CSCAMM Archives for Academic Year 2018

How Grain Boundaries Move

When: Wed, September 6, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. David Srolovitz (Department of Materials Science and Engineering, University of Pennsylvania) -
Abstract: 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.

Spreading in a kinetic reaction-transport equation for population dynamics

When: Wed, September 20, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Nils Caillerie (Department of Mathematics and Statistics, Georgetown University) -
Abstract: 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.

Mathematical Modeling of Cell-Cycle Structured Unicellular Populations, and a Numerical Continuation and Bifurcation Analysis - Joint CSCAMM/Ki-Net Seminar

When: Wed, September 27, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Markus Kirkilionis (Warwick Mathematics Institute, University of Warwick) -
Abstract: 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.

Stability of some super-resolution problems

When: Wed, October 4, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Dmitry Batenkov (Department of Mathematics, Massachusetts Institute of Technology) -
Abstract: 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.

A Hele-Shaw problem for tumor growth: Mathematical Analysis

When: Wed, October 18, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Antoine Mellet (Department of Mathematics, University of Maryland) -
Abstract: 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.

Hierarchical construction of images and the problem of Bourgain-Brezis

When: Wed, October 25, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Eitan Tadmor (CSCAMM, Mathematics & IPST, University of Maryland) -
Abstract: 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=Σjuj which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of "images" U.

Multiscale convergence properties for spectral approximations of a model kinetic equation

When: Wed, November 1, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Zheng Chen (Computer Science and Mathematics Division, Oak Ridge National Laboratory) -
Abstract: 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(εN+1). More surprisingly, for isotropic initial conditions, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the lth coefficient of the expansion scales like O(ε2N ) when l = 0 and O(ε2N+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.

Mathematics meets Biology: from Adaptive evolution to Zebrafish

When: Wed, November 8, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Alexander Lorz (Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions) -
Abstract: 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.

Steady-states Flows along Channels: Simulation and Numerical Solutions

When: Wed, November 15, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Jorge Balbas (Institute for Pure & Applied Math, University of California at Los Angeles) -

PDEs and semi-supervised learning

When: Wed, November 29, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Jeff Calder (School of Mathematics, University of Minnesota) -
Abstract: 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.

Weak and measure-valued solutions to general hyperbolic systems - Joint CSCAMM/Ki-Net Seminar

When: Wed, December 6, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Agnieszka Swierczewska-Gwiazda (Institute of Applied Mathematics and Mechanics, University of Warsaw) -
Abstract: 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

The role of the autologous immune response in chronic myelogenous leukemia

When: Wed, February 14, 2018 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Doron Levy (CSCAMM and Department of Mathematics, University of Maryland) -
Abstract: Tyrosine kinase inhibitors such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). Yet, most patients are not cured for undetermined reasons. In this talk we will describe our recent work on modeling the autologous immune response to CML. We will also discuss our previous results on cancer vaccines, drug resistance, and the dynamics of hematopoietic stem cells.

Boundary layers and the vanishing viscosity limit in incompressible flows

When: Wed, February 21, 2018 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Anna Mazzucato (Mathematics, Penn State University) -
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.