Fall 2017 Seminars

Grain boundaries are interfaces across which crystal orientation changes. Traditional analysis suggest that grain boundary migration is effectively motion by mean curvature.  However, this view is not in accordance with what we now know as the structure of grain boundaries on an atomic level.  Just as surfaces of crystals move and roughen through the dynamics of surface steps, grain boundary dynamics is controlled by the motion of line defects known as disconnections. Unlike surface steps, disconnections are sources of long range stress (i.e., they have both dislocation and step character).  In this talk, I will present an approach for understanding the motion of grain boundaries via disconnection motion and the relationship between disconnections and the underlying crystal structure. Next, I will discuss the homogenization of this type of disconnection-driven motion to yield a crystal-structure specific grain boundary equation of motion.  I will then show several atomistic and numerical examples of “tame" GB motion (i.e., in bicyrstals) and GB motion “in the wild” (within polycrystals). This is very much a work in progress so I will also outline approaches for generalizations to general interface controlled microstructure evolution.

In this talk, we will focus on a kinetic equation modeling the spatial dynamics of a set of particles subject to intra-specific competition. This equation is motivated by the study of the propagation of biological populations, such as the Escherichia coli bacterium or the cane toad Rhinella marina, for which the classical diffusion approximation underestimates the actual range expansion of the species. We will use the optics geometrics approach as well as Hamilton-Jacobi equations to study spreading results for this equation. As we will see, the multi-dimensional case engenders technical difficulties, and possible over-representation of fast individuals at the edge of the front.

In this talk I present a general framework to model cell-cycle structured
populations living in a chemostat. The main examples are E. coli, or yeast, both
model organisms which have been intensively investigated to understand cell-cycle
controls. In this simple case the cells' cell cycle influence each other only by the level
of nutrients found in the culture medium. Otherwise the cell cycle in each cell behaves
autonomously. As the cell-cycle depends on many cell-internal biochemical concentrations,
most importantly on the cyclin protein family, the dynamical system describing the internal
cell dynamics can be of arbitrary high dimension, making the model extremely complex.
In order to investigate the model behaviour we decided not to use numerical time-integration,
but numerical continuation and bifurcation techniques. The respective numerical algorithm
is again of immense complexity, and uses a cell cohort discretisation. The plan is to refine
the model in future, most importantly bringing it to a tissue level in order to describe
cancer dynamics.

We discuss several examples of inverse problems in computational super-resolution. The first one is a generalized version of well-known sparse sums-of-exponentials model, where we allow also for polynomial modulations. We derive upper bounds on the problem condition number and show that the attainable resolution exhibits Hölder-type continuity with respect to the noise level. A closely related problem is approximating  piecewise-smooth functions, including jump locations, from its Fourier coefficients, with high accuracy. We can show that the asymptotic accuracy of our approach is only dictated by the smoothness of the function between the jumps. Finally we describe some on-going work on the weighted extrapolation problem on the real line for functions of finite exponential type where we abandon the sparsity assumption. It turns out that the extrapolation range scales logarithmically with the noise level, while the pointwise extrapolation error exhibits again a Hölder-type continuity.

We consider a free boundary problem (of Hele-Shaw type) modeling tumor growth. Under certain conditions on the initial data, solutions can be obtained by passing to the stiff (incompressible) limit in a porous medium type problem with a Lotka-Volterra source term describing the evolution of the number density of cancerous cells. We will present several results concerning this derivation and the properties of the resulting free boundary problem. This is a joint work with B. Perthame and F. Quiros.

Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L²-data as the divergence of uniformly bounded vector fields. 

We use a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f's in the critical regularity space L²(T²). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U=Σ_ju_j which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of "images" U.

We prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form N^−q, where N is the number of modes and q depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter ε, which measures the ratio of the mean-freepath to the characteristic domain length. In particular, we show that the error in the spectral approximation is O(ε^N+1). More surprisingly, for isotropic initial conditions, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the l^th coefficient of the expansion scales like O(ε^2N ) when l = 0 and O(ε^2N+2−l) for all 1 ≤ l ≤ N. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on N, the time t, and the initial condition. We investigate specifically the dependence on N, in order to assess whether increasing N actually yields an additional factor of ε in the error. Numerical tests will also be presented to support the theoretical results.

In this talk, I focus on current biological problems and on how to use mathematical modeling to analyze a variety of pressing questions arising from oncology, developmental pattern formation and population ecology. I first discuss novel mathematical models for cancer growth dynamics and heterogeneity. These studies rely on evolutionary principles and shed light on 3D hepatic tumor dynamics, spatial heterogeneity and tumor invasion, and single cancer cell responses to antimitotic therapies. We also develop mathematical models that quantitatively demonstrate how the interplay between non-genetic instability, stress-induced adaptation, and selection leads to the transient and reversible phenotypic evolution of cancer cell populations exposed to therapy. Finally, we study control techniques for optimal therapeutic administration.

In this talk, we’ll discuss one-dimensional shallow-water flows along channels with varying geometry. The flows are modeled by the Saint-Venant equations, a system of hyperbolic PDEs that results from the vertical averaging of Navier-Stokes equations. In addition to the numerical schemes developed for simulating various types of flows along channels, we discuss the properties of non-stationary steady-state solutions and present an algorithm to calculate them with arbitrary precision. Flows at different regimes are calculated and compared to experimental flows so as to validate the mathematical model and algorithms for calculating its solution. 

Semi-supervised learning refers to machine learning algorithms that make use of both labeled data and unlabeled data for learning tasks. Examples include problems such as speech recognition, website classification, and discovering folding structure of proteins. In many problems there is an abundance of unlabeled data, while labeled data often requires expert labeling and is expensive to obtain. This has led to a resurgence of semi-supervised learning techniques, which use the topological or geometric properties of large amounts of unlabeled data to aid the learning task. In this talk, I will discuss some new rigorous PDE scaling limits for existing semisupervised learning algorithms and their practical implications. I will also discuss how these scaling limits suggest new ideas for fast algorithms for semi-supervised learning. 

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak solutions. We want to discuss what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality. An archetypal example of such result was derived for the incompressible Euler system by Constantin et al. ([1]) in the context of the seminal Onsager's conjecture. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

The second part of the talk will concern the problem of uniqueness. Strong solutions  are unique,  and as it has been observed for many systems, not only in the class of strong solutions, but also in a wider class of entropy weak or even entropy/dissipative measure-valued solutions. These properties are referred as weak-strong or measure-valued-strong uniqueness correspondingly. We will discuss dissipative measure-valued solutions to hyperbolic systems and we do not assume that a priori solutions satisfy any bounds, in particular, that  a solution consists  only of a classical Young measure. We do not exclude possibilties of concentration measures.

[1] P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys., 165(1):207–209, 1994.
[2] E. Feireisl, P. Gwiazda, A. Swierczewska-Gwiazda, and E. Wiedemann. Regularity and Energy Conservation for the Compressible Euler Equations.
Arch. Ration. Mech. Anal., 223(3):1–21, 2017
[3] P. Gwiazda, M. Michálek, A. Swierczewska-Gwiazda. A note on weak solutions of conservation laws and energy/entropy conservation, arXiv:1706.10154

This talk will aim to provide an overview of a recent series of papers, joint with Sung-Jin Oh,
devoted to the energy critical 4+1 dimensional hyperbolic Yang-Mills equation. These papers provide a
comprehensive analysis of the large data problem, ultimately providing a proof of the Threshold Conjecture
for Yang-Mills, and more.  We will cover an array of ideas, ranging from gauge theory to hard core pde estimates
to geometry and blow-up analysis.