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.
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.
Prof. Muruhan Rathinam, Department of Mathematics and Statistics, University of Maryland, Baltimore County
Analysis of Bounded Confidence Multidimensional Opinions in
Continuous Time
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.
Prof. Ulrich Krause, Department of Mathematics, University of Bremen, Germany
Swarm Formation and Flocking Dynamics via Discrete Positive Systems
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.
Prof. Amitabh Basu, Department of Applied Mathematics and Statistics, Johns Hopkins University
Projection: A unified approach to semi-infinite linear optimization and duality in convex optimization
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.
On weak solutions of Quantum Navier-Stokes and Korteweg Navier-Stokes equations
In this talk I'll present some results obtained in collaboration with Paolo Antonelli (GSSI - Gran Sasso Science Institute) regarding weak solutions of the Korteweg Navier-Stokes (KNS) equations and the special case of the Quantum Navier-Stokes (QNS) equations. Precisely, the compactness of weak solutions is proved for the (KNS) equations when the viscosity and the capillarity satisfy a particular non linear relation. Moreover, in the special case of (QNS) equations also global existence of weak solutions is proved. The main novelty of these results is that the vacuum region, i.e. the set when the density vanishes, is taken into account in the weak formulation.
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.
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.
Prof. Radu Balan, Department of Mathematics and CSCAMM, University of Maryland
Deterministic and Stochastic Bounds in the Phase Retrieval Problem
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
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.
Entropic structure and duality estimates for cross diffusion models
Cross diffusion models are designed to represent the tendency of species to avoid each other. For two species his leads typically to nonlinear parabolic systems where the diffusion pressure depends on the density of the other species. One of the difficulty is that maximum principle does not hold. We discuss conditions under which global weak solutions can be build and how we might build them generalizing a result of Chen and Jungel on linear cross diffusion pressure. One of the big difficulty is that the estimates we obtain on gradients of the solutions are often not enough to ensure existence and we need duality estimates (introduced by Michel Pierre and coauthors) to derive a complete existence result. Joint work with Laurent Desvillettes, Ayman Moussa and Ariane Trescases.
