Welcome to the Applied Mathematics Reading Seminar, organized by Nathan Soedjak, Han Yong Wunrow, and Edith Zhang.
The purpose of this seminar is to build community among the applied mathematics graduate students and to create a
space for students to teach and learn topics in applied mathematics. There will be an emphasis on applications to
supplement APAM coursework.
Graduate students will sign up for either 2-3 seminar dates to cover their proposed topic. Prior to each seminar,
speakers will email the attendees an abstract of their talk and any relevant textbook chapters or papers for
attendees to read prior to the seminar.
Our talks will be held in-person in Columbia University (Mudd 210) on Mondays from 4 p.m. to 5 p.m. EDT.
If you would like to come or to be added on the mailing list, please email ejz2120[at]columbia[dot]edu.
| Date | Presenter | Title/Abstract | Materials |
|---|---|---|---|
| 11/20 | Blake Sisson | Methods for continuous optimization: overview, challenges, and directions inspired by deep learning
The AI community regularly tackles extremely large optimization problems (while training neural networks for instance) with great success. Can the techniques they employ be rigorously adapted to solve hard nonconvex problems? We first overview the fundamental methods used in continuous optimization for unconstrained and constrained problems. Next, the challenges posed by high dimensional and nonconvex problems are clearly illustrated. We then provide some perspective on the AI community and the technical results they rely on. Finally, we give some insights on adapting their techniques to solve nonconvex problems. |
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| 11/13 | Edith Zhang | Large graphs in nonlocal analysis
This talk is on a forthcoming paper that relates graphons, which are a notion of a large-graph limit, and nonlocal calculus. I will define graphons, discuss several examples, and show how they are a natural functional-analytic object. Then I will introduce the graph min-cut problem, which is a motivating problem for our paper, and describe our main results. I will then present some future applications for graphons in applied graph theory, focusing on dynamics on graphs. Discussion is welcomed! The paper is joint with James Scott, Qiang Du, and Mason A. Porter. |
|
| 11/6 | Dion Ho | Reverse-engineering perfectly accurate Two-Stream Equations
Fast and accurate computation of atmospheric reflectance and transmittance is key for practically any form of climate modeling. The standard method for calculating reflectance and transmittance is through the Two-Stream Equations (TSE). Decades of analytic work have had limited success in deriving a TSE form that is both accurate and tractable. We take a novel approach of using numerical optimization and equation discovery to reverse-engineer TSE forms that can compute flux profiles to nigh perfect accuracy. We show that our TSE forms are tractable if we only require the reflectance and transmittance, and not the entire flux profile, to be accurate. |
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| 10/30 | Han Yong Wunrow | Estimating Time-Varying Parameters in Epidemiological Models using the Ensemble Adjustment Kalman
Filter The accurate estimation of time-varying parameters is pivotal in epidemiology for understanding and responding to disease outbreaks. This presentation explores the application of the Ensemble Adjustment Kalman Filter (EAKF) in this context. We delve into the principles of EAKF, the challenges of estimating time-varying parameters, solutions using adaptive inflation, and some numerical results. Our findings demonstrate the effectiveness of the EAKF in improving time-varying reproduction number estimates, offering significant potential for informing public health decision-making. |
Slides |
| 10/24 | Yan Cheng | Multiscale Full Waveform Inversion with Wavelet-Induced Data Misfit
This talk delves into a novel approach to Full Waveform Inversion (FWI) that leverages wavelet-induced data misfit. Traditional FWI, when performed with the \(\mathbb{L}^2\) loss function, is equivalent to solving a specific PDE-constrained optimization problem. This proposal introduces a wavelet-based objective function that aims to enhance the efficiency and accuracy of FWI. By judiciously choosing wavelets and integrating them into the FWI framework, this method offers potential advantages over classical settings. |
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| 10/16 | Jackson Turner | Existence of Energy Minimizers for the 1D Nonlinear Schrödinger Equation: An Exploration
via the Concentration-Compactness Principle (Part 2) We will quickly summarize the variational framework underlying the Schrödinger Equation's existence of energy minimizers (this was discussed last week in finer detail). We then will outline this theory to work in a focusing nonlinear setting and cover the role of the concentration-compactness principle in ensuring the existence of an admissible limit for a minimizing sequence. Additionally, we'll consider solutions in \(H^1_{\textrm{odd}}(\mathbb R)\), allowing us to seek odd solutions to the Nonlinear Schrödinger Equation with a symmetric linear potential. To render our discussion tangible, we present an illustrative example featuring a finite square potential well. This aids us in establishing an algebraic framework and visuals that reveal all possible solutions via a rescaling of Jacobi elliptic functions and delineate the necessary and sufficient conditions under which they exist. |
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| 10/9 | Jackson Turner | Existence of Energy Minimizers for the 1D Nonlinear Schrödinger Equation: An Exploration
via the Concentration-Compactness Principle In this talk, we will review the variational framework underlying the example Schrödinger Equation's existence of energy minimizers. We then will develop this theory to work in a focusing nonlinear setting. Central to our discussion will be the role of the concentration-compactness principle in ensuring the existence of an admissible limit for a minimizing sequence. Additionally, we'll explore specific linear subspaces of \(H^1(\mathbb R)\), such as \(H^1_{\textrm{odd}}(\mathbb R)\), pinpointing sufficient conditions for the existence of minimizers, allowing us to seek odd solutions to the Nonlinear Schrödinger Equation. To render our discussion tangible, we present an illustrative example featuring a finite square potential well. This aids us in establishing an algebraic framework and visuals that reveal all possible solutions via a rescaling of Jacobi elliptic functions and delineate the necessary and sufficient conditions under which they exist. |
|
| 10/2 | Dion Ho | Equation Discovery Equation Discovery is a subfield of machine learning wherein the goal is to learn explicit rules and laws from data. This contrasts with the usual machine learning goal of maximizing accuracy even if the resulting model is an uninterpretable "black box". We will first examine a research problem in Radiative Transfer and through it discuss the question of when Equation Discovery methods ought to be used. Next, we will discuss the paper Robust data-driven discovery of governing physical laws with error bars and the Bayesian Regression Equation Discovery method it proposes. Finally, we will loosely follow the paper Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges to discuss some of the successes and limitations encountered in the field of Equation Discovery. |
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| 9/25 | Nathan Soedjak | Recovering coefficients in a system of semilinear Helmholtz equations from internal data We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data. |
Slides |
| 9/18 | Blake Sisson | Fractional Packing Problems We present a fast algorithm that finds approximate solutions for the fractional packing problem. The algorithm employs a potential function reduction scheme and achieves a running time of \(\mathcal{O}(\epsilon^{-2})\) in terms of the tolerance. |
Outline notes |