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{\’e}rigot and Boris Thibert.

We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search

Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

Perspectives from computational algebra and numerical optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.

Given two probability measures and a transportation cost, the optimal transport problem asks to find the most cost efficient way to transport one measure to the other. Since its introduction in 1781 by Gaspard Monge, the optimal transport problem has found applications in logistics, economics, physics, PDEs, and more recently data science. However, despite sustained attention from the numerics community, solving optimal transport problems has been a notoriously difficult task. In this talk I will introduce the back-and-forth method, a new algorithm to efficiently solve the optimal transportation problem for a general class of strictly convex transportation costs. Given two probability measures supported on a discrete grid with n points, the method computes the optimal map in O(n log(n)) operations using O(n) storage space. As a result, the method can compute highly accurate solutions to optimal transportation problems on spatial grids as large as 4096 x 4096 and 384 x 384 x 384 in a matter of minutes. If time permits, I will demonstrate an extension of the algorithm to the simulation of a class of gradient flows. This talk is joint work with Flavien Leger.

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In the first part of this talk, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. In the second part of this talk, I will describe our recent effort in computing a nonlocal interface problem arising from segregation of two species with high competition. One species moves according to the classical diffusion and the other adopts a nonlocal strategy. A novel iterative scheme will be presented that constructs a sequence of supersolutions shown to be convergent to the viscosity solution of the interface problem.

One of the most fundamental questions in partial differential equations is that of regularity and the possible breakdown of solutions. We will discuss this question for solutions to a canonical example of a geometric wave equation; energy critical wave maps. Break-through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Rapha ̈el-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schro ̈dinger maps and Yang-Mills equations. A basic question is the following: • Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by it’s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.

Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. We present recent results on properties of these equilibria and long-time asymptotics of solutions in the setting where attractive and repulsive forces are in competition. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, fast algorithms will be presented to reduce the high computational cost from the numerical implementation of the nonlocal operators. Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here we will try to provide better perspectives of the origin of nonlocality from multiscale modeling and homogenization, which in turn may help the development of more effective numerical methods for homogenization.

In this talk, I will discuss three ingredients of optimization theory in the context of MCMC: Non-convexity, Acceleration, and Stochasticity.

I will focus on a class of non-convex objective functions arising from mixture models. For that class of objective functions, I will demonstrate that the computational complexity of a simple MCMC algorithm scales linearly with the model dimension, while optimization problems are NP-hard.

I will then study MCMC algorithms as optimization over the KL-divergence in the space of measures. By incorporating a momentum variable, I will discuss an algorithm which performs "accelerated gradient descent" over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained.

Finally, I will present a general recipe for constructing stochastic gradient MCMC algorithms that translates the task of finding a valid sampler into one of choosing two matrices. I will then describe how stochastic gradient MCMC algorithms can be applied to applications involving temporally dependent data, where the challenge arises from the need to break the dependencies when considering minibatches of observations.

Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. Technically, the heart of the argument are new laws of large numbers at the exponential scale, proved through an explicit combinatorics approach. Our result only requires weak regularity on the interaction kernel in negative Sobolev spaces, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes. For dissipative gradient flows, we may allow any singularity lower than the Poisson kernel. This talk corresponds to a joint work with Z. Wang and an upcoming work with D. Bresch and Z. Wang.