Computational Mathematics and Applications

Link for registration

https://docs.google.com/forms/d/1ra3TSRYtuXxZFPOYhLLyfezzCw6tIWoPHRwnRtjJ-fM

Detailed Program

(Abstracts below)

Day 1

Day 2

Break time or Chat with Speakers

Speakers and Abstracts

Fast Summation Methods for Particle Interactions

Abstract:

Many models in scientific computing involve interacting particles, for example charged particles in electrostatics or plasma dynamics. To address the cost of computing these interactions in large-scale computations, we present two fast summation methods based on barycentric Lagrange interpolation. The first is a tree code (BLTC) and the second is a fast multipole method where the interaction lists are formed by dual tree traversal (BLDTT). The methods are kernel-independent and a distributed memory implementation on multiple GPUs has been developed. The performance of the BLTC and BLDTT is demonstrated for several particle systems.

Minimax Problems: Motivating Examples, Challenges, and Algorithms

Abstract:

Minimax problems provide excellent mathematical tools to model many problems in different fields, including robust learning, distributionally robust optimization, game theory, and generative adversarial networks (GANs). However, these problems are still challenging to solve, especially in nonconvex settings. In this talk, we will discuss some of those examples, identify challenges, and present some solution approaches to solve minimax problems from an optimization perspective.

A Unified and Constructive Framework for the Universality of Neural Networks

Abstract:

One of the reasons that many neural networks are capable of replicating complicated tasks or functions is their universality property. The past few decades have seen many attempts in providing constructive proofs for single or class of neural networks. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activations including most of existing activations and beyond. At the heart of the framework is the concept of neural network approximate identity. It turns out that most of existing activations are neural network approximate identity, and thus universal in the space of continuous of functions on compacta. The framework induces several advantages. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is the first unified attempt that is valid for most of existing activations. Third, as a by product, the framework provides the first university proof for some of the existing activation functions including Mish, SiLU, ELU, GELU, and etc. Fourth, it discovers new activations with guaranteed universality property. Indeed, any activation\textemdash whose $\k$th derivative, with $\k$ being an integer, is integrable and essentially bounded\textemdash is universal. Fifth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with predetermined number of neuron, and the values of weights/biases.

Efficient time-Stepping Methods for Nonlinear Evolution Problems

Abstract:

Numerical modeling of geophysical flows is challenging due to the presence of various coupled processes that occur at different spatial and temporal scales. It is critical for the numerical schemes to capture such a wide range of scales in both space and time to produce accurate and robust simulations over long time horizons.

In this talk, we will discuss efficient time-stepping methods for the rotating shallow water equations discretized on spatial meshes with variable resolutions. Two different approaches will be considered: the first approach is a fully explicit local time-stepping algorithm based on the strong stability preserving Runge-Kutta schemes, which allows different time step sizes in different regions of the computational domain. The second approach, namely the localized exponential time differencing method, is based on spatial domain decomposition and exponential time integrators, which makes possible the use of much larger time step sizes compared to explicit schemes and avoids solving nonlinear systems. Numerical results on various test cases will be presented to demonstrate the performance of the proposed methods.

Control Systems and Applied Mathematics in Mechatronics and Automotive Applications

Abstract:

In this talk, I will discuss several control applications in mechatronics and automotive industries. Control systems is an engineering area that utilizes applied mathematics knowledge extensively. I will start with providing a global picture of control theory and its applications in industry. Next, the development cycle process of how a control system is developed starting from requirements, algorithms, proof-of-concept, to testing and operation will be presented. Through the applications, for example, optimal collision avoidance motion planning, I will discuss challenges, technologies and how they are connecting to mathematical formulations. Multiple aspects such as linear algebra, numerical optimization, linear/nonlinear control, data processing and learning, formal design specifications,… will be shown in a systematic framework from my personal views. Finally, several simulation and experimental demonstrations will be shown.

Deep Learning Methods for Stochastic Control and Partial Differential Equations

Abstract:

The numerical resolution of high-dimensional partial differential equations (PDEs) and stochastic control is a challenging problem in applied mathematics. Over the last five years, several deep neural networks-based algorithms have been proposed and have shown their great efficiency for tackling these issues. In this talk, we give an introduction to this field of research, review the main results in this literature, and present some new developments, notably regarding mean-field control problems and Master equation in Wasserstein space.

Stochastic Approximation and Deep Filtering

Abstract:

Because of the demand and current interests in using machine learning to solve estimation, control, and optimization problems, stochastic gradient algorithms have gained resurgent interests. These algorithms are rooted in stochastic approximation (SA). For a reference on SA, we mention the book of H.J. Kushner and G. Yin, 2nd Ed., Springer, 2013. In this talk, we will give a brief introduction to SA. Then we will present our recent work on deep filtering.

Orthogonality Sampling Methods for Electromagnetic Inverse Scattering

Abstract:

The electromagnetic inverse scattering problem aims to reconstruct the location and shape of an unknown object from measurements of the scattered electromagnetic field. It has applications in various areas such as radar, nondestructive testing, medical imaging, and geophysical exploration. However, solving this problem is challenging because it is highly nonlinear, severely ill-posed, and is associated with Maxwell's equations with possibly matrix-valued coefficients in the three dimensions. In this talk, we will discuss our recent results on a modified version of the Orthogonality Sampling Method (OSM) for solving the inverse scattering problem. The modified OSM is not only a fast and robust method, but also can work with more types of polarization associated with the scattering data in comparison to the original version. Numerical results testing against simulated data and experimental data will be presented. This talk is based on joint work with Dinh-Liem Nguyen, Hayden Schmidt, and Trung Truong.

Created by ICASM team