Towards a mathematical understanding of surface hopping algorithm
Surface hopping algorithm is widely used in chemistry for mixed quantum-classical dynamics, while it is not yet clear whether it can be derived asymptotically. We will discuss some recent progress in semiclassical asymptotics and understanding for the surface hopping algorithms.
Prof. Qin Li, Department of Mathematics, University of Wisconsin-Madison
Computation of linear transport equation
Linear kinetic transport equations are used to model many systems including rarefied gases and radiative transport. The standard computation methods include the source iteration method (with or without diffusion synthetic acceleration), and the even-odd parity decomposition. We first review and compare these methods, and then propose ours that combines the good properties of both, namely the low cost of the source iteration method and the asymptotic preserving property of the even-odd decomposition. The idea could easily get extended to the grey radiative transfer equation that contains the nonlinear coupling between the density and the temperature.
Adoption of new products that mainly spread through word-of-mouth is one of the key problems in Marketing. Ideally, given the sales data of the first few months, one should be able to predict both future sales and the overall market potential. In this talk I will introduce and analyze agent-based models for the adoption of new products. Unlike previous studies, adopters are allowed to ``recover’’ after some time (i.e., to stop influencing their friends to adopt the product). I will discuss the effect of the social network and of adopters’ recovery on the diffusion.
Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement
Line defects appear in the microscopic structure of crystalline materials (e.g. metals) as well as liquid
crystals, the latter an intermediate phase of matter between liquids and solids. Mathematically, their study is challenging since they correspond to topological singularities that result in blow-up of total energies of finite bodies when utilizing most commonly used classical models of energy density; as a consequence, formulating nonlinear dynamical models (especially pde) for the representation and motion of such defects is a challenge as well. I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. The model alleviates the nasty singularities mentioned above and it will also be shown that incorporating a conservation law for the topological charge of line defects allows for the correct prediction of some important features of defect dynamics that would not be possible just with the knowledge of an energy function.
Population dynamics and therapeutic resistance: mathematical models
We are interested in the Darwinian evolution of a population structured by a phenotypic trait. In the model, the trait can change by mutations and individuals compete for a common resource e.g. food. Mathematically, this can be described by non-local Lotka-Volterra equations. They have the property that solutions concentrate as Dirac masses in the limit of small diffusion.
We review results on long-term behaviour and small mutation limits.
A promising application of these models is that they can help to quantitatively understand how resistances against treatment develop.
The population of cells is structured by how resistant they are against a therapy. We describe the model, give first results and discuss optimal control problems arising in this context.
Prof. Roman Shvydkoy, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago
Homogeneous solutions to the incompressible Euler equation
In this talk we describe recent results on classification and rigidity
properties of stationary homogeneous solutions to the 3D and 2D Euler
equations. The problem is motivated by recent exclusions of self-similar
blowup for Euler and its relation to Onsager conjecture and
intermittency. In 2D the problem also arises in several other areas such as isometric immersions
and optimal transport. A full classification of two dimensional solutions
will be given. In 3D we reveal several new classes of solutions and prove their
rigidity properties. In particular, irrotational solutions are characterized by vanishing of
the Bernoulli function; and tangential flows are necessarily 2D
axisymmetric pure rotations. In several cases solutions are excluded
altogether. The arguments reveal geodesic features of the Euler equation on
the sphere. We further discuss the case when homogeneity corresponds to
the Onsager-critical state. We will show that anomalous energy flux at
the singularity vanishes, which is suggestive of absence of extreme
Direct numerical simulations of canonical shock-turbulence interaction
The numerical requirements for stable and accurate computations of turbulence and shock waves are contradictory, a fact which has driven the popularity of hybrid numerical methods that apply different numerical schemes in different regions of the domain.The talk will describe the speaker's journey in this area, on the road from an algorithm on paper to large-scale simulations of shock-turbulence interaction several years later. The stability of the coupled scheme will be discussed, and the errors induced by the shock-capturing numerics on the turbulence statistics. The talk will conclude by discussing some of the interesting physics of shock-turbulence interaction discovered through these large-scale computational studies.
A Lagrangian fluctuation-dissipation relation for scalar turbulence
A common approach to calculate the solution of a scalar advection-diffusion
equation is by a Feynman-Kac representation which averages over stochastic Lagrangian
trajectories going backward in time to the initial conditions and boundary data. The trajectories
are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing
the scalar diffusivity. In this framework we present an exact formula for scalar dissipation in terms
of the variance of the scalar values acquired along each random trajectory. As an important
application, we study the connection between anomalous scalar dissipation in turbulent flows for
large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle
trajectories. The latter property corresponds to the Lagrangian trajectories remaining random
in the limit Re,Pe→∞, when the backward Itō term formally vanishes but the advecting velocity
field becomes non-Lipschitz. For flows on domains without boundaries (e.g. tori, spheres) and
for wall-bounded flows with no-flux Neumann conditions for the scalar, we prove that spontaneous
stochasticity is necessary and sufficient for anomalous scalar dissipation. The fluctuation-dissipation
relation provides a Lagrangian representation of scalar dissipation also in turbulent flows where
present experiments suggest that dissipation is tending to zero as Re,Pe→∞. We discuss an
illustrative example of Rayleigh-Bénard convection with imposed heat-flux at the top and bottom
plates. Our formula here shows that the scalar dissipation is given by the variance of the local
time densities of the stochastic particles at the heated boundaries. The ``ultimate regime'' of
turbulent convection predicted by Kraichnan-Spiegel occurs when the near-wall particle densities
are mixed to their asymptotic uniform values in a large-scale turnover time. The current observations
of vanishing scalar dissipation require that fluid particles be trapped at the wall and remain unmixed
for many, many large-scale turnover times. This talk presents joint work with Theodore Drivas.
It is known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.
In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov’s h-principle. In this lecture I will explain this connection and outline recent progress concerning Onsager’s conjecture.
Filtered spherical harmonic methods for radiation transport
We present a filtering approach to improve the robustness of spherical harmonic methods in the simulation of radiation transport. Although the filter is applied to the angular variable, it provides significant improvement to the spatial profile of the numerical solution. After describing the filter, we will give several numerical examples along with some initial convergence results. We also introduce a limiter which enforces positivity of the spherical harmonic approximation without affecting convergence properties.
Prof. Eytan Ruppin, Center for Bioinformatics and Computational Biology, University of Maryland
Harnessing genetic interactions to advance cancer treatment
Much of the current focus in cancer research is on studying genetic aberrations in cancer driver genes. However, recent work has revealed that interactions between genes can be highly useful for predicting patient survival and drug response. This talk will focus on two fundamental types of genetic interactions in cancer: The first are the well-known Synthetic Lethal (SL) interactions, describing the relationship between two genes whose combined inactivation is lethal to the cell. SLs have long been considered for developing selective anticancer treatments, with a few combinations already in trails and in the clinic. The second type are Synthetic Rescues (SR) interactions, where a change in the activity of one gene is lethal to the cell but an alteration in its SR partner ‘rescues’ cell viability. SRs, though receiving very little attention up until now, may play an important in tumor relapse and emergence of resistance to therapy. I shall describe new approaches for data-driven identification of these two types of genetic interactions (GIs). Applying them to analyze 10,000 tumor samples from the Cancer Genome Atlas (TCGA) we have identified the first pan-cancer SL and SR networks in cancer, and validated subsets of these predictions via existing and new experimental in vitro screens. We find that: (1) the identified GIs successfully predict patient survival and response to drug treatments. (2) The SL networks expose specific cancer vulnerabilities that provide new drug target candidates. (3) The SR networks predict the likelihood of emerging resistance to drugs and point to new ways to mitigate resistance. Importantly, these results are derived directly from patient data and hence more likely to have translational impact.
Joint work with Livnat Jerby, Avinash Das, Joo Sang Lee, Sridhar Hannenhalli and with the experimental labs of Eyal Gottlieb, Paul Clemons, Emma Shanks, Talia Golan and Silvio Gutkind.
Measure-valued solutions to compressible models of fluid dynamics
Measure-valued solutions to hyperbolic conservation laws
were introduced by DiPerna. He showed for scalar conservation laws in one space dimension that measure-valued solutions exist and are, under the assumption of entropy admissibility, in fact concentrated at one point, i.e. they
can be identified with a distributional (entropy) solution.
In other words, in this case the formation of fast oscillations, which corresponds to a measure with positive variance, can be excluded.
In many other physically relevant systems, however, no such
compactness arguments are available, and existence of admissible weak solutions seems hopeless. In such cases, the existence of measure-valued solutions is the best one can hope for. For the incompressible Euler equations, DiPerna and Majda showed the global existence of measure-valued solutions for any initial data with
finite energy. The main point of their work was to introduce the so-called
generalised Young measures which take into account not only oscillations, but also concentrations. I will discuss the issue of weak - strong uniqueness of of measure-valued solutions in the sense of generalised Young measures.
In the second part of my talk I will discuss the model describing granular flows. The theory for gravity driven avalanche flows is qualitatively similar to that of compressible fluid dynamics. I will present one of the models describing flow of granular avalanches - the Savage-Hutter model. The evolution of granular avalanches along an inclined slope is described by the mass conservation law and momentum balance law.
Originally the model was derived in one-dimensional setting. Our interest is mostly directed to two-dimensional extension. Solutions of the Savage-Hutter system develop shock waves and other singularities characteristic for hyperbolic system of conservation laws. Accordingly, any mathematical theory based on the classical concept of smooth solutions fails as soon as we are interested in global-in-time solutions to the system.
Finally I will shortly describe the problem of weak - strong uniqueness of measure-valued solutions to compressible Navier-Stokes equations.
The talk is based on the following results
 P. Gwiazda. On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model. Math. Methods Appl. Sci. 28 (2005), no. 18, 2201-2223.
 E. Feireisl, P. Gwiazda, and A. Świerczewska-Gwiazda. On weak solutions to the 2d Savage-Hutter model of the motion of a gravity driven avalanche
flow, to appear in Comm. Partial Diff. Eq.
 E. Feireisl, P. Gwiazda, A. ŚSwierczewska-Gwiazda and E. Wiedemann.
Dissipative measure-valued solutions to the compressible Navier-Stokes system,
 P. Gwiazda, A. Świerczewska-Gwiazda, and E. Wiedemann. Weak-strong uniqueness for measure-valued solutions of the Savage-Hutter equations, Nonlinearity, 28 (2015) 3873--3890
On ill-posedness of Euler system with non-local terms
The talk will concern the issue of existence of weak solutions to the Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity appearing in collective behavior patterns.
We consider several modifications of the Euler system of fluid dynamics including its pressureless
variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension N=2,3. These models arise in the study of self-organisation in collective behavior modeling of animals and crowds.
We adapt the method of convex integration, adapted to the incompressible Euler system by De Lellis and Szèkelyhidi, to show the existence of infinitely
many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality).
The discussed result is in a certain sense negative result concerning stability of particular solutions. It turns out that the solutions must be sought in a stronger class than that of weak and/or dissipative solutions. We essentially show that there are infinitely many weak solutions for any initial data and that there is a vast class of velocity fields that gives rise to infinitely many admissible (dissipative) weak solutions. We may therefore infer that the class of weak solutions is not convenient for analysing certain qualitative properties such as stability and formation of the flock patterns. However, we also show that the strong solutions are robust in a larger class of all admissible (dissipative) weak solutions leading to the possibility of establishing certain stability results of flock solutions. We establish a weak-strong uniqueness principle for
the pressure driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.
The talk is based on the following result:
J. A. Carrillo, E. Feireisl, P. Gwiazda, and A. Świerczewska-Gwiazda. Weak solutions for Euler systems with non-local interactions, arXiv:1512.03116
On mechanical models for tumor growth: Modeling, analysis and simulations
We investigate the evolution of tumor growth relying on a non-linear model of partial differential equations which incorporates mechanical
laws for tissue compression combined with rules for nutrients availability and
drug application. Rigorous analysis and simulations are presented which show
the role of nutrient and drug application in the progression of tumors. We
construct an explicit convergent numerical scheme to approximate solutions
of the nonlinear system. Extensive numerical
tests show that solutions exhibit a necrotic core when the nutrient level
falls below a critical level in accordance with medical observations. The same
numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug
both shrinkage and growth of tumors can occur. This is joint work with F. Weber.