CSCAMM Archives for Academic Year 2016

Sparsifying Preconditioners (joint CSCAMM/KI-Net seminar)

When: Wed, September 2, 2015 - 2:00pm
Where: 4122 CSIC Bldg
Speaker: Prof. Lexing Ying (Department of Mathematics, Stanford University) -
Abstract: High frequency wave propagation has been a longstanding challenge in scientific computing. For the time-harmonic problems, integral formulations and/or efficient numerical discretization often lead to dense linear systems. Such linear systems are extremely difficult to solve for standard iterative methods since they are highly indefinite. In this talk, we consider several such examples with important applications. For each one, we construct a sparsifying preconditioner that reduces the dense linear system to a sparse one and solves the problem within a small number of iterations.

Sparse Mean-Field Optimal Control (joint CSCAMM/KI-Net seminar)

When: Wed, September 9, 2015 - 2:00pm
Where: 4122 CSIC Bldg
Speaker: Prof. Massimo Fornasier (Faculty of Mathematics, Technical University of Munich, Germany) -
Abstract: Starting with the seminal papers of Reynolds (1987), Vicsek et. al. (1995) Cucker-Smale (2007), there has been a flood of recent works on models of self-alignment and consensus dynamics. Self-organization has been so far the main driving concept. However, the evidence that in practice self-organization does not necessarily occur leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most “economical” manner to achieve a certain outcome. The first part of this talk precisely addresses the issue of finding the sparsest control strategy for finite dimensional models in order to lead the dynamics optimally towards a given outcome. In the second part of the talk we introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Gamma-limit of the finite dimensional sparse optimal control problems. Additionally we derive corresponding first order optimality conditions for the infinite dimensional optimal control problem in the form of Hamiltonian flows in the Wasserstein space of probability measures, which correspond to natural limits of the finite dimensional Pontryagin Maximum principles. We conclude the talk by mentioning recent results in sparse optimal control of high-dimensional dynamical systems.

Analysis of Bounded Confidence Multidimensional Opinions in Continuous Time (joint CSCAMM/KI-Net seminar)

When: Wed, September 16, 2015 - 2:00pm
Where: 4122 CSIC Bldg
Speaker: Prof. Muruhan Rathinam (Department of Mathematics and Statistics, University of Maryland, Baltimore County) -
Abstract: We analyze the Hegselmann-Krause model for continuous time and multidimensional opinions where agents have heterogeneous but symmetric confidence bounds and prove that all trajectories approach an equilibrium in infinite time. We investigate two forms of stability of equilibria. We prove the Lyapunov stability of all equilibria in the relative interior of the set of equilibria. We provide a necessary condition and a sufficient condition for a form of structural stability of the equilibria. This structural stability is a notion of stability introduced by Blondel et al. where a new agent with an arbitrarily small weight is introduced to a system in equilibrium.

Swarm Formation and Flocking Dynamics via Discrete Positive Systems (joint CSCAMM/KI-Net seminar)

When: Wed, September 30, 2015 - 2:00pm
Where: 4122 CSIC Bldg
Speaker: Prof. Ulrich Krause (Department of Mathematics, University of Bremen, Germany) -
Abstract: In the talk a general model for swarm formation of birds (or other agents) will be presented. Swarm formation means that birds approach asymptotically the same velocity , whereby distances among them converge. The main result offers conditions on the local interaction of the birds for swarm formation to happen. Roughly speaking, the structure of interaction should not be "too loose" and the intensity of interaction should not decay "too fast". Furthermore, the various flight regimes, e.g. echelons, occuring in swarm formation, will be analysed. The talk addresses also the famous, albeit somewhat particular, Cucker - Smale model of flocking as well as more recent models which allow for the non-symmetric interaction pertinent to swarms of birds. What if the interaction of birds (or other agents) is too loose and/or decays too fast? In this case a further result will describe the flocking dynamics of the birds and how the ensemble of birds splits into separate flocks (or sub-swarms). The model presented applies to other kinds of animal flocking, to opinion dynamics, to distributed sensor fusion, to mobile robots coordination and others. Considered in discrete time, all these are cases of discrete positive dynamical systems. The theory of the latter provides useful results, in particular on the convergence behavior of infinite products of stochastic matrices and, more general, on the (inhomogeneous) iteration of maps given by averaging.

Projection: A unified approach to semi-infinite linear optimization and duality in convex optimization

When: Wed, October 7, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Amitabh Basu (Department of Applied Mathematics and Statistics, Johns Hopkins University) -
Abstract: Fourier-Motzkin elimination was invented as an explicit projection algorithm for convex polyhedra and can be used to encode a lot of information in instances of linear optimization with finitely many constraints. We extend Fourier-Motzkin elimination to semi-infinite linear programs, i.e., linear optimization problems with infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap. Our approach yields a new classification of variables that is used to determine the existence of duality gaps. Our approach has interesting applications in finite-dimensional convex optimization, such as completely new proofs of Slater's condition for strong duality.

Graph Limits and Dynamics of Large Networks (Joint CSCAMM/KI-Net Seminar)

When: Wed, October 21, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Georgi Medvedev (Department of Mathematics, Drexel University) -
Abstract: The continuum limit is an approximate procedure, by which coupled dynamical systems on large graphs are replaced by an evolution integral equation on a continuous spatial domain. This approach has been used for studying dynamics of diverse networks throughout physics and biology.

We use the combination of ideas and results from the theories of graph limits and nonlinear evolution equations to develop a rigorous justification for using the continuum limit for a variety of dynamical models on deterministic and random graphs. As an application, we discuss stability of spatial patterns in the Kuramoto model on certain Cayley and random graphs.

The truth about diffusion (in liquids) --- Joint CSCAMM/KI-Net Seminar

When: Wed, October 28, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Aleksandar Donev (Courant Institute of Mathematical Sciences, New York University) -
Abstract: Diffusion is one of the most ubiquitous transport processes and is often thought to be one of the simplest dissipative mechanisms. Fick's law of diffusion is derived in most elementary textbooks, and relates diffusive fluxes to the gradient of chemical potentials via a diffusion coefficient that is typically thought of as an independent material property. In this talk we will discuss the miscroscopic and mesoscopic mechanism of diffusion in liquids, for both molecular diffusion and diffusion of colloidal particles. Through a combination of theory and simulations I will demonstrate that diffusion in liquids is, in fact, a rather subtle process due to the crucial contribution of hydrodynamic correlations and fluctuations.

Using multiscale analysis we derive a closed form stochastic diffusion equation that captures both Fick's law for the ensemble-averaged mean and also the long-range correlated giant fluctuations in individual realizations of the mixing process. These giant fluctuations, observed in experiments, are shown to be the result of the long-ranged hydrodynamic correlations among the diffusing particles. Through a combination of Eulerian and Lagrangian numerical experiments we demonstrate that mass transport in liquids can be modeled at all scales, from the microscopic to the macroscopic, not as dissipative Fickian diffusion, but rather, as non-dissipative random advection by thermal velocity fluctuations. Our model gives effective dissipation with a diffusion coefficient that is not a material constant as its value depends on the scale of observation. Our work reveals somewhat unexpected connections between flows at small scales, dominated by thermal fluctuations, and flows at large scales, dominated by turbulent fluctuations.

Deterministic and Stochastic Bounds in the Phase Retrieval Problem

When: Wed, November 4, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Radu Balan (Department of Mathematics and CSCAMM, University of Maryland) -
Abstract: The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redundant system (frame), we want to reconstruct the unknown input. This problem has first occurred in X-ray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this talk I present Lipschitz extension results as well as Cramer-Rao Lower Bounds that govern any reconstruction algorithm. In particular we show that the left inverse of the nonlinear analysis map can be extended to the entire measurement space with a small increase in the Lipschitz constant independent of the space dimension or the frame redundancy.

Numerical methods for the Dirac equation in the non-relativistic limit regime

When: Wed, November 18, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Yongyong Cai (Department of Mathematics, Purdue University) -
Abstract: Dirac equation, proposed by Paul Dirac in 1928, is a relativistic version of the Schroedinger equation for quantum mechanics. It describes the evolution of spin-1/2 massive particles, e.g. electrons. Due to its applications in graphene and 2D materials, Dirac equations has drawn considerable interests recently. We are concerned with the numerical methods for solving the Dirac equation in the non-relativistic limit regime, involving a small parameter inversely proportional to the speed of light. We begin with commonly used numerical methods in literature, including finite difference time domain and time splitting spectral, which need very small time steps to solve the Dirac equation in the non-relativistic limit regime. We then propose and analyze a multi-scale time integrator pseudospectral method for the Dirac equation, and prove its uniform convergence in the non-relativistic limit regime.

Towards a mathematical understanding of surface hopping algorithm

When: Wed, January 27, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Jianfeng Lu (Department of Mathematics, Duke University) -
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.

Computation of linear transport equation

When: Wed, February 3, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Qin Li (University of Wisconsin-Madison, Department of Mathematics) -
Abstract: 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.

Diffusion of new products with recovering agents

When: Wed, February 10, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Gadi Fibich (Tel Aviv University, Department of Applied Mathematics) -

Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement (Joint CSCAMM / Applied Math Seminar)

When: Thu, February 11, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Amit Acharya (Carnegie Mellon University, Civil & Environmental Engineering) -
Abstract: 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.

This is joint work with Chiqun Zhang, Xiaohan Zhang, Dmitry Golovaty, and Noel Walkington.

Population dynamics and therapeutic resistance: mathematical models (Joint CSCAMM/KI-Net Seminar)

When: Wed, February 24, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Alexander Lorz (Université Pierre et Marie Curie) -
Abstract: 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.

Homogeneous solutions to the incompressible Euler equation

When: Wed, March 2, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Roman Shvydkoy (University of Illinois at Chicago) -
Abstract: 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 be 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 0-dimensional intermittencies.

Direct numerical simulations of canonical shock-turbulence interaction

When: Wed, March 9, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Johan Larsson (University of Maryland, Department of Mechanical Engineering) -
Abstract: 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.

The h-principle in fluid mechanics, The Avron Douglis Memorial Lecture

When: Thu, March 31, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. László Székelyhidi Jr. (Institute of Mathematics, University of Leipzig) -
Abstract: 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, Joint CSCAMM/KI-Net Seminar

When: Wed, April 6, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Cory Hauck (Department of Mathematics, University of Tennesee and ORNL) -
Abstract: 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.

Harnessing genetic interactions to advance cancer treatment

When: Wed, April 13, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Eytan Ruppin (Center for Bioinformatics and Computational Biology, University of Maryland) -
Abstract: 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 (Joint CSCAMM/KI-Net Seminar)

When: Wed, April 20, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Piotr Gwiazda (Institute of Mathematics, Polish Academy of Sciences) -
Abstract: 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
[1] 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.
[2] 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.
[3] E. Feireisl, P. Gwiazda, A. ŚSwierczewska-Gwiazda and E. Wiedemann. Dissipative measure-valued solutions to the compressible Navier-Stokes system, arXiv:1512.04852
[4] 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 (Joint Ki-Net/Applied Math Seminar)

When: Thu, April 21, 2016 - 3:30pm
Where: Math 3206
Speaker: Prof. Agnieszka Świerczewska-Gwiazda (Institute of Applied Mathematics and Mechanics, University of Warsaw) -
Abstract: 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

When: Wed, April 27, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Konstantina Trivisa (Department of Mathematics, University of Maryland) -
Abstract: 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.