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An Energy Stable High-Order Cut Cell Discontinuous Galerkin Method with State Redistribution for Wave Propagation

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arXiv 2024年

作者： Taylor, Christina G. Wilcox, Lucas Chan, Jesse Department of Computational Applied Mathematics and Operations Research Rice University United States Department of Applied Mathematics Naval Postgraduate School

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is L2 energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably L2 energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme’s L2 stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems. Copyright © 2024, The Authors. All rights reserved.

关键词： Wave propagation

Thermal analysis of ternary hybrid nanofluid flow in convergent/divergent channel using Box-Behnken design of response surface methodology

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Results in Physics 2025年 74.0卷

作者： Mahariq, Ibrahim Mohana, C.M. Kumar, B. Rushi Mohamed, Kezzar SARI, Mohamed Rafik El-Wahed Khalifa, Hamiden Abd Nagarathnam, Sunitha University College Korea University Seoul 02481 South Korea Najjad Zeenni Faculty of Engineering Al Quds University Jerusalem Palestine Department of Medical Research China Medical University Hospital China Medical University Taichung Taiwan Applied Science Research Center Applied Science Private University Amman Jordan GUST Engineering and Applied Innovation Research Center (GEAR) Gulf University for Science and Technology Mishref Kuwait Department of Mathematics School of Advanced Sciences Vellore Institute of Technology Tamil Nadu Vellore India Materials and Energy Engineering Laboratory (LMGE) Technology Department Faculty of Technology 20 Aout 1955 University of Skikda PO Box 26 Skikda 21000 Algeria Mechanics of Materials and Plant Maintenance Research Laboratory (LR3MI) Mechanical Engineering Department Faculty of Engineering Badji Mokhtar University of Annaba (UBMA) PO Box 12 Annaba 23052 Algeria Department of Operations and Management Research Faculty of Graduate Studies for Statistical Research Cairo University Giza 12613 Egypt Centre for Research in Computational and Applied Mechanics University of Cape Town Rondebosch 7701 South Africa

Convergent/divergent channels represent a critical advancement in thermal management systems, offering enhanced heat transfer capabilities through geometrically optimized flow paths. This study investigates the thermal and hydrodynamic characteristics of UO₂-ZnO-Fe₃O₄/EG-water ternary hybrid nanofluid in rotating convergent/divergent channels with magnetohydrodynamic and radiation effects, a critical area for enhancing heat transfer efficiency in modern thermal management systems. Employing the Runge-Kutta-Fehlberg method coupled with a Box-Behnken design approach, we analyzed how key parameters affect flow behavior and heat transfer performance. The numerical simulations reveal that increasing the nanoparticle volume fraction from 0.01 to 0.06 enhances the Nusselt number by 27.3 %, while an increase in the magnetic parameter reduces the friction coefficient by 41.2 %. The Reynolds number demonstrates a strong positive correlation with the Nusselt number (R² = 0.978). Maximum heat transfer (Nu = 33.4859) achieved at: α = 1°, Re = 40, φ = 3 %, Ha = 50, Rd = 1, Kn = 0.04, De = 0.5, Ro = 50, ε = 0, λ = 0.5 whereas a minimum flow resistance (Cf = -2.60454) occurred at: α = -3°, Re = 60, φ = 6 %, Ha = 25, Rd = 1, Kn = 0, De = 0.5, Ro = 50, ε = 0.2, λ = 0.5. For the convergent channel configuration (α = -3°), the friction coefficient was 28.3 % higher than in the divergent channel (α = 5°) under identical flow conditions. This work advances the field beyond previous studies by comprehensively analyzing the synergistic effects of rotation, magnetohydrodynamic, radiation, and channel geometry on the performance of ternary hybrid nanofluids, providing crucial insights for designing more efficient thermal systems in electronics cooling, heat exchangers, and industrial processing applications. © 2025 The Authors

关键词： Box-Behnken Design Convergent/Divergent Channels Heat Transfer Enhancement Non-linear Solar Radiation Ternary Hybrid Nanofluids Variable thermal conductivity

Riemannian Bilevel Optimization

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arXiv 2024年

作者： Li, Jiaxiang Ma, Shiqian Department of Electrical and Computer Engineering University of Minnesota Twin Cities United States Department of Computational Applied Math and Operations Research Rice University United States

In this work, we consider the bilevel optimization problem on Riemannian manifolds. We inspect the calculation of the hypergradient of such problems on general manifolds and thus enable the utilization of gradient-based algorithms to solve such problems. The calculation of the hypergradient requires utilizing the notion of Riemannian cross-derivative and we inspect the properties and the numerical calculations of Riemannian cross-derivatives. Algorithms in both deterministic and stochastic settings, named respectively RieBO and RieSBO, are proposed that include the existing Euclidean bilevel optimization algorithms as special cases. Numerical experiments on robust optimization on Riemannian manifolds are presented to show the applicability and efficiency of the proposed *** Codes 90C26 Copyright © 2024, The Authors. All rights reserved.

关键词： Numerical methods

MEAN FIELD GAMES WITH COMMON NOISE VIA MALLIAVIN CALCULUS

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arXiv 2024年

作者： Tangpi, Ludovic Wang, Shichun Princeton University Department of Operations Research and Financial Engineering and Program in Computational and Applied Mathematics United States

We present a simpler proof of the existence of equilibria for a class of mean field games with common noise, where players interact through the conditional law given the current value of the common noise rather than its entire path. By extending a compactness criterion for Malliavin-differentiable random variables to processes, we establish existence of strong equilibria, where the conditional law and optimal control are adapted to the common noise filtration and defined on the original probability space. Notably, our approach only requires measurability of the drift and cost functionals with respect to the state *** Codes 91A16, 60H07 Copyright © 2024, The Authors. All rights reserved.

关键词： Cost functions

FEDERATED LEARNING on RIEMANNIAN MANIFOLDS

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Applied Set-Valued Analysis and Optimization

applied Set-Valued Analysis and Optimization 2023年第2期5卷 213-232页

作者： Li, Jiaxiang Ma, Shiqian Department of Mathematic University of California Davis United States Computational Applied Mathematics and Operations Research Rice University Houston United States

Federated learning (FL) has found many important applications in smart-phone-APP based machine learning applications. Although many algorithms have been studied for FL, to the best of our knowledge, algorithms for FL with nonconvex constraints have not been studied. This paper studies FL over Riemannian manifolds, which finds important applications such as federated PCA and federated kPCA. We propose a Riemannian federated SVRG (RFedSVRG) method to solve federated optimization over Riemannian manifolds. We analyze its convergence rate under different scenarios. Numerical experiments are conducted to compare RFedSVRG with the Riemannian counterparts of FedAvg and FedProx. We observed from the numerical experiments that the advantages of RFedSVRG are significant. © 2023 applied Set-Valued Analysis and Optimization.

关键词： Machine learning

The knowledge gradient for sequential decision making with stochastic binary feedbacks 33

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The knowledge gradient for sequential decision making with s...

33rd International Conference on Machine Learning, ICML 2016

作者： Wang, Yingfei Wang, Chu Powell, Warren Department of Computer Science Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08544 United States

ISBN: (纸本)9781510829008

We consider the problem of sequentially making decisions that are rewarded by "successes" and "failures" which can be predicted through an unknown relationship that depends on a partially controllable vector of attributes for each instance. The learner takes an active role in selecting samples from the instance pool. The goal is to maximize the probability of success, either after the offline training phase or minimizing regret in online learning. Our problem is motivated by real-world applications where observations are time consuming and/or expensive. With the adaptation of an online Bayesian linear classifier, we develop a knowledge-gradient type policy to guide the experiment by maximizing the expected value of information of labeling each alternative, in order to reduce the number of expensive physical experiments. We provide a finite-time analysis of the estimated error and demonstrate the performance of the proposed algorithm on both synthetic problems and benchmark UCI datasets.

关键词： Benchmarking

Tuning-Free Bilevel Optimization: New Algorithms and Convergence Analysis

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arXiv 2024年

作者： Yang, Yifan Ban, Hao Huang, Minhui Ma, Shiqian Ji, Kaiyi Department of Computer Science and Engineering University at Buffalo United States AI Research for Monetization Meta United States Department of Computational Applied Math and Operations Research Rice University United States

Bilevel optimization has recently attracted considerable attention due to its abundant applications in machine learning problems. However, existing methods rely on prior knowledge of problem parameters to determine stepsizes, resulting in significant effort in tuning stepsizes when these parameters are unknown. In this paper, we propose two novel tuning-free algorithms, D-TFBO and S-TFBO. D-TFBO employs a double-loop structure with stepsizes adaptively adjusted by the "inverse of cumulative gradient norms" strategy. S-TFBO features a simpler fully single-loop structure that updates three variables simultaneously with a theory-motivated joint design of adaptive stepsizes for all variables. We provide a comprehensive convergence analysis for both algorithms and show that D-TFBO and S-TFBO respectively require (Equation presented) and (Equation presented) iterations to find an ϵ-accurate stationary point, (nearly) matching their well-tuned counterparts using the information of problem parameters. Experiments on various problems show that our methods achieve performance comparable to existing well-tuned approaches, while being more robust to the selection of initial stepsizes. To the best of our knowledge, our methods are the first to completely eliminate the need for stepsize tuning, while achieving theoretical guarantees. Copyright © 2024, The Authors. All rights reserved.

关键词： Optimization algorithms

Combinatorial Disjunctive Constraints for Obstacle Avoidance in Path Planning

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arXiv 2023年

作者： Garcia, Raul Hicks, Illya V. Huchette, Joey The Department of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Faculty of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Google Research CambridgeMA02142 United States

We present a new approach for modeling avoidance constraints in 2D environments, in which waypoints are assigned to obstacle-free polyhedral regions. Constraints of this form are often formulated as mixed-integer programming (MIP) problems employing big-M techniques—however, these are generally not the strongest formulations possible with respect to the MIP’s convex relaxation (so called ideal formulations), potentially resulting in larger computational burden. We instead model obstacle avoidance as combinatorial disjunctive constraints and leverage the independent branching scheme to construct small, ideal formulations. As our approach requires a biclique cover for an associated graph, we exploit the structure of this class of graphs to develop a fast subroutine for obtaining biclique covers in polynomial time. We also contribute an open-source Julia library named ClutteredEnvPathOpt to facilitate computational experiments of MIP formulations for obstacle avoidance. Experiments have shown our formulation is more compact and remains competitive on a number of instances compared with standard big-M techniques, for which solvers possess highly optimized procedures. Copyright © 2023, The Authors. All rights reserved.

关键词： Motion planning

Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms

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arXiv 2023年

作者： Li, Jiaxiang Balasubramanian, Krishnakumar Ma, Shiqian Department of Mathematics University of California Davis United States Department of Statistics University of California Davis United States Department of Computational Applied Math and Operations Research Rice University United States

We present Zeroth-order Riemannian Averaging Stochastic Approximation (Zo-RASA) algorithms for stochastic optimization on Riemannian manifolds. We show that Zo-RASA achieves optimal sample complexities for generating ϵ-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. We improve the algorithm’s practicality by using retractions and vector transport, instead of exponential mappings and parallel transports, thereby reducing per-iteration complexity. Additionally, we introduce a novel geometric condition, satisfied by manifolds with bounded second fundamental form, which enables new error bounds for approximating parallel transport with vector *** Codes 90C56 Copyright © 2023, The Authors. All rights reserved.

关键词： Approximation algorithms