## PDE & Applied math seminar

Wednesday 10:00 -10:50 am, https://ucr.zoom.us/j/97606227247

Organizers : Weitao Chen / Heyrim Cho / Yat Tin Chow / Qixuan Wang / Jia Guo / Mykhailo Potomkin

Past Organizers : Mark Alber / James Kelliher / Amir Moradifam

**Winter 2021 Schedule **

Jan 6 10:00 (Wed) Organization meeting

Jan 13 10:00 (Wed) **Dr. Katy Craig**, University of California, Santa Barbara

Jan 20 11:00* (Wed) **Dr. Samy Wu Fung**, University of California, Los Angeles (** Note unusual time!*)

Jan 27 10:00 (Wed) **Dr. Justin T. Webster**, University of Maryland, Baltimore County

Feb 3 10:00 (Wed) **Dr. Siming He**, Duke University

Feb 10 10:00 (Wed) **Dr. Marissa Renardy**, University of Michigan

Feb 17 10:00 (Wed) **Dr. Matthew Mizuhara**, The College of New Jersey

Feb 24 10:00 (Wed) **D****r. Yi Man**, University of Southern California

Mar 3 10:00 (Wed) **Dr. Di Kang**, McMaster University

Mar 10 10:00 (Wed) **Dr. Bhargav Ram Karamched**, Florida State University

### Next talk:

March 3rd, 10-10:50am PST

**Dr. Di Kang** (McMaster University)

**Title**: Searching for singularities in Navier-Stokes flows using variational optimization methods

**Abstract**: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. More specifically, such an optimization formulation allows one to identify "extreme" initial data which, subject to certain constraints, leads to the most singular flow evolution. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it makes it possible to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 1D and 2D flows characterized by the maximum possible growth of different Sobolev norms of the solutions. As regards 3D flows, we focus on the enstrophy which is a well-known indicator of the regularity of the solution. We find a family of initial data with fixed enstrophy which leads to the largest possible growth of this quantity at some prescribed final time. Since even with such worst-case initial data the enstrophy remains finite, this indicates that the 3D Navier-Stokes system reveals no tendency for singularity formation in finite time. This is joint work with Dongfang Yun and Bartosz Protas.

### Talk Titles & Abstracts:

January 13th, 10-10:50am PST

**Dr. Katy Craig** (University of California, Santa Barbara)

**Title**: A blob method for diffusion and applications to sampling and two layer neural networks

**Abstract**: Given a desired target distribution and an initial guess of that distribution, composed of finitely many samples, what is the best way to evolve the locations of the samples so that they more accurately represent the desired distribution? A classical solution to this problem is to allow the samples to evolve according to Langevin dynamics, the stochastic particle method corresponding to the Fokker-Planck equation. In today’s talk, I will contrast this classical approach with a deterministic particle method corresponding to the porous medium equation. This method corresponds exactly to the mean-field dynamics of training a two layer neural network for a radial basis function activation function. We prove that, as the number of samples increases and the variance of the radial basis function goes to zero, the particle method converges to a bounded entropy solution of the porous medium equation. As a consequence, we obtain both a novel method for sampling probability distributions as well as insight into the training dynamics of two layer neural networks in the mean field regime. This is joint work with Karthik Elamvazhuthi (UCLA), Matt Haberland (Cal Poly), and Olga Turanova (Michigan State).

January 20th, 11-11:50am PST (**Note unusual time!**)

**Dr. Samy Wu Fung** (University of California, Los Angeles)

**Title**: Adversarial Projections for Inverse Problems

**Abstract**: We present a new mechanism, called *adversarial projection*, that projects a given signal onto the intrinsically low dimensional manifold of true data. This operation can be used for solving inverse problems, which consists of recovering a signal from a collection of noisy measurements. Rather than attempt to encode prior knowledge via an analytic regularizer, we leverage available data to project signals directly onto the (possibly nonlinear) manifold of true data (i.e., regularize via an indicator function of the manifold). Our approach avoids the difficult task of forming a direct representation of the manifold. Instead, we directly learn the projection operator by solving a sequence of unsupervised learning problems, and we prove our method converges in probability to the desired projection. This operator can then be directly incorporated into optimization algorithms in the same manner as Plug and Play methods, but now with added theoretical guarantees. Numerical examples are provided.

January 27th, 10-10:50am PST

**Dr. Justin T. Webster** (University of Maryland, Baltimore County)

**Title**: Mathematical Aeroelasticity: The Analysis of Flow-Structure Interactions

**Abstract**: This talk focuses on the underlying mathematics of the aeroelastic phenomenon flutter - i.e., the way that an elastic structure may become unstable in the presence of an adjacent flow of air. Under certain circumstances, a feedback occurs between elastic deformations and pressure dynamics in the airflow, resulting in sustained oscillations. A canonical example was seen in the Tacoma Narrows bridge (Washington, USA), which collapsed in 1940 while fluttering in 65 kph winds. Flutter is typically discussed in the context of aero-mechanical systems: buildings and bridges in wind, and flight systems. However, applications also arise in biology (snoring and sleep apnea), and in alternative energy technologies (piezoelectric energy harvesters).

We will look at a variety of flow-structure interaction models which are partial differential equation systems coupled via an interface. After a brief discussion of relevant modeling, we will examine well-posedness and long-time behavior properties of PDE solutions for three different physical configurations that can exhibit aeroelastic flutter: (1) projectile paneling, (2) a bridge deck, (3) an elastic energy harvester. From a rigorous point of view, we attempt to capture the mechanism that gives rise to the flutter instability. Additionally, when flutter occurs, we attempt to describe its qualitative features through a dynamical systems approach, as well as how to prevent it or bring it about (stability).

February 3rd, 10-10:50am PST

**Dr. Siming He** (Duke University)

**Title**: Fluid Flows, Chemotaxis and Reactions

**Abstract**: In this talk, I will present some results on suppressing chemotactic blow-ups and enhancing chemical reactions through passive fluid flows. Enhanced dissipation effect of mixing flows and shear flows play significant roles. I will also discuss a new coupled Chemotaxis-Fluid system (Patlak-Keller-Segel-Navier-Stokes) and the corresponding optimal critical mass result.

February 10th, 10-10:50am PST

**Dr. Marissa Renardy** (University of Michigan)

**Title**: Structural identifiability analysis of PDEs: A case study in continuous age-structured epidemic models

**Abstract**: Identifiability analysis is crucial for interpreting and determining confidence in biologically relevant predictions from computational and mathematical models that rely on estimated parameter values. While structural identifiability analysis has been extensively applied in the context of ordinary differential equation (ODE) models in epidemiology, it has not yet been widely explored for age-structured partial differential equation (PDE) models. These models present additional difficulties due to increased number of variables and partial derivatives as well as the presence of boundary conditions. In this work, we establish a pipeline for structural identifiability analysis of age-structured PDE models using a differential algebra framework and derive identifiability results for specific age-structured epidemic models. In our application of the identifiability analysis pipeline, we focus on a Susceptible-Exposed-Infected model for which we compare identifiability results for a PDE and corresponding ODE system and explore the effects of age-dependent parameters on identifiability. We also show how practical identifiability analysis can be applied in this example.

February 17th, 10-10:50am PST

**Dr. Matthew Mizuhara** (The College of New Jersey)

**Title**: Bifurcations and pattern formation in random oscillatory systems

**Abstract**: A common occurrence in biological and physical systems is the emergence of spontaneous, collective behavior from a group of individual agents. A particularly striking example is the ability of certain species of fireflies to synchronize their flashes. The Kuramoto model is a non-linear dynamical system widely used to explore such synchronization and other types of pattern formation in many such oscillatory systems. We will briefly introduce several classical results, and present recent advances in the study of bifurcations of the Kuramoto model on random graphs. In particular, we will show how one can construct bifurcations from incoherence to a variety of patterns including clusters and chimera states. This work is in collaboration with Georgi Medvedev and Hayato Chiba.

February 24th, 10-10:50am PST

**D****r. Yi Man** (University of Southern California)

**Title**: Synchronization in biological filaments

**Abstract**: Locomotion is a fundamental characteristic of life, and has been pervasive throughout the history of science and philosophy. Recent technological advances have made it extremely exciting time to explore the motion in the microscopic world, which leaves many interesting questions for biology, physics, mathematics and engineering. My research lies squarely at the intersection of these disciplines. In this talk, I will focus on the eukaryotic flagella and present my recent work about the synchronization in biological filaments. Experiments revealed that a pair of isolated eukaryotic flagella, coupled solely via the fluid medium, display synchronization with different phase lags. Using an elasto-hydrodynamic filament model in conjunction with numerical simulations and a Floquet-type theoretical analysis, I show that it is possible to reach synchronization states with multiple phase lags by varying the intrinsic activity of the filament and the strength of hydrodynamic coupling between the two filaments. In particular, I find that non-trivial phase lag corresponds to asymmetric synchronization even though the activity of the two filaments is identical. These novel results could have significant implications in the locomotion of bi-flagellated cells.

March 3rd, 10-10:50am PST

**Dr. Di Kang** (McMaster University)

**Title**: Searching for singularities in Navier-Stokes flows using variational optimization methods

**Abstract**: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. More specifically, such an optimization formulation allows one to identify "extreme" initial data which, subject to certain constraints, leads to the most singular flow evolution. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it makes it possible to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 1D and 2D flows characterized by the maximum possible growth of different Sobolev norms of the solutions. As regards 3D flows, we focus on the enstrophy which is a well-known indicator of the regularity of the solution. We find a family of initial data with fixed enstrophy which leads to the largest possible growth of this quantity at some prescribed final time. Since even with such worst-case initial data the enstrophy remains finite, this indicates that the 3D Navier-Stokes system reveals no tendency for singularity formation in finite time. This is joint work with Dongfang Yun and Bartosz Protas.