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WELL-POSEDNESS AND INVERSE PROBLEMS FOR SEMILINEAR NONLOCAL WAVE EQUATIONS

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arXiv 2024年

作者： Lin, Yi-Hsuan Tyni, Teemu Zimmermann, Philipp Department of Applied Mathematics National Yang Ming Chiao Tung University Hsinchu Taiwan Unit of Applied and Computational Mathematics University of Oulu Finland Department of Mathematics ETH Zurich Zürich Switzerland Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f (x, u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n ∈ .MSC Codes Primary 35R30, secondary 26A33, 42B37 Copyright © 2024, The Authors. All rights reserved.

关键词： Wave equations

Packet Scheduling in the Underwater Network Using Active Priority and QLS-Based Energy-Efficient Backpressure Technique 6th

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Packet Scheduling in the Underwater Network Using Active Pri...

6th International Conference on Innovative Computing and Communication, ICICC 2023

作者： Caroline Mary, A. Senthil Kumar, A.V. Saleh, Omar S. Research Scholar Department of Computer Science Hindusthan College of Arts and Science Coimbatore India Assistant Professor Department of Applied Mathematics and Computational Sciences PSGCollege of Technology Coimbatore India Professor and Director Department of MCA Hindusthan College of Arts and Science Coimbatore India Ministry of Higher Education and Scientific Research Baghdad Iraq

ISBN: (纸本)9789819940707

Underwater communications are playing a major role in recent times as underwater happenings have gained interest. Especially in scenarios based on natural calamities like tsunamis, pollution detection, and surveillance, underwater communications are indispensable. Parameters like battery life span and capabilities of sensor nodes play a vital role in the underwater environment. QoS has to be taken care of in underwater communication. Congestion avoidance helps in using energy efficiently in underwater communication. Simulation results prove that this algorithm overcomes the drawback of a normal backpressure-based, static priority algorithm where packet loss, delay, the energy of nodes, and routing overhead are reduced overall, and packet delivery ratio and throughput are increased in special cases of real-time important packets. As underwater nodes are battery-operated, it is essential to have energy-efficient scheduling algorithm. The proposed algorithm solves the drawback of the backpressure algorithm by having energy-efficient queue length stabilizer (QLS) technique, where the backlog is adjusted such that no queues are affected and the energy consumption is less. And real-time packets are treated effectively such that real-time applications can be supported using this scheduling algorithm. The novelty of the proposed method is congestion control methodology with active priority and energy efficiency for an underwater network for effective scheduling of packets where queue backlog size is also adjusted virtually based on energy. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2024.

关键词： Scheduling algorithms

Relaxations and Duality for Multiobjective Integer programming

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arXiv 2023年

作者： Dunbar, Alex Sinha, Saumya Schaefer, Andrew J. Department of Mathematics Emory University AtlantaGA United States Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States

Multiobjective integer programs (MOIPs) simultaneously optimize multiple objective functions over a set of linear constraints and integer variables. In this paper, we present continuous, convex hull and Lagrangian relaxations for MOIPs and examine the relationship among them. The convex hull relaxation is tight at supported solutions, i.e., those that can be derived via a weighted-sum scalarization of the MOIP. At unsupported solutions, the convex hull relaxation is not tight and a Lagrangian relaxation may provide a tighter bound. Using the Lagrangian relaxation, we define a Lagrangian dual of an MOIP that satisfies weak duality and is strong at supported solutions under certain conditions on the primal feasible region. We include a numerical experiment to illustrate that bound sets obtained via Lagrangian duality may yield tighter bounds than those from a convex hull relaxation. Subsequently, we generalize the integer programming value function to MOIPs and use its properties to motivate a set-valued superadditive dual that is strong at supported solutions. We also define a simpler vector-valued superadditive dual that exhibits weak duality but is strongly dual if and only if the primal has a unique nondominated *** Codes 90C100, 90C29, 90C46 © 2023, CC BY-NC-ND.

关键词： Integer programming

Multi-Track Music Generation Network Based on a Hybrid Learning Module

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Multi-Track Music Generation Network Based on a Hybrid Learn...

2023 IEEE International Conference on Multimedia and Expo Workshops, ICMEW 2023

作者： Zhang, Donghui Hu, Ziqing Li, Xiaobing Tie, Yun Qi, Lin School of Electrical and Information Engineering Zhengzhou University Zhengzhou China University of Notre Dame Department of Applied and Computational Mathematics and Statistics South Bend on Lake MI United States Central Conservatory of Music Department of Music Artificial Intelligence Beijing China

ISBN: (纸本)9798350313154

Generative models has been widely used for symbolic music generation. However, the quality of the music produced is hindered by the inadequate modeling of the harmonic and rhythmic relationships among various instruments, as well as the neglect of musical principles. To address the aforementioned challenges, we propose an attention-based hybrid learning multitrack music generation(HL-MTMG) model that incorporates regularization of musical rules and simultaneous exploitation between intra-track and inter-track information, resulting in the generation of high-quality symbolic music that conforms to established musical principles. Through extensive examinations of musical structures, specifically rhythm, and assessments by domain experts, our study demonstrates that HL-MTMG exhibits significant improvements in musical integrity and fluency in comparison to symbolic music generation models, such as, MuseGAN and HRNN. © 2023 IEEE.

关键词： Music

Review of the Mean Field Models for Predicting the Spread of Viral Infections

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Review of the Mean Field Models for Predicting the Spread of...

Cognitive Sciences, Genomics and Bioinformatics (CSGB)

作者： Viktoriya Petrakova Olga Krivorotko Andrei Neverov Department of Computational Mathematics Institute of Computational Modeling SB RAS Krasnoyarsk Russia Department of Applied Inverse Problems Sobolev Institute of Mathematics SB RAS Novosibirsk Russia International Mathematical Center Sobolev Institute of Mathematics SB RAS Novosibirsk Russia

Models for the dynamics and control of disease spread are now developing rapidly and play a central role in making decisions about the introduction of various antiviral measures. One of the approaches to solving such problems may be the use of mean field models, which, on the one hand, allow to take into account some stochastic processes occurring in the population and, on the other hand, are expressed in the form of systems with a small number of equations, the solution of which does not require large computing power. This work is devoted to a review of mean field models now used to solve epidemiological problems, as well as a comparison of four specific approaches that can be formulated within the framework of mean field theory.

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ATTENTION IS A SMOOTHED CUBIC SPLINE

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arXiv 2024年

作者： Lai, Zehua Lim, Lek-Heng Liu, Yucong Department of Mathematics University of Texas AustinTX78712 United States Computational and Applied Mathematics Initiative Department of Statistics University of Chicago ChicagoIL60637 United States School of Mathematics Georgia Institute of Technology AtlantaGA30332 United States

We highlight a perhaps important but hitherto unobserved insight: The attention module in a transformer is a smoothed cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old notion deeply entrenched in classical approximation theory. More precisely, we show that with ReLU-activation, attention, masked attention, encoder–decoder attention are all cubic splines. As every component in a transformer is constructed out of compositions of various attention modules (= cubic splines) and feed forward neural networks (= linear splines), all its components — encoder, decoder, and encoder–decoder blocks;multilayered encoders and decoders;the transformer itself — are cubic or higher-order splines. If we assume the Pierce–Birkhoff conjecture, then the converse also holds, i.e., every spline is a ReLU-activated encoder. Since a spline is generally just C2, one way to obtain a smoothed C∞-version is by replacing ReLU with a smooth activation;and if this activation is chosen to be SoftMax, we recover the original transformer as proposed by Vaswani et al. This insight sheds light on the nature of the transformer by casting it entirely in terms of splines, one of the best known and thoroughly understood objects in applied *** Codes 26B40, 41A15, 65D07, 68T01, 14P10, 13J30 © 2024, CC BY.

关键词： Approximation theory

ERROR ANALYSIS OF A BACKWARD EULER POSITIVE PRESERVING STABILIZED SCHEME FOR A CHEMOTAXIS SYSTEM

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arXiv 2022年

作者： Chatzipantelidis, P. Pervolianakis, C. Department of Mathematics and Applied Mathematics University of Crete Heraklion71003 Greece Institute of Applied and Computational Mathematics FORTH Heraklion70013 Greece

For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in L2 for the cell density and H1 for the chemical concentration. We also present numerical experiments to illustrate the theoretical *** Codes 65M60, 65M15 © 2022, CC BY-NC-ND.

关键词： Finite element method

Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

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arXiv 2025年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Department of Computational Applied Mathematics & Operations Research Rice University United States Department of Mathematics The University of Arizona United States

We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as 104 compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 Copyright © 2025, The Authors. All rights reserved.

关键词： Stochastic systems

BOTTOM-UP TRANSIENT TIME MODELS IN COARSE-GRAINING MOLECULAR SYSTEMS

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arXiv 2023年

作者： Baxevani, Georgia Harmandaris, Vagelis Kalligiannaki, Evangelia Tsantili, Ivi Department of Mathematics and Applied Mathematics University of Crete Heraklion Greece Computation-Based Science and Technology Research Center The Cyprus Institute Nicosia2121 Cyprus Department of Statistics and Actuarial-Financial Mathematics University of the Aegean Greece Institute of Applied and Computational Mathematics FORTH Heraklion Greece

MSC Codes 62M20, 82C31, 82M37, 82B30This work presents a systematic methodology for describing the transient dynamics of coarse-grained molecular systems inferred from all-atom simulated data. We suggest Langevin-type dynamics where the coarse-grained interaction potential depends explicitly on time to efficiently approximate the transient coarse-grained dynamics. We apply the path-space force matching approach at the transient dynamics regime to learn the proposed model parameters. In particular, we parameterize the coarse-grained potential both with respect to the pair distance of the coarse-grained particles and the time, and we obtain an evolution model that is explicitly time-dependent. Moreover, we follow a data-driven approach to estimate the friction kernel, given by appropriate correlation functions directly from the underlying all-atom molecular dynamics simulations. To explore and validate the proposed methodology we study a benchmark system of a moving particle in a box. We examine the suggested model’s effectiveness in terms of the system’s correlation time and find that the model can approximate well the transient time regime of the system, depending on the correlation time of the system. As a result, in the less correlated case, it can represent the dynamics for a longer time interval. We present an extensive study of our approach to a realistic high-dimensional water molecular system. Posing the water system initially out of thermal equilibrium we collect trajectories of all-atom data for the, empirically estimated, transient time regime. Then, we infer the suggested model and strengthen the model’s validity by comparing it with simplified Markovian models. Copyright © 2023, The Authors. All rights reserved.

关键词： Molecular dynamics