The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\â??e}rigot and Boris Thibert.

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. However, solution of Keller-Segel equation can exhibit dramatic collapsing behavior. In other words, there exist initial data leading to finite time blow up. In this talk, we will discuss the possible effects resulting from interaction of chemotactic and fluid transport processes, namely we will consider the Keller-Segel equation with additional advection term modeling ambient fluid flow. We will prove that the presence of fluid can prevent the singularity formation. We will discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers.

An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.

The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.

In this talk, we will introduce a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV). Using optimal control theory, we will illustrate that optimal dynamic multidrug therapies can produce a drug dosing strategy that exhibits structured treatment interruption, a regimen in which patients are cycled on and off therapy. In addition, sensitivity analysis of the model including both classical sensitivity functions and generalized sensitivity functions will be presented. Finally, we will describe how stochastic estimation can be used to filter and estimate states and parameters from noisy data. In the course of this analysis it will be shown that automatic differentiation can be a powerful tool for this type of study.

Given a number of labeled and unlabeled images, it is possible to determine the class membership of each unlabeled image by creating a sequence of such image transformations that connect it, through other unlabeled images, to a labeled image. In order to measure the total transformation, a robust and reliable metric of the path length is proposed, which combines a local dissimilarity between consecutive images along the path with a global connectivity-based metric. For the local dissimilarity we use a symmetrized version of the zero-order image deformation model (IDM) proposed by Keysers et al. For the global distance we use a connectivity-based metric proposed by Chapelle and Zien in [2]. Experimental results on the MNIST benchmark indicate that the proposed classifier out-performs current state-of-the-art techniques, especially when very few labeled patterns are available.

There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

In 1953, the physicist P.E. Rouse proposed to model polymers in dilute solution by taking the polymer to be a series of beads connected by Gaussian springs. Neglecting inertia, the dynamics are set by a balance between the thermal fluctuations in the fluid and the elastic restoring force of the springs. One year later, B. Zimm noted that a polymer will interact with itself through the fluid in a qualitatively meaningful way. In this talk, we consider a more recent Langevin equation approach to dealing with hydrodynamic self-interaction. This involves coupling the continuum scaling limit of the Rouse model with stochastically forced time-dependent Stokes equations. The resulting pair of parabolic SPDE, with non-linear coupled forcing, presents a number of mathematical challenges. On the way to providing an existence and uniqueness result, we shall take time to develop relevant stochastic tools, and consider the modeling implications of certain technical results.

Sparsity plays a central role in recent developments of many fields such as signal and image processing, compressed sensing, statistics, and optimization. In practice, sparsity is promoted through the additional of an L1 norm (or related quantity) as a constraint or penalty in a variational model. Motivated by the success of sparsity-inducing methods in imaging and information sciences, there is a growing interest in exploiting sparsity in dynamical systems and partial differential equations. In this talk, we will investigate the connections between compressed sensing, sparse optimization, and numerical methods for nonlinear differential equations. In particular, we will discuss about sparse modeling as well as the advantage of sparse optimization in solving various differential equations arising from physical and data sciences.

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.

Nonsmooth and nonconvex loss functions are often used to model physical phenomena, provide robustness, and improve stability. While convergence guarantees in the smooth, convex settings are well-documented, algorithms for solving large-scale nonsmooth and nonconvex problems remain in their infancy.

I will begin by isolating a class of nonsmooth and nonconvex functions that can be used to model a variety of statistical and signal processing tasks. Standard statistical assumptions on such inverse problems often endow the optimization formulation with an appealing regularity condition: the objective grows sharply away from the solution set. We show that under such regularity, a variety of simple algorithms, subgradient and Gauss Newton like methods, converge rapidly when initialized within constant relative error of the optimal solution. We illustrate the theory and algorithms on the real phase retrieval problem, and survey a number of other applications, including blind deconvolution and covariance matrix estimation.

One of the main advantages of smooth optimization over its nonsmooth counterpart is the potential to use a line search for improved numerical performance. A long-standing open question is to design a line-search procedure in the stochastic setting. In the second part of the talk, I will present a practical line-search method for smooth stochastic optimization that has rigorous convergence guarantees and requires only knowable quantities for implementation. While traditional line-search methods rely on exact computations of the gradient and function values, our method assumes that these values are available up to some dynamically adjusted accuracy that holds with some sufficiently high, but fixed, probability. We show that the expected number of iterations to reach an approximate-stationary point matches the worst-case efficiency of typical first-order methods, while for convex and strongly convex objectives it achieves the rates of deterministic gradient descent.

Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.

The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. It is conjectured by Osher (and proven in certain special cases) that the Green�s function for the biharmonic equation on regions with corners has infinitely many oscillations in the vicinity of each corner. In this talk, we show that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of elementary functions which oscillate with a frequency proportional to the logarithm of the distance from the corner. These representations are used to construct highly accurate and efficient Nyström discretizations, significantly reducing the number of degrees of freedom required for solving the corresponding integral equations. We illustrate the performance of our method with several numerical examples.

After a short introduction presenting our group and our main research topics, I will address the problem of audio-visual speech recognition, i.e. a typical example of multimodal signal analysis, when we want to extract and exploit information coming from two different but complementary signals: an audio and a video channel. We will discuss two important aspects of this analysis. We will first present a new feature extraction algorithm based in information theoretical principles, and show its performances, compared to other classical approaches, in our multimodal context. Then we will discuss multimodal information fusion, i.e. how to combine information from those two channels for optimal classification.

Many common data sets can be modeled by a mixture of manifolds, e.g., facial images of several human subjects under different angles and illumination conditions. However, effective modeling of such data along with careful theoretical analysis is a challenging mathematical problem. In this talk I will introduce the Spectral Curvature Clustering (SCC) algorithm for solving the problem of hybrid linear modeling, i.e., modeling data using linear manifolds, and discuss possible extensions to general multi-manifold data modeling. Our analysis shows that, given data sampled sufficiently around a collection of well separated affine subspaces, the SCC will succeed with high probability. Numerical techniques as well as an application to motion segmentation are also presented.

Large complex networks naturally represent relationships in a variety of settings, e.g. social interactions, computer/communication networks, and genomic sequences. A significant challenge in analyzing these networks has been understanding the Â?intermediate structureÂ? Â? those properties not captured by metrics which are local (e.g. clustering coefficient) or global (e.g. degree distribution). It is often this structure which governs the dynamic evolution of the network and behavior of diffusion-like processes on it. Although there is a large body of empirical evidence suggesting that complex networks are often Â?tree-likeÂ? at intermediate to large size-scales (e.g. work of Boguna et al in physics, Kleinberg on internet routing, and Chung & Lu on power-law graphs), it remains a challenge to take algorithmic advantage of this structure in data analysis. We discuss several approaches and heuristics for quantifying and elucidating tree-like structure in networks, including various tree-decompositions and Gromov's delta hyperbolicity. These approaches were developed with very different "tree-like" applications in mind, and thus we discuss the strengths and short-comings of each in the context of complex networks and how each might aid in identifying intermediate-scale structure in these graphs.

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.