Where: CSIC 4122

Speaker: Andrea Prosperetti (Department of Mechanical Engineering, Johns Hopkins University) - http://www.me.jhu.edu/prosper/

Abstract: The talk will summarize work on several different problems involving particles and bubbles. It will begin with a brief description of the physics underlying a numerical method for the simulation of laminar and turbulent fluid flows with particles. Applications to rotating spheres, sedimenting particles, model porous media and particles in turbulence will be described. The talk will then address some physical aspects of vapor bubbles generated by a pulse of laser or electrical energy, to finish with a gas-bubble-based "acoustic fish".

Where: CSIC 4122

Speaker: Dr. Bin Cheng (School of Mathematical and Statistical Sciences Arizona State University) - http://math.la.asu.edu/~cheng/

Abstract: Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a PDE-theoretical framework for studying time-averages of dynamical systems that evolve in both fast and slow scales. Patterns arise upon time-averaging, which in turn affects long term dynamics via nonlinear coupling. We apply this framework to geophysical fluid dynamics in spherical and bounded domains subject to strong Coriolis force and/or Lorentz force.

Where: CSIC 4122

Speaker: Prof. Brenton LeMesurier (College of Charleston: Mathematics) - http://lemesurierb.people.cofc.edu/

Abstract: A variety of problems in modeling of large biomolecules and nonlinear optics lead to large, stiff, mildly nonlinear systems of ODEs that have Hamiltonian form.

This talk describes a discrete calculus approach to constructing unconditionally stable, time-reversal symmetric discrete gradient conservative schemes for such Hamiltonian systems (akin to the methods developed by Simo, Gonzales, et al), an iterative scheme for the solution of the resulting nonlinear systems which preserves unconditional stability and exact conservation of quadratic first integrals, and methods for increasing the order of accuracy. Some comparisons are made to the more familiar momentum conserving symplectic methods.

As an application, some models of pulse propagation along protein and DNA molecules and related numerical observations will be described, with some consequences for the search for continuum limit PDE approximations.

Where: CSIC 4122

Speaker: Prof. Elias Balaras (Dept. of Mechanical and Aerospace Engineering, George Washington University) - http://www.seas.gwu.edu/~balaras/

Abstract: Mechanical hemolysis is a critical element in the design of cardiovascular devices where abnormal, disturbed flow patterns are often unavoidable. A characteristic example is that of ventricular assist devices (VAD), which are used to treat more than 50000 patients with ailing hearts in the US alone. Today, computational fluid dynamics (CFD) are increasingly used to optimize the hydrodynamic performance of biomedical devices, such as VAD’s, but improvements on blood damage and blood aggregation characteristics is hampered by the lack of predictive mechanical hemolysis and thrombosis models. Although in most of these devices the flow is highly three-dimensional and unsteady, currently available models for hemolysis are usually based steady shearing experiments and utilize global measures of the stress scalar magnitude and duration of exposure. More recent strain-based models are conceptually well suited for unsteady configurations, but still the instantaneous red blood cell (RBC) deformation estimates are as accurate as the steady experiments utilized. In this talk we will first present a brief survey of the existing models, which are based on either “lumped” descriptions of stress or analytical-numerical RBC descriptions relying on simple geometrical assumptions. We will also introduce a new approach, which is based on an existing coarse-grained particle dynamics method. We will then explore the rationale and RBC physics within each method through model “virtual”, numerical experiments. Finally, we apply all models to simulate the expected level of RBC damage using pathlines calculated for a realistic artificial heart valve. As we will show, our results shed light on the strengths and weaknesses of each approach and identify the key gaps that should be addressed in the development of new models.

Where: CSIC 4122

Speaker: Prof. Maria Gualdani (Department of Mathematics, George Washington University) - http://home.gwu.edu/~gualdani/

Abstract: We present an abstract method for deriving decay estimates on the resolvents of non-symmetric operators in Banach spaces in terms of estimates in another – typically smaller – reference Hilbert space. We then apply this approach to several equations of statistical physics, such as the Fokker-Planck equation and the linear and non-linear Boltzmann equation. The main outcome of the method is the first constructive proof of exponential decay towards global equilibrium for the fully nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness assumptions.

This is a joint work with Stephane Mischler and Clement Mouhot.

Where: CSIC 4122

Speaker: Prof Andre Tits (Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland) - http://www.ece.umd.edu/~andre/

Abstract: Constraint reduction is a technique by which each search direction is computed based only on a small subset of the inequality constraints (when the problem is expressed in standard dual form), containing those deemed most likely to be active at the solution. A dramatic reduction in computing time may result for severely imbalanced problems.

In this talk, we survey developments made at UMPC over the past few years, in a group led by Dianne O'Leary and the presenter. The power of constraint reduction is demonstrated on classes of randomly generated problems and on real-world applications. Numerical comparison with both simplex and "unreduced" interior point is reported.

Note: This talk has a significant overlap with a talk given by the author in the UMCP NA Seminar Series in April 2012.

Where: CSIC 4122

Speaker: Prof. Gitta Kutyniok (Department of Mathematics, Technische Universität Berlin) - http://www.tu-berlin.de/?108957

Abstract: One main problem in data processing is the reconstruction of missing data. In the situation of image data, this task is typically termed image inpainting. Recently, inspiring algorithms using sparse approximations and ℓ1 minimization have been developed and have, for instance, been applied to seismic images. The main idea is to carefully select a representation system which sparsely approximates the governing features of the original image -- curvilinear structures in case of seismic data. The algorithm then computes an image, which coincides with the known part of the corrupted image, by minimizing the ℓ1 norm of the representation coefficients.

In this talk, we will develop a mathematical framework to analyze why these algorithms succeed and how accurate inpainting can be achieved. We will first present a general theoretical approach. Then we will focus on the situation of images governed by curvilinear structures, in which case we analyze both wavelets as well as shearlets as the chosen representation system. Using the previously developed general theory and methods from microlocal analysis, under certain conditions on the size of the missing parts we will prove that such images can be arbitrarily well reconstructed.

This is joint work with Emily King and Xiaosheng Zhuang.

Where: CSIC 4122

Speaker: Benoit Perthame ( Laboratoire J.-L. Lions, Universit\'e P. et M. Curie, CNRS, INRIA and Institut Universitaire de France) - http://www.ann.jussieu.fr/~perthame/

Abstract: Many integro-differential equations are used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the most wellknown and describe spiking rates in different locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. It has also been proposed to describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time s elapsed since its last discharge.

We will compare these models and perform some mathematical analysis. A striking observation is that solutions to the I&F can blow-up in finite time, a form of synchronization. We can also show that for small or large connectivity the 'elapsed time model' leads to desynchronization. For intermediate regimes, sustained periodic activity occurs which profile is compatible with observations. A common tool is the use of the relative entropy method.

Where: MATH 3206

Speaker: Prof. Mario Milman (Department of Mathematics, Florida Atlantic University) - http://math.fau.edu/milman/

Abstract: I will discuss a new approach to Sobolev inequalities in the context of metric spaces, based on new rearrangement inequalities that involve the associated isoperimetric pro le. This leads to a reformulation, as well as an extension, of the classical equivalence between the Gagliardo-Nirenberg inequality and the isoperimetric inequality.

In this fashion the isoperimetric profile associated with a given geometry determines the corresponding Sobolev inequalities. For example, in the Euclidean case we recover the usual Sobolev inequalities while for Gaussian measure we obtain logarithmic Sobolev inequalities. We also obtain new fractional Besov-Sobolev inequalities and formulate a generalized Morrey-Sobolev theorem.

Despite the considerable techno jargon of the previous discussion I will try to focus on the general ideas and make the talk understandable to non specialists.

Where: CSIC 4122

Speaker: Various CSCAMM Faculty (University of Maryland, College park ) -

Where: CSIC 4122

Speaker: Prof. Michael Shearer (Department of Mathematics, North Carolina State University) - http://www4.ncsu.edu/~shearer/home0.html

Abstract: I will discuss two models of two-phase ﬂuid ﬂow in which undercompressive shock waves have been dis-covered recently. In the ﬁrst part of talk, the focus is on two-phase ﬂow in porous media. Plane waves are modeled by the one-dimensional Buckley-Leverett equation, a scalar conservation law. The Gray-Hassanizadeh model for rate-dependent capillary pressure adds dissipation and a BBM-type dis-persion, giving rise to undercompressive waves. Two-phase ﬂow in porous media is notoriously subject to ﬁngering instabilities, related to the classic Saffman-Taylor instability. However, a two dimensional linear stability analysis of sharp planar interfaces reveals a criterion predicting that weak Lax shocks may be stable or unstable to long-wave two-dimensional perturbations. This surprising result depends on the hyperbolic-elliptic nature of the system of linearized equations. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. In the second part of the talk, I describe a phase ﬁeld model of capillary effects in a thin tube, in which a resident ﬂuid is displaced by injected air. PDE simulations reveal the appearance of a combination of rarefaction wave together with an undercompressive shock that terminates at the spherical cap tip of the injected air. The shock can be understood through a singular dynamical system whose trajectories yield travelling wave solutions.

Where: CSIC 4122

Speaker: Dr. Amit Einav (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge) - http://www.math.gatech.edu/users/aeinav

Abstract: In 1956 Marc Kac introduced a binary stochastic N-particle model from which, under suitable condition on the initial datum (what we now call 'Chaoticity') a caricature of the famous Boltzmann equation, in its spatially homogeneous form, arose as a mean field limit. The ergodicity of the evolution equation resulted in convergence to equilibrium as time goes to infinity, for any N. Kac expressed hopes that investigation of the rate of convergence can be expressed independently in N and result in an exponential trend to equilibrium for his caricature of Boltzmann equation. Later on, in 1967, McKean extended Kac's model to a more realistic d-dimensional one from which the actual Boltzmann equation arose, extending Kac's results and hopes to the real case.

Kac's program reached its conclusion in the 2000s in a series of papers by Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, however it was known long before that the linear L2 based approach of Kac will not yield the desired result. A new method was devised, one that draws its ideas from a conjecture by Cercignani's for the real Boltzmann equation: investigate the entropy, and entropy production in Kac's model, in hope to get a better rate of convergence.

In our talk we will discuss Kac models of any dimension, recall the spectral gap problem and its conclusions as well as describe Cercignani's many body conjecture. We will show that, while the entropy and entropy production are more suited to deal with Kac's models, in full generality the rate they produce is not much better than that of the linear approach. We will conclude that more restrictions are need, and share a few insights we may have in the subject.

Where: CSIC 4122

Speaker: Prof. Henk Dijkstra (Department of Physics and Astronomy, Utrecht University)

Abstract: Results will be presented of a study on the interaction of noise and nonlinear dynamics in a quasi-geostrophic model of the wind-driven ocean circulation. The recently developed framework of dynamically orthogonal field theory is used to determine the statistics of the flows which arise through successive bifurcations of the system as the ratio of forcing to friction is increased. Focus will be on the understanding of the role of the spatial and temporal coherence of the noise in the wind-stress forcing. For example, when the wind-stress noise is additive and temporally white, the statistics of the stochastic ocean flow does not depend on the spatial structure and amplitude of the noise. This implies that a spatially inhomogeneous noise forcing in the wind stress field only has an effect on the dynamics of the flow when the noise is temporally colored. The latter kind of stochastic forcing may cause more complex or more coherent dynamics depending on its spatial correlation properties.

Where: 4122 CSIC

Speaker: Prof. Bill Rand, Robert H. Smith School of Business, University of Maryland - http://www.rhsmith.umd.edu/marketing/faculty/rand.aspx

Abstract: The dramatic feature of social media is that it gives everyone a voice; anyone can speak out and express their opinion to a crowd of followers with little or no cost or effort, which creates a loud and potentially overwhelming marketplace of ideas. The good news is that the organizations have more data than ever about what their consumers are saying about their brand. The bad news is that this huge amount of data is difficult to sift through. We will look at developing methods that can help sift through this torrent of data and examine important questions, such as who do users trust to provide them with the information that they want? Which entities have the greatest influence on social media users? Using agent-based modeling, machine learning and network analysis we begin to examine and shed light on these questions and develop a deeper understanding of the complex system of social media.

Where: MATH 3206

Speaker: Prof. Razvan Fetecau, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge - http://people.math.sfu.ca/~van/

Abstract: We consider the aggregation equation ρt − ∇ • (ρ∇K ∗ ρ) = 0 in Rn, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials with repulsion given by a Newtonian potential and attraction in the form of a power law. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. The equilibria have biologically relevant features, such as finite densities and compact support with sharp boundaries.

This is joint work with Yanghong Huang and Theodore Kolokolnikov.

Where: CSIC 4122

Speaker: Dr. Carola-Bibiane Schönlieb, Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge - http://www.damtp.cam.ac.uk/user/cbs31/Home.html

Abstract: A key issue in image denoising is an adequate choice of the correct noise model. In a variational approach this amounts to the choice of the data fidelity and its weighting. Depending on this choice, different results are obtained.

In this talk I will discuss a PDE-constrained optimization approach for the determination of the noise distribution in total variation (TV) image denoising. An optimization problem for the determination of the weights correspondent to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems. The talk is furnished with numerical examples computed on simulated data.

This is joint work with Juan Carlos De Los Reyes

Where: CSIC 4122

Speaker: Prof. Anne Gelb, School of Mathematical and Statistical Sciences, Arizona State University - http://math.la.asu.edu/~ag/

Abstract: In this talk I discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR).

Two standard reconstruction techniques, convolutional gridding (the non-uniform FFT) and uniform resampling, are summarized, and some of the difficulties are discussed. It is then demonstrated how spectral reprojection can be used to mollify both the Gibbs phenomenon and the error due to the non-uniform sampling. It is further shown that incorporating prior information, such as the internal edges of the underlying function, can greatly improve the reconstruction quality. Finally, an alternative approach to the problem that uses Fourier frames is proposed.

Where: 4122 CSIC

Speaker: Prof. Weizhu Bao, Department of Mathematics, Center for Computational Science & Engineering, National University of Singapore - http://www.math.nus.edu.sg/~bao/

Abstract: In this talk, I begin with a brief derivation of the nonlinear Schrodinger/Gross-Pitaevskii equations (NLSE/GPE) from Bose-Einstein condensates (BEC) and/or nonlinear optics. Then I will present some mathematical results on the existence and uniqueness as well as non-existence of the ground states of NLSE/GPE under different external potentials and parameter regimes. Dynamical properties of NLSE/GPE are then discussed, which include conservation laws, soliton solutions, well-posedness and/or finite time blowup. Efficient and accurate numerical methods will be presented for computing numerically the ground states and dynamics. Extension to NLSE/GPE with an angular momentum rotation term and/or non-local dipole-dipole interaction will be presented. Finally, applications to collapse and explosion of BEC, quantum transport and quantized vortex interaction will be investigated.

Where: 4122 CSIC

Speaker: Prof. Alex Mahalov, School of Mathematical & Statistical Sciences, Arizona State University - http://math.asu.edu/people/alex-mahalov

Abstract: We consider stochastic three-dimensional rotating Navier–Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of precise mechanisms of relevant three-dimensional nonlinear dynamics. We prove multi-scale averaging theorems for the stochastic dynamics and describe its effective covariance operator.

Where: 4122 CSIC

Speaker: Dr. Stephan Martin

Abstract: In my talk I will first review some modeling concepts describing the behavior of individuals in an animal swarm of e.g. fish or birds, and focus on a model of self-propelled interacting particles. It is a well-known fact that even minimalistic interactions rules allow for the emergence of coherent macroscopic patterns observed in nature, when applied to all members of a swarm. In the mean-field limit approach, a kinetic PDE is used to model the evolution of a particle density rather than tracing individuals separately. Its macroscopic closure allows for a compact description of some coherent patterns, such as flocks or mills.

I will then discuss the possibility to explicitly compute the stationary density profile of such states using a particular type of interaction potential called Quasi-Morse. Flock and mill profiles can be predicted with a cheap numerical procedure that does not necessitate particle simulations.

Finally, I will present a result on the stability of flock solutions, where we are able to show that under mild assumptions the stability of the interaction potential (in a first-order aggregation model) inherits to the family of flock solutions in our second-order model.

Where: 4122 CSIC

Speaker: Prof. Siddhartha Mishra, Seminar für Angewandte Mathematik, ETH - http://www.sam.math.ethz.ch/people/smishra

Abstract: We start by arguing through numerical examples as to why entropy measure valued solutions are the appropriate solution concept for systems of conservation laws in several space dimensions. Two classes of numerical schemes are presented that can be shown to converge to entropy measure valued solutions. The first class are finite volume schemes based on entropy conservative fluxes and numerical diffusion operators, using a ENO reconstruction. The second class are space-time shock capturing discontinuous Galerkin (STDG) schemes. The schemes are compared on a set of numerical experiments. The lecture concludes with a discussion of efficient ways to compute measure valued solutions using multi-level monte carlo methods, that were originally developed for uncertainty quantification in conservation laws.

Where: CSIC 4122

Speaker: Prof. Joanna Wares, Department of Mathematics, University of Richmond http://math.richmond.edu/faculty/jwares/

Abstract: Antibiotic-resistant bacteria present an enormous challenge in hospitals and other health care settings. Infection rates are high and new strains are constantly emerging. Mortality from nosocomial infections of certain gram-negative strains approaches 60%. Mathematical models consisting of systems of differential equations or computer simulations of agent-based models can be used to study interventions that can lessen transmission and infection rates. In this talk, I will discuss techniques for studying nosocomial infections and look at some results from our recent work.

Where: CSIC 4122

Speaker: Prof. Chi-Wang Shu, Division of Applied Mathematics, Brown University

Abstract: Discontinuous Galerkin (DG) methods are finite element methods with features from high resolution finite difference and finite volume methodologies and are suitable for solving hyperbolic equations with nonsmooth solutions. In this talk we will first give a survey on DG methods, then we will describe our recent work on the study of DG methods for solving hyperbolic equations with singularities in the initial condition, in the source term, or in the solutions. The type of singularities include both discontinuities and δ-functions. Especially for problems involving δ-singularities, many numerical techniques rely on modifications with smooth kernels and hence may severely smear such singularities, leading to large errors in the approximation. On the other hand, the DG methods are based on weak formulations and can be designed directly to solve such problems without modifications, leading to very accurate results. We will discuss both error estimates for model linear equations and applications to nonlinear systems including the rendez-vous systems and pressureless Euler equations involving δ-singularities in their solutions. This is joint work with Qiang Zhang, Yang Yang and Dongming Wei.

Where: CSIC 4122

Speaker: Prof. Yann Brenier, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique

Abstract: The ``relative entropy method'' is well-known in several fields of mathematical physics, PDEs and probabilities (systems of hyperbolic conservation laws, systems of particles, kinetic equations...). It typically leads to the rigorous derivation of some asymptotic models having smooth enough solutions.

Here, we report on examples of geophysical fluid dynamics (with rotation and convection effects, at different aspect ratio and time scales), for which the relative entropy method applies in a non-standard way because the entropy functional depends on the asymptotic solution itself.

Where: CSIC 4122

Speaker: Prof. Donatella Donatelli, Department of Mathematics, Università degli Studi L'Aquila - http://univaq.it/~donatell/

Abstract: We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in 3 − D. We show that as the Debye length goes to zero the velocity field strongly converges towards an incompressible velocity vector field and the density fluctuation weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis.

Where: CSIC 4122

Speaker: Prof. Andrea Bertozzi, Department of Mathematics, University of California Los Angeles - http://www.math.ucla.edu/~bertozzi/

Abstract: The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community. This talk will cover a survey of the speaker’s research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments. We conclude with a discussion of some interesting problems for the applied mathematics community.

Where: 4122 CSIC

Speaker: Prof. Laurent Desvillettes, École Normale Supérieure de Cachan

Abstract: Sprays are complex flows consisting of an underlying gas and a large quantity of small liquid droplets. They appear in many industrial devices (engines, nuclear industry) and natural phenomena (clouds, lungs). Their modeling, first proposed by Williams in the 70s, can be performed through the coupling of a kinetic equation of Vlasov type (for the droplets) and a fluid equation (viscous or inviscid, compressible or incompressible) for the gas. The mathematical theory for this coupling mixes the kinetic theory (control of moments as in the Vlasov-Poisson equation) and the Euler/Navier-Stokes theory (strong local solutions, weak solutions). We wish to present during the seminar some of the latest results obtained for sprays.

Where: 4122 CSIC

Speaker: Prof. Alexis Vasseur, University of Texas at Austin

Abstract: The relative entropy method is a powerful tool for the study of conservation laws. It provides, for example, the weak/strong uniqueness principle, and has been used in different context for the study of asymptotic limits. Up to now, the method was restricted to the comparison to Lipschitz solutions. This is because the method is based on the strong stability in L2 of such solutions. Shocks are known to not be strongly L2 stable. We show, however that their profiles are strongly L2 stable up to a drift. We provide a first application of this stability result to the study of asymptotic limits.