High performance transducers utilizing piezoceramic, electrostrictive, magnetostrictive or shape memory elements offer novel control capabilities in applications ranging from flow control to precision placement for nanoconstruction. To achieve the full potential of these materials, however, models, numerical methods and control designs which accommodate the constitutive nonlinearities and hysteresis inherent to the compounds must be employed. Furthermore, it is advantageous to consider material characterization, model development, numerical approximation, and control design in concert to fully exploit the novel sensor and actuator capabilities of these materials in coupled systems.

In this presentation, the speaker will discuss recent advances in the development of model-based control strategies for high performance smart material systems. The presentation will focus on the development of unified nonlinear hysteresis models, inverse compensators, reduced-order approximation techniques, and nonlinear control strategies for high precision or high drive regimes. The range for which linear models and control methods are applicable will also be outlined. Examples will be drawn from problems arising in structural acoustics, high speed milling, deformable mirror design, artificial muscle development, tendon design to minimize earthquake damage, and atomic force microscopy.

In this talk, I will present a kernel-free boundary integral (KFBI) method for the variable coefficient elliptic partial differential equation on complex domains. The KFBI method is a generalization of the standard boundary integral method. But, unlike the standard boundary integral method, the KFBI method does not need to know an analytical expression for the kernel of the boundary integral operator or the Green's function associated with the elliptic PDE. So it is not limited to the constant-coefficient PDEs. The KFBI method solves the discrete integral equations by an iterative method, in which only part of the matrix vector multiplication involves the discretization of the boundary integral. With the KFBI method, the evaluation of the boundary integral is replaced by interpolation from a structured grid based solution to an equivalent interface problem, which is solved quickly by a Fourier transform or geometric multigrid based fast elliptic solver. Numerical examples for Dirichlet and Neumann BVPs, interface problems with different conductivity constants and the Poisson-Boltzmann equations will be presented.

We study the large-time behavior of hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with alignment which makes the large time behavior very different from the original Cucker-Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of \emph{quadratic} potentials, we are able to treat a large class of admissible interaction kernels, $\phi(r) \gtrsim (1+r^2)^{-\beta}$ with `thin' tails $\beta \leq 1$ --- thinner than the usual `fat-tail' kernels encountered in CS flocking $\beta\leq\nicefrac{1}{2}$: we discover unconditional flocking with exponential convergence of velocities \emph{and} positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a necessary stability condition, requiring large enough alignment kernel to avoid crowd dispersion. We prove, by hypocoercivity arguments, that both the velocities \emph{and} positions of smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.

A very classical shape optimization problem consists in optimizing the topology and geometry of an elastic structure subjected to fixed boundary loads. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal designs in engineering, but e.g. also helps to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)

There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk I will discuss some recent works on the existence of global solutions.

Adaptive acquisition of correct features from massive datasets is at the core of modern data analysis. One particular interest in medicine is the extraction of hidden dynamics from an observed time series composed of multiple oscillatory signals. The mathematical and statistical problems are made challenging by the structure of the signal which consists of non-sinusoidal oscillations with time varying amplitude and time varying frequency, and by the heteroscedastic nature of the noise. In this talk, I will discuss recent progress in solving this kind of problem. Based on the cepstrum-based nonlinear time-frequency analysis and manifold learning technique, a particular solution will be given along with its theoretical properties. I will also discuss the application of this method to two medical problems Â? (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) the simultaneous extraction of the instantaneous heart rate and instantaneous respiratory rate from a PPG signal during exercise. If time permits, an extension to multiple-time series will be discussed.

At the beginning of this talk, We will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS including tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then We will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

The Boltzmann equation models the evolution of the probability density of gas particles that interact through predominantly binary collisions. The equation consists of a transport operator and a collision operator. The latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that the kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. Recently, however, it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting. We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated. The result uses Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

When a fluid layer is heated from below with temperature larger than a certain critical value, the convective motion appears in the fluid. The convection caused by the thermocapillary effect is called Benard-Marangoni heat convection. The thermocapillary effect is the dependence of the surface tension on the temperature. Near a hot spot on a free surface of fluid a thermocapillary tangential stress generates a fluid motion. In this talk the mathematical model system for this convection is explained. The Oberbeck-Boussinesq approximation is used for the system and the upper boundary is a free surface with surface tension which depends on the temperature. After formulating the linearized problem around the conductive state, we derive the resolvent estimates which guarantee the sectorial property. Stationary and Hopf bifurcations (periodic solutions) are proved to exist depending on the parameters (Raylegh and Marangoni numbers).

One of the pressing problems in the analysis of reaction-diffusion equations is obtaining accurate and reliable estimates of the error of numerical solutions. Recently, we made significant progress using a new approach that at the heart is computational rather than analytical. I will describe a framework for deriving and analyzing a posteriori error estimates, discuss practical details of the implementation of the theory, and illustrate the error estimation using a variety of well-known models. I will also briefly describe an application of the theory to the class of problems that admit invariant rectangles and discuss the preservation of invariant rectangles under discretization.