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Matthew Dawson

Publications and source records attributed to Matthew Dawson.

15 recordsLinked to original sources
arXiv2024

Adapting Atmospheric Chemistry Components for Efficient GPU Accelerators

Atmospheric models demand a lot of computational power and solving the chemical processes is one of its most computationally intensive components. This work shows how to improve the computational performance of the Multiscale Online Nonhydrostatic AtmospheRe CHemistry model (MONARCH), a chemical weather prediction system developed by the Barcelona Supercomputing Center. The model implements the new flexible external package Chemistry Across Multiple Phases (CAMP) for the solving of gas- and aerosol-phase chemical processes, that allows multiple chemical processes to be solved simultaneously as a single system. We introduce a novel strategy to simultaneously solve multiple instances of a chemical mechanism, represented in the model as grid-cells, obtaining a speedup up to 9x using thousands of cells. In addition, we present a GPU strategy for the most time-consuming function of CAMP. The GPU version achieves up to 1.2x speedup compared to CPU. Also, we optimize the memory access in the GPU to increase its speedup up to 1.7x.

physics.comp-ph↗
arXiv2024

Quantum Harmonic Analysis on the Unweighted Bergman Space of the Unit Ball

We study quantum harmonic analysis (QHA) on the Bergman space $\mathcal{A}^2(\mathbb{B}^n)$ over the unit ball in $\mathbb{C}^n$. We formulate a Wiener's Tauberian theorem, and characterizations of the radial Toeplitz algebra over $\mathcal{A}^2(\mathbb{B}^n)$. We discuss the $\alpha$-Berezin transform and investigate the question of approximations by Toeplitz operators.

math.FA↗
arXiv2024

A Representation Theoretic Approach to Toeplitz Quantization on Flag Manifolds

In this paper, we study Toeplitz operators on generalized flag manifolds of compact Lie groups using a representation-theoretic point of view. We prove several basic properties of these Toeplitz operators, including an abstract formula for their matrix coefficients in terms of the decomposition of certain tensor product representations. We also show how to identify large commuting families of Toeplitz operators based on invariance of their symbols under certain subgroups. Finally, we realize the Berezin transform as a convolution with certain functions that form an approximate identity on the generalized flag manifold, which allows us to prove a Szeg\"o Limit Theorem using certain results due to Hirschman, Liang, and Wilson.

math.RT↗
arXiv2024

Optimized thread-block arrangement in a GPU implementation of a linear solver for atmospheric chemistry mechanisms

Earth system models (ESM) demand significant hardware resources and energy consumption to solve atmospheric chemistry processes. Recent studies have shown improved performance from running these models on GPU accelerators. Nonetheless, there is room for improvement in exploiting even more GPU resources. This study proposes an optimized distribution of the chemical solver's computational load on the GPU, named Block-cells. Additionally, we evaluate different configurations for distributing the computational load in an NVIDIA GPU. We use the linear solver from the Chemistry Across Multiple Phases (CAMP) framework as our test bed. An intermediate-complexity chemical mechanism under typical atmospheric conditions is used. Results demonstrate a 35x speedup compared to the single-CPU thread reference case. Even using the full resources of the node (40 physical cores) on the reference case, the Block-cells version outperforms them by 50%. The Block-cells approach shows promise in alleviating the computational burden of chemical solvers on GPU architectures.

cs.AR↗
arXiv2023

Optimizing Hierarchical Queries for the Attribution Reporting API

We study the task of performing hierarchical queries based on summary reports from the {\em Attribution Reporting API} for ad conversion measurement. We demonstrate that methods from optimization and differential privacy can help cope with the noise introduced by privacy guardrails in the API. In particular, we present algorithms for (i) denoising the API outputs and ensuring consistency across different levels of the tree, and (ii) optimizing the privacy budget across different levels of the tree. We provide an experimental evaluation of the proposed algorithms on public datasets.

cs.DS↗
arXiv2020

Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce \textbf{AIDN}, \textit{Algebraically-Informed Deep Networks}. \textbf{AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a set of deep neural networks. The deep networks obtained via \textbf{AIDN} are \textit{algebraically-informed} in the sense that they satisfy the algebraic relations of the presentation of the algebraic structure that serves as the input to the algorithm. Our proposed network can robustly compute linear and non-linear representations of most finitely-presented algebraic structures such as groups, associative algebras, and Lie algebras. We evaluate our proposed approach and demonstrate its applicability to algebraic and geometric objects that are significant in low-dimensional topology. In particular, we study solutions for the Yang-Baxter equations and their applications on braid groups. Further, we study the representations of the Temperley-Lieb algebra. Finally, we show, using the Reshetikhin-Turaev construction, how our proposed deep learning approach can be utilized to construct new link invariants. We believe the proposed approach would tread a path toward a promising future research in deep learning applied to algebraic and geometric structures.

cs.LG↗
arXiv2020

Rank Dynamics for Functional Data

The study of the dynamic behavior of cross-sectional ranks over time for functional data and the ranks of the observed curves at each time point and their temporal evolution can yield valuable insights into the time dynamics of functional data. This approach is of interest in various application areas. For the analysis of the dynamics of ranks, estimation of the cross-sectional ranks of functional data is a first step. Several statistics of interest for ranked functional data are proposed. To quantify the evolution of ranks over time, a model for rank derivatives is introduced, where rank dynamics are decomposed into two components. One component corresponds to population changes and the other to individual changes that both affect the rank trajectories of individuals. The joint asymptotic normality for suitable estimates of these two components is established. The proposed approaches are illustrated with simulations and three longitudinal data sets: Growth curves obtained from the Zürich Longitudinal Growth Study, monthly house price data in the US from 1996 to 2015 and Major League Baseball offensive data for the 2017 season.

stat.ME↗
arXiv2019

Toeplitz operators on the domain $\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\}$ with $\mathrm{U}(2)\times\mathbb{T}^2$-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators.

math.FA↗
arXiv2018

Direct Proofs of the Fundamental Theorem of Calculus for the Omega Integral

When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not relying on the Riemann integral. This note provides those proofs.

math.CA↗
arXiv2017

Radial Toeplitz operators on the weighted Bergman spaces of Cartan domains

Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.

math.FA↗
arXiv2017

Dynamic Modeling with Conditional Quantile Trajectories for Longitudinal Snippet Data, with Application to Cognitive Decline of Alzheimer's Patients

Longitudinal data are often plagued with sparsity of time points where measurements are available. The functional data analysis perspective has been shown to provide an effective and flexible approach to address this problem for the case where measurements are sparse but their times are randomly distributed over an interval. Here we focus on a different scenario where available data can be characterized as snippets, which are very short stretches of longitudinal measurements. For each subject the stretch of available data is much shorter than the time frame of interest, a common occurrence in accelerated longitudinal studies. An added challenge is introduced if a time proxy that is basic for usual longitudinal modeling is not available. This situation arises in the case of Alzheimer's disease and comparable scenarios, where one is interested in time dynamics of declining performance, but the time of disease onset is unknown and the chronological age does not provide a meaningful time reference for longitudinal modeling. Our main methodological contribution is to address this problem with a novel approach. Key quantities for our approach are conditional quantile trajectories for monotonic processes that emerge as solutions of a dynamic system, and for which we obtain uniformly consistent estimates. These trajectories are shown to be useful to describe processes that quantify deterioration over time, such as hippocampal volumes in Alzheimer's patients.

stat.ME↗
arXiv2015

Conical Representations for Direct Limits of Symmetric Spaces

We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.

math.RT↗
arXiv2014

Commuting Toeplitz Operators on Bounded Symmetric Domains and Multiplicity-Free Restrictions of Holomorphic Discrete Series

For any given bounded symmetric domain, we prove the existence of commutative $C^*$-algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known multiplicity-free results for restrictions to certain subgroups of the holomorphic discrete series. In the compact case we completely characterize the subgroups that define invariant symbols that yield commuting Toeplitz operators in terms of the multiplicity-free property.

math.OA↗
arXiv2012

Direct Systems of Spherical Functions and Representations

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $ϕ(x) = $ where $e$ is a $K_\infty$--fixed unit vector for $π$. Specifically, we look at representations $π_\infty = \varinjlim π_n$ of $G_\infty$ where $π_n$ is $K_n$--spherical, so the spherical representations $π_n$ and the corresponding spherical functions $ϕ_n$ are related by $ϕ_n(x) = $ where $e_n$ is a $K_n$--fixed unit vector for $π_n$, and we consider the possibility of constructing a $K_\infty$--spherical function $ϕ_\infty = \lim ϕ_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_\infty$--fixed vector $e$, and (ii) $G_\infty/K_\infty$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$--planes in $\F^\infty$ where $p < \infty$ and $\F$ is $\R$, $\C$ or $\H$). In that finite rank case we also prove the functional equation $ϕ(x)ϕ(y) = \lim_{n\to \infty} \int_{K_n}ϕ(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.

math.RT↗