Department of Mathematics, Colorado State University
Instantaneous Time Mirrors and Time Reversal
Instantaneous time mirrors (ITMs) were recently introduced by M. Fink and collaborators as a new avenue for time reversal. The latter allows for the focusing of
waves, whether acoustic, electromagnetic or elastic, and has found many important applications in medical imaging, non-destructive testing, and telecommunications for
instance. The main practical difficulty of standard time reversal is the recording/reversal process which necessitates a quite complex apparatus.
ITMs offer on the contrary a simplified experimental alternative that does not require any measurements, provided there is some control over the medium of propagation.
We will review in this talk the basics of the time reversal of waves introduced in the nineties, and discuss the ITMs and some of their properties.
In addition to the experimental setup proposed by M. Fink et al, we will describe another physically realizable system based on surface plasmons.
Department of Computer Science, Durham University, UK
Who said using GPUs was straightforward
With OpenMP and oneAPI's SYCL, we have two programming languages on the table which promise that you can write
scientific codes once, while they run on both CPUs and GPUs. We report on our efforts to migrate ExaHyPE, a
hyperbolic PDE solver which we use for earthquake simulations and gravitational wave research, onto GPUs.
Hereby, we focus on three major challenges: The orchestration of compute steps on the GPU, the memory
management on GPUs, and the realisation of multitasking for tasks that can either run on a CPU or an
accelerator. For us, none of these three areas seem to work out-of-the-box with current technologies.
For all the stumbling blocks that we encounter, we present workarounds and solutions.
Yet, we can only sketch ideas how to answer the overarching research questions: Do these languages provide
the right abstraction level for the realisation of modern numerical codes, do we have to rephrase key
ingredients of scientific codes in different ways to make them fit for GPUs, and can the technologies on
the table allow us to write performance-portable code and not only platform-portable realisations?
Compressed Finite Element Methods and Linear Solvers for Extreme-Scale Computing
To push finite element method applications to larger scales, many have considered approaches such as sum factorization to combat the dimension-based exponent on the cost for both evaluating and storing finite element function data. While this allows one to solve higher dimensional problems more efficiently, we instead focus on structure detection and pattern analysis across mesh cells to reduce redundancy in finite element basis evaluations. We show that preprocessing a mesh for redundancy by considering the de Rham complex increases the speed of a simulation while allowing for larger simulations. We perform mathematical analysis to understand the error incurred in the operator by performing compression. We additionally perform similar compression and analysis for domain-decomposition based linear solvers. We provide numerical results demonstrating the effectiveness of our approach, including a Trilinos-based finite element simulation of 1 billion elements on a single computer node and timings from simulations on Sandia's supercomputers. We conclude by presenting goals for future directions and potential intern projects.
Robust finite element methods for poroelasticity and its coupled equations
Poroelasticity equations arise from many applications in geophysics and biomechanics, so numerical simulations of poroelasticity equations are of great interest nowadays. In this talk I present advanced finite element methods for poroelasticity and related problems.
In the first part, I introduce parameter-robust discretization of poroelasticity, and explain that efficient preconditioners can be easily obtained for the system by the operator preconditioning approach.
In the second part, I present hybridizable discontinuous Galerkin (HDG) methods for the problems that Stokes/Navier-Stokes equations and porous/poroelastic equations are coupled with interface. The HDG methods give numerical solutions which preserve many important physical quantities such as the compressibilities of fluid and poroelastic matrix, and the fluid mass in poroelastic domain.
The talk is based on joint works with K.-A. Mardal (University of Oslo), M. E. Rognes (Simula Research Laboratory), A. Cesmelioglu (Oakland University) S. Rhebergen (University of Waterloo), and other collaborators.
Department of Applied Mathematics & Statistics, Colorado School of Mines
Sparse Gradients in Derivative-Free Optimization.
Derivative-Free Optimization (DFO) is concerned with minimizing a function without using derivatives, and is used in applications where
computing gradients is impossible, intractable, or expensive. While DFO has been classically applied to problems with 10^3 variables or
fewer, emerging applications in machine learning require solving DFO problems with 10^5 variables or more. In this talk I'll survey
some of my recent work in extending DFO to this high-dimensional regime, primarily by exploiting sparsity in gradients to constuct good
gradient approximations cheaply.
Department of Applied Mathematics, University of Colorado Boulder
Randomization methods for big-data
In this era of big-data, we must adapt our algorithms to handle large datasets. One obvious issue is that the number of floating-point operations (flops) increases as the input size increases, but there are many less obvious issues as well, such as the increased communication cost of moving data between different levels of computer memory. Randomization is increasingly being used to alleviate some of these issues, as those familiar with random mini-batch sampling in machine learning are well aware of. This talk goes into some specific examples of using randomization to improve algorithms. We focus on special classes of structured random dimensionality reduction, including the count sketch, tensorSketch, Kronecker fast Johnson-Lindenstrauss sketch, and pre-conditioned sampling. These randomized techniques can then be applied to speeding up the classical Lloyd's algorithm for K-means and for computing tensor decompositions, for example. If time permits, we will also show extensions to optimization, including a gradient-free method that uses random finite differences and a method for solving semi-definite programs in an optimal low-memory fashion.
School of Sciences and Engingeering, University of Missouri - Kansas City
Structure-conforming Operator Learning via Transformers
GPT, Stable Diffusion, AlphaFold 2, etc., all these
state-of-the-art deep learning models use a neural architecture called
"Transformer". Since the emergence of "Attention Is All You Need" paper by
Google, Transformer is now the ubiquitous architecture in deep learning. At
Transformer's heart and soul is the "attention mechanism". In this talk, we
shall dissect the "attention mechanism" through the lens of Galerkin
methods. We shall give a specific example to try to answer a fundamental but
critical question: whether and how one can benefit from the theoretical
structure of a mathematical problem to develop task-oriented and
structure-conforming deep neural networks? An attention-based deep direct
sampling method is proposed for solving Electrical Impedance Tomography
(EIT), a class of boundary value inverse problems. Progresses within
different communities will be briefed to answer some open problems on the
mathematical properties of the attention mechanism in Transformers. This is
joint work with Ruchi Guo (UC Irvine) and Long Chen (UC Irvine).