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Hardware Acceleration for HPS Algorithms in Two and Three Dimensions

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arXiv 2025年

作者： Melia, Owen Fortunato, Daniel Hoskins, Jeremy Willett, Rebecca Department of Computer Science University of Chicago United States Center for Computational Mathematics Flatiron Institute United States Center for Computational Biology Flatiron Institute United States Data Science Institute University of Chicago United States Computational and Applied Mathematics Department of Statistics University of Chicago United States

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU;in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [2019] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems. © 2025, CC BY.

关键词： Graphics processing unit

The Fokas method for evolution partial differential equations

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Partial Differential Equations in applied mathematics 2025年 14卷

作者： Chatziafratis, A. Fokas, A.S. Kalimeris, K. Department of Mathematics and Statistics School of Pure and Applied Sciences University of Cyprus Cyprus Institute of Applied and Computational Mathematics Foundation for Research and Technology Crete Greece Department of Applied Mathematics and Theoretical Physics University of Cambridge United Kingdom Mathematics Research Center Academy of Athens Greece

In the late 1990s a novel methodology was introduced for solving boundary value problems for linear and integrable nonlinear PDEs. This new approach is known as the Unified Transform or the Fokas method. Here we discuss important developments regarding the implementation of this methodology to evolution PDEs. In particular, we analyse linear PDEs, including the heat equation, the Sobolev–Barenblatt pseudo-parabolic model, the Rubinshtein–Aifantis double-diffusion system, and the linearized Cahn–Hilliard model, as well as certain integrable nonlinear PDEs, like the nonlinear Schrödinger equation. Regarding the latter equation, emphasis is placed on the so-called linearizable boundary conditions, which remarkably, include the well-studied x-periodic initial value problem. © 2025 The Authors

关键词： Evolution PDEs Fokas method Integrable PDEs Unified transform

Advancements on multi-fidelity random Fourier neural networks: application to hurricane modeling for offshore wind energy

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Advancements on multi-fidelity random Fourier neural network...

AIAA Science and Technology Forum and Exposition, AIAA SciTech Forum 2025

作者： Davis, Owen Geraci, Gianluca Wentz, Jacqueline M. King, Ryan N. Cortiella, Alexandre Rybchuk, Alex Gomez, Miguel Sanchez Deskos, Georgios Motamed, Mohammad Optimization and Uncertainty Quantification Sandia National Laboratories AlbuquerqueNM United States Computational Data Science Sandia National Laboratories LivermoreCA United States National Wind Technology Center National Renewable Energy Laboratory GoldenCO United States Department of Applied Mathematics and Statistics University of New Mexico AlbuquerqueNM United States

ISBN: (数字)9781624107238

ISBN: (纸本)9781624107238

Multi-fidelity approaches are emerging as effective strategies in computational science to handle otherwise intractable tasks like Uncertainty Quantification (UQ), training of Machine Learning (ML) models, and optimization, for expensive high-fidelity applications in which the amount of available simulations or data is limited. The main idea is simple: large datasets generated for low-fidelity approximations of the problem at hand are fused with a much sparser dataset for the target (high-fidelity) system. In this paper, we build on our recent success in designing random Fourier Neural Networks (rFNNs) [1] to target problems arising in wind energy applications and in particular problems of interest for hurricane modeling. In this context, data for the high-fidelity models are limited and lower fidelity alternatives are needed. In this work, we introduce a novel multi-fidelity training approach for our rFNNs and demonstrate its use on a simple verification problem and on a hurricane modeling problem in which high-fidelity data are generated via Large-Eddy Simulations (LES), while low-fidelity data are given by a mesoscale model. Initial results demonstrate how the multi-fidelity training approach can improve the quality of the resulting surrogate. © 2025, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.

关键词： Large eddy simulation

Asymptotic normality of the Conditional Value-at-Risk based Pickands estimator

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statistics and Probability Letters 2025年 223卷

作者： Li, Yizhou Polak, Paweł Department of Applied Mathematics and Statistics Stony Brook University United States Institute for Advanced Computational Science Stony Brook University United States

We show weak convergence of the empirical Conditional Value-at-Risk (CVaR) in functional space and the asymptotic normality of the CVaR-based Pickands estimator from Chen (2021). These results demonstrate that the CVaR-based estimator has significantly lower asymptotic variance than analogous VaR-based constructions. © 2025 Elsevier B.V.

关键词： Asymptotic normality Conditional Value-at-Risk Extreme value index Pickands estimator Second-order regular variation

Towards Fair and Robust Face Parsing for Generative AI: A Multi-Objective Approach

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arXiv 2025年

作者： Abraham, Sophia J. Hauenstein, Jonathan D. Scheirer, Walter J. Department of Computer Science and Engineering University of Notre Dame Notre Dame IN46556 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame IN46556 United States

Face parsing is a fundamental task in computer vision, enabling applications such as identity verification, facial editing, and controllable image synthesis. However, existing face parsing models often lack fairness and robustness, leading to biased segmentation across demographic groups and errors under occlusions, noise, and domain shifts. These limitations affect downstream face synthesis, where segmentation biases can degrade generative model outputs. We propose a multi-objective learning framework that optimizes accuracy, fairness, and robustness in face parsing. Our approach introduces a homotopy-based loss function that dynamically adjusts the importance of these objectives during training. To evaluate its impact, we compare multi-objective and single-objective U-Net models in a GAN-based face synthesis pipeline (Pix2PixHD). Our results show that fairness-aware and robust segmentation improves photorealism and consistency in face generation. Additionally, we conduct preliminary experiments using ControlNet, a structured conditioning model for diffusion-based synthesis, to explore how segmentation quality influences guided image generation. Our findings demonstrate that multi-objective face parsing improves demographic consistency and robustness, leading to higher-quality GAN-based synthesis.1 © 2025, CC BY-NC-ND.

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EUCLIDEAN DISTANCE DEGREE IN MANIFOLD OPTIMIZATION

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arXiv 2025年

作者： Lai, Zehua Lim, Lek-Heng Ye, Ke Department of Mathematics University of Texas AustinTX78712 United States Computational and Applied Mathematics Initiative Department of Statistics University of Chicago ChicagoIL60637-1514 United States KLMM Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we will also determine the Euclidean distance degree of an important class of Schubert varieties that often appear in applications. Our technique goes further than furnishing the value of the Euclidean distance degree;it will also yield closed-form expressions for all stationary points of the Euclidean distance function in each instance. We will discuss the implications of these results on the tractability of manifold optimization *** Codes 14M15, 15A24, 58C05, 58K05, 90C26 © 2025, CC BY.

关键词： Optimization

Internal layer solutions and coefficient recovery in time-periodic reaction-diffusion-advection equations

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arXiv 2025年

作者： Chaikovskii, Dmitrii Zhang, Ye Liubavin, Aleksei MSU-BIT-SMBU Joint Research Center of Applied Mathematics Shenzhen MSU-BIT University Shenzhen518172 China School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China Department of Computational Mathematics and Cybernetics MSU-BIT-SMBU Joint Research Center of Applied Mathematics Shenzhen MSU-BIT University Shenzhen518172 China

This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary conditions. A periodic asymptotic solution featuring an inner transition layer is proposed, advancing the mathematical modeling of reaction-diffusion-advection dynamics. Building on this asymptotic analysis, we develop a simple yet effective numerical algorithm to address ill-posed nonlinear inverse problems aimed at reconstructing coefficient functions that depend solely on spatial or temporal variables. Conditions ensuring the existence and uniqueness of solutions for both forward and inverse problems are established. The proposed method’s effectiveness is validated through numerical experiments, demonstrating high accuracy in reconstructing coefficient functions under varying noise conditions. © 2025, CC BY.

关键词： Advection

SPECIAL ORTHOGONAL, SPECIAL UNITARY, AND SYMPLECTIC GROUPS AS PRODUCTS OF GRASSMANNIANS

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arXiv 2025年

作者： Lim, Lek-Heng Lu, Xiang Ye, Ke Computational and Applied Mathematics Initiative University of Chicago ChicagoIL60637 United States Department of Statistics University of Chicago ChicagoIL60637 United States KLMM Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We describe a curious structure of the special orthogonal, special unitary, and symplectic groups that has not been observed, namely, they can be expressed as matrix products of their corresponding Grassmannians realized as involution matrices. We will show that SO(n) is a product of two real Grassmannians, SU(n) a product of four complex Grassmannians, and Sp(2n, R) or Sp(2n, C) a product of four symplectic Grassmannians over or ℂ *** Codes 14M15, 22E10, 22E15, 51A50 © 2025, CC BY.

关键词： Matrix algebra

No Imputation Without Representation 35th

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No Imputation Without Representation

35th Benelux Conference on Artificial Intelligence and Machine Learning, BNAIC/Benelearn 2023

作者： Lenz, Oliver Urs Peralta, Daniel Cornelis, Chris Leiden Institute of Advanced Computer Science Leiden University Leiden Netherlands Research Group for Computational Web Intelligence Department of Applied Mathematics Computer Science and Statistics Ghent University Ghent Belgium Department of Information Technology Ghent University Ghent Belgium

ISBN: (纸本)9783031746499

By filling in missing values in datasets, imputation allows these datasets to be used with algorithms that cannot handle missing values by themselves. However, missing values may in principle contribute useful information that is lost through imputation. The missing-indicator approach can be used in combination with imputation to instead represent this information as a part of the dataset. There are several theoretical considerations why missing-indicators may or may not be beneficial, but there has not been any large-scale practical experiment on real-life datasets to test this question for machine learning predictions. We perform this experiment for three imputation strategies and a range of different classification algorithms, on the basis of twenty real-life datasets. In a follow-up experiment, we determine attribute-specific missingness thresholds for each classifier above which missing-indicators are more likely than not to increase classification performance. And in a second follow-up experiment, we evaluate numerical imputation of one-hot encoded categorical attributes. We reach the following conclusions. Firstly, missing-indicators generally increase classification performance. Secondly, with missing-indicators, nearest neighbour and iterative imputation do not lead to better performance than simple mean/mode imputation. Thirdly, for decision trees, pruning is necessary to prevent overfitting. Fourthly, the thresholds above which missing-indicators are more likely than not to improve performance are lower for categorical attributes than for numerical attributes. Lastly, mean imputation of numerical attributes preserves some of the information from missing values. Consequently, when not using missing-indicators it can be advantageous to apply mean imputation to one-hot encoded categorical attributes instead of mode imputation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： Adversarial machine learning