UCSB Applied Math/PDE/Data Science Seminar

Abstract: Rigid structures such as cars or any other solid objects are often represented by finite clouds of ordered points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose a continuous and complete invariant of unordered clouds under rigid motion in any Euclidean space, called the Simplexwise Centered Distribution (SCD). This invariant extends the distribution of pairwise distances by Boutin and Kemper (Adv Appl Math 2004), which is generically complete but cannot distinguish 2D clouds of 4 points depending on 4 free parameters, and the more recent Pointwise Distance Distribution (PDD) by Widdowson and Kurlin (NeurIPS 2022), which was proved to be generically complete in the much harder case of periodic point sets but cannot distinguish 3D clouds of 6 points depending on 3 free parameters. For a fixed dimension, the new metric for the SCD invariant is computable in polynomial time in the number of points. All results appeared in the paper at http://kurlin.org/research-papers.php#CVPR2023.

Host: Paul Atzberger

Abstract: Fluids with free surface are ubiquitous in our daily lives, and how they would evolve for certain initial datum is of great practical and mathematical interest. In this talk, we will discuss 2D free boundary incompressible Euler equations with surface tension, where the fluid domain is periodic in the horizontal variable and has finite depth. We construct initial data with a flat free boundary and arbitrarily small velocity, such that the gradient of vorticity grows at least double-exponentially for all times during the lifespan of the associated solution. This work generalizes the celebrated result by Kiselev--Šverák to the free boundary setting. The free boundary introduces some major challenges in the proof due to the deformation of the fluid domain and the fact that the velocity field cannot be reconstructed from the vorticity using the Biot-Savart law. We overcome these issues by deriving uniform-in-time control on the free boundary and obtaining pointwise estimates on an approximate Biot-Savart law. This is joint work with Chenyun Luo and Yao Yao. 

Host: Elie Abdo

Abstract: A fundamental problem in machine learning is the following. Given data of the form {(xj, yj )}, find a functional relationship f so that f(xj ) ≈ yj. The manifold hypothesis in machine learning assumes that the points {xj} are random samples from a probability measure supported on a small dimensional unknown sub-manifold of a high dimensional space. We describe the construction of a “diffusion polynomial” multiscale to approximate and analyze the local features of functions on such manifolds. 

Host: Sui Tang

Abstract: How long will a confined Brownian particle take to hit an exceedingly small target? It is a classical result that the expected value of this first hitting time (FHT) blows up as the size of the target vanishes in two or more spatial dimensions. This is an example of a "strongly localized perturbation'' in the sense that small geometric defects have large global effects. If Brownian motion is replaced with Lévy flights, a spatially discontinuous jump process, then the FHT has the potential to blow up even in the case of one spatial dimension. In this talk, I will discuss how matched asymptotic expansions yield a computationally inexpensive method for computing the FHT in the case of Lévy flights by reducing the problem to that of solving a linear system of equations. Moreover, we will see that depending on the fractional order of the Lévy flight, the FHT is qualitative similar to that for Brownian motion in one or more spatial dimensions. In addition to analyzing FHT problems, matched asymptotic expansions have also been highly successful in studying localized solutions to singularly perturbed reaction-diffusion systems. I will conclude by outlining how matched asymptotic expansions similarly yield nonlinear algebraic systems, globally coupled eigenvalue problems, and differential algebraic equations that describe the structure and dynamical properties of localized solutions. 

Host: Paul Atzberger 

Abstract: In this talk, we propose to use random batch methods to construct the gradient used for iterations in seismic tomography. Specifically, we use the frozen Gaussian approximation to compute seismic wave propagation, and then construct stochastic gradients by random batch methods. The method inherits the spirit of stochastic gradient descent methods for solving high-dimensional optimization problems. The proposed idea is general in the sense that it does not rely on the usage of the frozen Gaussian approximation, and one can replace it with any other efficient wave propagation solvers, e.g., Gaussian beam methods and spectral element methods. We prove the convergence of the random batch method in the mean-square sense, and show the numerical performance of the proposed method by two-dimensional and three-dimensional examples of wave-equation-based travel-time inversion and full-waveform inversion, respectively. 

Host: Xu Yang

Abstract: Learning algorithms have achieved significant advances in recent years, but their robustness and reliability remains problematic. Many mathematical approaches have been proposed for mitigating this issue: the first aim of this talk will be to give an overview of different means used to achieve this goal. The second part of this talk will turn towards a particular approach, stemming from classical statistical theory, which seeks to generalize quantiles and medians to higher dimension: these are known as data depths. Recent work, in collaboration with Martin Molina-Fructuoso, connects certain types of data depths (specifically Tukey/Halfspace depths) with Hamilton-Jacobi equations, a first-order partial differential equation that is fundamental to control theory. We'll discuss various numerical algorithms stemming from this work which enable computation of depth functions in high dimensions, which was previously not feasible. This talks will aim to be accessible to a broad audience in applied math, and will not assume expertise in statistics or partial differential equations. 

Host: Matt Jacobs

Abstract: I will talk about the long time dynamics of solutions of the Nernst-Planck-Navier-Stokes system, which models the electrodiffusion of charged particles in fluids. We show that the long time dynamics are intricately related to the choice of boundary condition. We discuss what is known about the long time behavior for equilibrium vs nonequilibrium boundary conditions, and in each of these cases, we study properties of the global attractor and rates of convergence. This talk is based on joint works with Peter Constantin, Mihaela Ignatova, and Elie Abdo.  

Host: Elie Abdo

Abstract: Generative Adversarial Networks (GANs) are popular tools for generation tasks, but they face challenges such as mode collapse and training instability. Existing strategies partially address these issues but may require careful architecture selection and empirical tuning. To address these issues, we propose a deep generative model that uses the geometry-preserving encoder via a Gromov-Monge cost. Our model identifies the low-dimensional structure of the underlying measure of the data and maps it into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. Furthermore, we show that the induced generative map exhibits $c$-cyclical monotonicity, where $c$ is an intrinsic embedding cost employed by the encoder. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images,  exhibiting robustness to mode collapse and training instability.   

Host: Matt Jacobs