Abstract: Conventional computational methods often create a dilemma for fluid–structure interaction problems. Typically, solids are simulated using a Lagrangian approach with a grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems, and several examples in two and three dimensions will be presented.

• October 28, 11:30am-12:30pm (CDT) online via zoom Modelling Electrochemical Systems with Continuum Thermodynamics - from Fundamental Electrochemistry to Porous Intercalation Electrodes

Manuel Landstorfer, WIAS Berlin

Abstract: In this talk I will give an overview on the modeling of electrochemical systems on the basis of non-equilibrium thermodynamics. On the basis of a mixture theory for electrolyte species, which explicitly accounts for solvation effects [1], I will show how to deduce a self-consistent model for the electrochemical double layer [2], with a profound experimental validation [5]. Some aspects of incomplete dissociation in the context of solvation are further discussed [3]. Via asymptotic analysis, double layer effects can be incorporated in effective boundary conditions [4,5], yielding transport equations for electro neutral electrolytes. These can be employed to model, for example, lithium ion batteries. Within a porous battery electrode, at least three scales arise: the macroscopic porous media scale, the micro-structure scale, and the double layer scale. Each of these scales fundamentally contributes to the overall performance of the battery cell, and several strategies exist to overcome the scales in a mathematical model [6-9]. On the basis of general balance equations and non-equilibrium thermodynamics, as well as material models for the intercalation phase, homogenization techniques are employed to deduce the transport equations for the porous media scale as well as the geometric cell problem for the micro-structure [10-12]. For various material models of the intercalation phase we discuss some aspects of the open circuit potential on the porous media scale [10]. Numerical simulations show the validity of the overall approach, and potential extensions regarding ageing effects are finally discussed.

References:

[1] W. Dreyer, C. Guhlke, and M. Landstorfer. Electrochemistry Communications, 43 (2014).

[2] M. Landstorfer, C. Guhlke, W. Dreyer, Electrochim. Acta, 201 (2016)

[3] M. Landstorfer. Electrochemistry Communications, 92 (2018).

[4] W. Dreyer, C. Guhlke, R. Müller, Phys. Chem. Chem. Phys., 18 (2016)

[5] M. Landstorfer. J. Electrochem. Soc., 164, (2017)

[6] M. Landstorfer and T. Jacob, Chem. Soc. Rev., 42 (2013)

[7] M. Doyle, T. F. Fuller, J. Newman, J. Electrochem. Soc. 140 (1993), 6, 1526

[8] J. Newman, K. Thomas. Electrochemical Systems. John Wiley & Sons, 2014.

[9] T. R. Ferguson and M. Z. Bazant, J. Electrochem. Soc., 159 (2012)

[10] U. Hornung. Homogenization and porous media. Springer Science & Business Media, 2012

[11] G. Allaire. SIAM J. Math. Anal., 23 (1992)

[12] M. Landstorfer M. Ohlberger S. Rave M. Tacke, WIAS Preprint 2882 (2021)

• October 21, 11:30am-12:30pm (CDT) Survey of Solution Landscapes for Two-Dimensional Confined Nematic Liquid Crystals

Apala Majumdar, University of Strathclyde Glasgow

Abstract: Nematic liquid crystals are quintessential examples of partially ordered materials that combine fluidity with the orientational order or directionality of solids. Nematics have long been the working material of choice for a range of electro-optic devices and there are now new applications in nano-sciences, biomimetic materials, metamaterials, polymers and artificial intelligence, to name a few. We survey some recent work on the mathematical modelling, analysis and simulations of nematic liquid crystals confined to two-dimensional geometries, that mimic thin systems. We work within the celebrated Landau-de Gennes theory for nematic liquid crystals and illustrate how the geometry, boundary conditions and material properties tailor the solution landscapes of these confined systems, with emphasis on the multiplicity of stable solutions and their singular sets. The singular sets act as distinguished sites for materials design and responses, and we conclude by reviewing some modelling of recent experimental work on nematic shells. All collaborations will be acknowledged throughout the talk.

• October 7, 11:30am-12:30pm (CDT) in RE 027 Energetic variational inference: a variational approach to Bayesian statistics and beyond

Yiwei Wang, Department of Applied Mathematic, Illinois Institute of Technology

Abstract: Bayesian inference is one of the most important techniques in modern statistics and data science. In this talk, we present a variational framework to Bayesian methods, called energetic variational inference (EVI), which views a posterior as a minimizer of a functional and uses an energy-dissipation law to characterize the minimization procedure. The framework is motivated by non-equilibrium physics. Using the EVI framework, we can derive many existing flow-based Variational Inference (VI) methods, including the popular Stein Variational Gradient Descent (SVGD) approach. More importantly, many new algorithms can be created under this framework. As an example, we propose a new particle-based VI (ParVI) scheme, which performs the particle-based approximation of the density first and then uses the approximated density in the variational procedure. The “approximation-then-variation” procedure enables us to construct a numerical algorithm based on a minimizing movement scheme (implicit Euler scheme). Numerical experiments show the proposed method outperforms some existing ParVI methods in terms of fidelity to the target distribution. The framework can be applied to a wide class of unsupervised learning problems beyond the Bayesian methods, such as the generative model and the density estimation.

This is joint work with Prof. Chun Liu and Prof. Lulu Kang.

• October 1, 11:15am-12:15pm (CDT) online via zoom Ion-dependent DNA Configuration in Bacteriophage Capsids

Pei Liu, College of Science and Engineering, University of Minnesota

Abstract: The conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure. Experiments have confirmed that the details of the hexagonal packing depend on the electrochemistry of the capsid and its environment. We propose a biophysical model that quantifies the relationship between DNA configurations inside bacteriophage capsids and the types and concentrations of ions present in a biological system. We introduce an expression for the free energy which combines the electrostatic energy with contributions from bending of individual segments of DNA and Lennard--Jones-type interactions between these segments. The equilibrium points of this energy solve a partial differential equation that defines the distributions of DNA and the ions inside the capsid. We develop a computational approach that allows us to simulate much larger systems than what is currently possible using the existing simulations, typically done at a molecular level. In particular, we are able to estimate bending and repulsion between DNA segments as well as the full electrochemistry of the solution, both inside and outside of the capsid. The numerical results show good agreement with existing experiments and molecular dynamics simulations for small capsids. We also perform a bifurcation analysis about the preferred coiling state in the form of concentric circles or helical structures.

• September 9 and September 16, 2021, 11:30am-12:30pm (CDT) in RE 027 Energetic Variational Approach

Chun Liu, Department of Applied Mathematic, Illinois Institute of Technology

Abstract: In this talk we introduce the Energetic Variational Approach (EnVarA) for the analysis of mechanical systems, kinematics and transport theory, and principles for Hamiltonian and dissipative systems. We also give basic examples of simple solid and fluid systems and discuss what characterizes solid and fluid materials.

• May 28, 2021, 11m-12pm (CDT) A surface moving mesh PDE method

Prof. Weizhang Huang, Department of Mathematics, University of Kansas

Abstract: We will present a surface moving mesh method for general surfaces with or without explicit parameterization. The method is an extension of the moving mesh partial differential equation (MMPDE) method that has been developed for bulk meshes. The surface moving mesh equation is defined as the gradient system of a meshing energy function, with the nodal mesh velocities being projected onto the underlying surface. Like the bulk mesh situation, we show that any mesh generated by the surface moving mesh method remains nonsingular if it is so initially. Moreover, a surface meshing energy function is presented based on mesh equidistribution and alignment. The main challenges in the development come from the fact that the Jacobian matrix of the affine mapping between the reference element and a simplicial surface element is not square. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples will be presented.

• April 16, 2021, 11m-12pm (CDT) Numerical Approximation of Parabolic SPDE’s

Professor Noel J. Walkington, Department of Mathematical Sciences, Carnegie Mellon University

Convergence theory for numerical schemes to approximate solutions of stochastic parabolic equations of the form form $du + A(u) dt = f dt + g dW, u(0) = u^0$ , will be reviewed. Here u is a random variable taking values in a function space U, $A : U → U'$ is partial differential operator, $W = {Wt}t≥0$ a Wiener process, and f, g, and u^0 are data. This talk will illustrate how techniques from stochastic analysis and numerical partial differential equations can be combined to obtain a realization of the Lax–Richtmeyer meta–theorem: A numerical scheme converges if (and only if) it is stable and consistent. Structural properties of the partial differential operator(s) and probabilistic methods will be developed to establish stability and a version of Donsker’s theorem for discrete processes in the dual space U' . This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).

• March 26, 2021, 11am- 12pm (CDT) From discrete Moran processes to continuous Kimura equations via optimal transport

Professor Léonard Monsaingeon, IECL Univ. Lorraine & GFM Univ. Lisboa

Abstract: The Moran process and the Kimura equation are two well-established models in evolutionary genetics. The former is a discrete Markov process

with fixed population size $N$, while the latter is a continuous one-dimensional parabolic PDE. Both share the particularity that they possess absorbing states and boundary conditions. In this talk I will show that these two evolution equations can be considered as variational gradient flows of the entropy, each with respect to some non-standard optimal transport geometry. The analysis involves a singular conditioning of the underlying stochastic processes in order to compensate for the absorption. The discrete and continuous geometries are moreover compatible in the hydrodynamic limit $N\to\infty$, and one thus expects at least formally a Gamma-convergence of gradient-flows in the spirit of Sandier-Serfaty.

This is a joint work with Fabio Chalub, Ana Ribeiro, and Max Souza.

• March 12, 2021, 11am-12pm (CT) Some recent developments on SAV approaches for complex dissipative systems

Professor Jie Shen, Department of Mathematics, Purdue University

Abstract: Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme.

I will present some recent advances on using the scalar auxiliary variable (SAV) approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

• March 11, 2021, 3:30pm- 4:30pm (CT) Well-posedness of the Abels-Garcke-Grün model for two-phase flows

Doctor Andrea Giorgini, Department of Mathematics, Indiana University, Bloomington

Abstract: In the last decades, the Diffuse Interface-Phase Field theory has made enormous progresses in the description of multi-phase flows from modeling to numerical analysis. A particularly active research topic has been the development of thermodynamically consistent extensions of the well-known Model H for the case of unmatched (homogeneous) densities of the fluids. In the talk, I will focus on the AGG model proposed by H. Abels, H. Garcke and G. Grün in 2012. The model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a concentration-dependent density and an additional flux term due to interface diffusion. Using the method of matched asymptotic expansions, it was shown that the sharp interface limit of the AGG model corresponds to the two-phase Navier-Stokes equations. In the literature, the analysis of the AGG system has so far been focused on the existence of weak solutions. During the seminar, I will present some recent results concerning the existence, uniqueness and stability of strong solutions for the AGG model in two dimensions.

• February 19, 2021, 11:30am-12:30pm (CT) Uncertainty quantification in inverse PDE problems with shape constraints

Professor Matthew Dixon, Department of Applied Mathematics, Illinois Institute of Technology

• February 12, 2021, 11am-12pm (CT) Self-generating lower bounds for the Boltzmann equation

Professor Andrei Tarfulea, Department of Mathematics, Louisiana State University

Abstract: The Boltzmann equation models the space and velocity distribution of the particles in a diffuse gas. The particles collide with each other at microscopic scales, leading to a quadratic, nonlocal (in velocity), collision operator that behaves somewhat like a fractional Laplacian. In recent years there has been substantial progress on the regularity and continuation program for the Cauchy problem. Notably, a smooth and unique solution exists for as long as the so-called hydrodynamic quantities remain "under control'': the mass, energy, and entropy densities must stay bounded above uniformly in space and the mass density must stay bounded below uniformly in space. The last condition is crucial for smoothing since it gives the collision operator elliptic properties in certain velocity directions. We show that the solution to the Boltzmann equation (even starting from initial data that contains large regions of vacuum) instantaneously fills space. That is, the gas diffuses and spreads positive mass to every space and velocity coordinate at any positive times. We obtain this result dynamically through barrier arguments for moving mass through space and a De Giorgi type iteration for spreading mass to arbitrary velocities. A consequence is that the above continuation criterion can now be weakened; it is no longer necessary to assume that the mass density is bounded from below for continuation of smooth solutions. Joint work with Christopher Henderson and Stanley Snelson.

• January 29, 2021, 11am-12pm (CT) On hydrodynamic fluctuations from nonlinear heat equation given by stochastic Carleman particles

Professor Jin Feng, Department of Mathematics, University of Kansas, Lawrence

Abstract: The deterministic Carleman equation can be considered as a one dimensional two speed fictitious gas model. Its associated (2-scale, skipping the kinetic scale) hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as mean field, we derive a macroscopic fluctuation theory (MFT) associated with it. Our main focus is on methodology which is new and different than existing ones. The new method has potential to be used on deterministic Hamiltonian particle systems. It is formulated in terms of Hamilton-Jacobi equation in the space of probability measures. It relies on characterization and convergence (2 scale) using abstract viscosity solutions and weak KAM theory extended to infinite dimensions (particles). Also important is proper choices of coordinates by using the density-flux description of the problem.

• January 28, 2021, 1pm-2pm (CT) Adhesion of droplets with surfactant: Onsager’s principle, variational inequality, computations

Doctor Yuan Gao, Duke University

Abstract: The capillary effect caused by the interfacial energy dominates the dynamics of small droplets, particularly the contact lines (where three phases meet). With insoluble surfactant laid on the capillary surface, the adhesion of droplets to some textured substrates becomes more complicated: (i) insoluble surfactant moves with the evolutionary capillary surface (ii) the surfactant-dependent surface tension will in turn drive the full dynamics of droplets, particularly the moving contact lines. Using Onsager’s principle with different Rayleigh dissipation functionals, we derive and compare both the geometric motion of the droplets and the viscous flow model inside the droplets. To enforce impermeable obstacle constraint, the full dynamics of the droplet can be formulated as a gradient flow on a manifold with boundary, which leads to effective numerical methods. We propose unconditionally stable first/second order numerical schemes based on explicit moving boundaries and arbitrary-Lagrangian Eulerian method. After adapting a projection method for a variational inequality with phase transition information, we use those numerical schemes to simulate the surfactant effect, contact angle hysteresis and unavoidable topological changes of droplets on inclined textured substrates.

• December 18, 2020, 11am-12pm (CT) Coarse Graining and its application to the slip-link model for entangled star polymers

Professor Jay Schieber, Center for Molecular Study of Condensed Soft Matter, Illinois Institute of Technology

Abstract: The dynamics of entangled linear polymers have been described reasonably successfully by both the slip-link models and various tube models. The linear viscoelasticity of entangled monodisperse stars have likewise been explained. The molecular weight scaling of the longest relaxation time of linear chains is not affected by constraint release, but only the prefactor. In contrast, constraint dynamics plays a much larger role for star polymers, and has been essential in describing data even qualitatively. Experimentally, the relative role of constraint dynamics can be examined by blending star and linear chains, which has been accomplished on several systems. However, recent work has shown that the tube models are incapable of explaining these blends, whereas the slip-link model agrees well without any parameter adjustment. These observations raise the question of whether slip-link models are capable of repairing the discrepancy of tube models with data. We first summarize the essential elements necessary for coarse graining. We then propose an efficient means of obtaining the necessary elements while simultaneously testing the quality of the coarse-grained level of description, with a simple (successful) example. We then attempt to coarse grain the slip-link model to a tube level of description. We find that the two models predict very different molecular weight dependence on the relaxation time of star arms without constraint dynamics. Moreover, we find that the tube level of description is incapable of capturing the physics in the slip-link model. We conclude that the two theories are incompatible, and at least one of them should be eliminated by comparison with data.

• December 4, 2020, 10am-11am (CT) Mathematical Justification of Slender Body Theory

Professor Yoichiro Mori, Department of Mathematics, University of Pennsylvania

Abstract: Systems in which thin filaments interact with the surrounding fluid abound in science and engineering. The computational and analytical difficulties associated with treating thin filaments as 3D objects has led to the development of slender body theory, in which filaments are approximated as 1D curves in a 3D fluid. In the 70-80s, Keller, Rubinow, Johnson and others derived an expression for the Stokesian flow field around a thin filament given a one-dimensional force density along the center-line curve. Through the work of Shelley, Tornberg and others, this slender body approximation has become firmly established as an important computational tool for the study of filament dynamics in Stokes flow. An issue with slender body approximation has been that it is unclear what it is an approximation to. As is well-known, it is not possible to specify some value along a 1D curve to solve the 3D exterior Stokes problem. What is the PDE problem that slender body approximation is approximating? Here, we answer this question by formulating a physically natural PDE problem with non-conventional boundary conditions on the filament surface, which incorporates the idea that the filament must maintain its integrity (velocity along filament cross sections must be constant). We prove that this PDE problem is well-posed, and show furthermore that the slender body approximation does indeed provide an approximation to this PDE problem by proving error estimates. This is joint work with Laurel Ohm, Will Mitchell and Dan Spirn.

• November 20, 2020, 10am-11am (CT) The Brinkman-Fourier System with Ideal Gas Equilibrium

Jan-Eric Sulzbach, PhD candidate, Department of Applied Mathematics, Illinois Institute of Technology

Abstract: In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistency between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existence of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.

• November 13, 2020, 4pm-5pm (CT) Learning thermodynamically stable partial differential equations for non-equilibrium flows

Professor Wen-An Yong, Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China

Abstract: In this talk, I will introduce our CDF (conservation-dissipation formalism) theory of irreversible thermodynamics. As a learning-based methodology, CDF produces evolution first-order PDEs modeling irreversible flow processes. It will be seen that this theory provides a tailored framework for using machine learning to discover Galilean invariant and thermodynamically stable PDEs. Some details of the learning process will be explained.

• October 30, 2020, 10am-11am (CT) On classical solutions for some Oldroyd-B model of viscoelastic fluids

Francesco De Anna, PhD , Institute of Mathematics, University of Würzburg

Abstract: This talk is devoted to the analysis of an Oldroyd-B system of PDEs which models the evolution of certain viscoelastic fluids. A particular emphasis is spent on the so called "corotational" model. We are interested in the well-posedness theory of classical solutions. We show in a bidimensional setting that, without any restriction on the initial data, the solutions exist globally in time and they are unique. This result is due to the particular structure of the system which allows to propagate high regularities of the solutions, in particular a Lipschitz regularity of the velocity field. A specific toolbox of Fourier Analysis is presented to address the mentioned result.

• October 23, 2020, 2pm-3pm (CT) Permanent charge effects on ionic flow

Professor Weishi Liu, Department of Mathematics, University of Kansas

Abstract: Permanent charge is the most important structure of an ion channel. In this talk, we will report our studies toward an understanding of permanent charges on ionic flow via a quasi-one-dimensional Poisson-Nernst-Planck (PNP) model. The permanent charges are limited to a special case of piecewise constant with one non-zero portion. For ionic mixtures with one cation species and one anion species, a fairly rich behavior of permanent charge effects is revealed from rigorous analyses based on a geometric framework for PNP and from numerical simulations guided by the analytical results. For ionic mixtures with two cation species and one anion species, richer behavior is expected and our preliminary analytical results identify, in concrete manner, a number of these, including some not-so-intuitive ones.

• October 16, 2020, 2pm-3pm (CT) Preconditioned Nesterov's accelerated gradient descent method and its applications for some nonlinear PDEs

Jeahyun Park, Department of Mathematics, University of Tennessee

Abstract: In this talk, we discuss an intuitive understanding of Nesterov’s accelerated gradient descent method, a minimizing scheme that performs better than the gradient descent method, and its convergence result. The treatment is done so that it is suitable for numerical PDEs: the objective functional is required to be only locally Lipschitz smooth rather than globally Lipschitz smooth and the existence of an invariant set is guaranteed. The theory and the intuition are examined by a numerical experiment. Finally, some examples of its application to real-world problems are briefly discussed.

• September 24, 2020, 2pm-3pm (CT) Friction Stir Welding, Reduced Order Models and Neural Networks

Professor Huaxiong Huang, York University and Beijing Normal University

Abstract: Friction stir welding is a preferred method for joining pieces of metal in the manufacturing processes. In order to find optimal operating conditions, computational models are used to complement experiments. However, searching over high-dimensional parameter space is computationally expensive. In this talk, we present a combined machine learning and model reduction technique to solve a 2D model for friction stir welding that consists of the Navier-Stokes equations with a heat equation. Numerical results will be given to illustrate the significant reduction of computational time and effectiveness of the method. This is joint work with my former postdocs X. Cao, Z. Song, in collaboration with the National Research Council of Canada.

• September 18, 2020, 2pm-3pm (CT) Weak compactness of nematic liquid flows in dimension two

Professor Tao Huang, Department of Mathematics, Wayne State University

Abstract: For any bounded smooth domain in dimension two, we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argument for the Ginzburg-Landau type nematic liquid crystal flow. As a byproduct, we obtain a similar weak compactness result for full Ericksen-Leslie system.

• July 24, 2020, 8:30am-9:30am (CT) Fokker-Planck equations of neuron networks: rigorous justification and numerical simulation

Professor Zhennan Zhou, Peking University, China

Abstract: In this talk, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. In the first part of the talk, we present an alternative way to derive such Fokker-Planck equations from the microscopic model based on a novel iterative expansion. With this formulation, we prove that the probability density function of the “leaky integrate-and-fire” type stochastic process is a classical solution to the Fokker-Planck equation. Secondly, we propose a conservative and positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models.

• July 20, 2020, 11am-12pm (CT) Parameter estimation for semilinear SPDEs from local measurements

Professor Igor Cialenco, Department of Applied Mathematics, Illinois Institute of Technology

Abstract: We will start with a broad discussion of questions arising in the statistical analysis for stochastic partial differential equations (SPDEs), briefly review some existing results and depict some key features specific to the infinite dimensional nature of the considered SPDEs. The second part of the talk will focus on estimating the diffusivity or drift coefficient of a large class of nonlinear SPDEs driven by an additive noise, assuming that the solution is measured locally in space and over a finite time interval. We will introduce the notion of the augmented maximum likelihood estimator and show that this estimator is consistent and asymptotically normal. The proof relies on splitting of the solution argument that exploits some fine regularity properties of the solution and the Gaussian structure of corresponding linear equation. Finally, we will present how general results can be applied to particular classes of equations, including stochastic reaction-diffusion equations and stochastic Burgers equation. The presentation will be kept at high level assuming only basic knowledge from statistics, probability, and PDEs.

• July 13, 2020, 11am-12pm (CT) Tumor growth models with Energetic Variational Approaches (EnVarA)

Min-Jhe Lu, PhD candidate, Department of Applied Mathematics, Illinois Institute of Technology

Abstract: The existing continuum models for tumor growth can be considered as three categories: Hele-Shaw type models, phase-field models and compressible models, where the first is the sharp interface limit and the incompressible limit of the other two, respectively. The derivations and the relations of these models have attracted much research interest recently. In this talk, we will first focus on previous works in the literature on these two aspects, and then we will illustrate how to apply Energetic Variational Approaches (EnVarA) to derive a thermodynamically consistent diffuse interface type tumor growth model and introduce the corresponding sharp interface limit. This is a joint work with Dr. Yiwei Wang.

• July 6, 2020, 11am-12pm Reaction-diffusion equations in biology: from pattern formation to Alzheimer's disease

Professor Wenrui Hao, Department of Mathematics, Pennsylvania State University

Abstract: In this talk, I will cover some examples of chemical reactions arising from biology. The first example is from pattern formation which involves computing multiple solutions; I will introduce a new computational framework to compute these multiple solutions. The second example is a chemical reaction network to model the blood clotting process; the sensitivity analysis of such a network is critical to understand the biological process. The third example is an application of the mass action law to model Alzheimer's disease; I will incorporate the clinical biomarkers data to provide a personalized Alzheimer's risk.

• June 26, 2020, 10am-11am (CT) On the boundary conditions at the fluid-gel interface

Professor Yuan-Nan Young, Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract: Biological fluids are often suspensions of deformable particles and filaments. For example, inside a living cell the fluid is filled with filaments (such as actins and microtubules) that form deformable networks, giving rise to the gel-like fluid mechanics inside the cell. A two-phase flow model has been used to describe the hydrodynamics of such gel. In this talk I will focus on the boundary conditions on the interface between a viscous fluid and a gel. Two sets of boundary conditions will be derived and some examples will be used to illustrate which boundary conditions should be more realistic. We will also show how such different boundary conditions may give rise to different small-deformation dynamics of a poroelastic drop under linear flows. This work is in collaboration with Jimmy Feng from UBC, Canada.

• June 15, 2020, 10am-11am (CT) Generalized law of mass action (LMA) with energetic variational approaches (EnVarA)

Professor Chun Liu, Department of Applied Mathematics, Illinois Institute of Technology

Abstract: In this talk, we'll present a systematic variational derivation to generalize the mass action kinetics of chemical reactions with detailed balance using an energetic variational approach. Our approach starts with an energy dissipation law for a chemical reaction system,which could be argued to carry all the information of the dynamics. The dynamics of the system is determined by both the choice of the free energy, as well as the dissipation, the entropy production. This approach enables us to capture the coupling and competition of various mechanisms, including mechanical effects such as diffusion, drift in an electric field, as well as the thermal effects. We will also discuss several practical examples under this approach, in particular, the modeling of wormlike micellar solutions. This is a joint work with Bob Eisenberg, Pei Liu, Yiwei Wang and Tengfei Zhang.