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Double diffusion structure of logarithmically damped wave equations with a small parameter

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arXiv 2021年

作者： Piske, Alessandra Charão, Ruy Coimbra Ikehata, Ryo Graduate Program in Pure and Applied Mathematics Department of Mathematics Federal University of Santa Catarina Florianopolis88040-270 Brazil Department of Mathematics Division of Educational Sciences Graduate School of Humanities and Social Sciences Hiroshima University Higashi-Hiroshima739-8524 Japan

We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter 0 1/2. We study the Cauchy problem for this model in Rnto the case θ ∈ (0, 1/2), and we obtain an asymptotic profile and optimal estimates in time of solutions as t → ∞ in L2-sense. An important discovery in this research is that in the case when n = 1, we can present a threshold θ∗= 1/4 of the parameter θ ∈ (0, 1/2) such that the solution of the Cauchy problem decays with some optimal rate for θ ∈ (0,θ∗) as t → ∞, while the L2-norm of the corresponding solution never decays for θ ∈ [θ∗, 1/2), and in particular, in the case θ ∈ [θ∗, 1/2) it shows an infinite time L2-blow up of the corresponding solutions. The former (i.e., θ ∈ (0, θ∗) case) indicates an usual diffusion phenomenon, while the latter (i.e., θ ∈ [θ∗, 1/2) case) implies, so to speak, a singular diffusion phenomenon. Such a singular diffusion in the one dimensional case is a quite novel phenomenon discovered through our new model produced by logarithmic damping with a small parameter θ. It might be already prepared in the usual structural damping case such as (-Δ)θutwith θ ∈ (0, 1/2), however unfortunately nobody has ever just pointed out even in the structural damping *** Codes 35B40, 35L05, 35B20, 35R12, 35S05 © 2021, CC BY.

关键词： Damping

Spatio-temporal characterization of nonlinear forcing and response in turbulent channel flow

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arXiv 2025年

作者： Huang, Yuting Toedtli, Simon S. Chini, Gregory P. McKeon, Beverley J. Mechanical and Civil Engineering California Institute of Technology PasadenaCA91125 United States National Center for Atmospheric Research BoulderCO80301 United States Program in Integrated Applied Mathematics Department of Mechanical Engineering University of New Hampshire DurhamNH03824 United States Department of Mechanical Engineering Center for Turbulence Research Stanford University StanfordCA94305 United States

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a nonlinear forcing to the linear resolvent operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. A framework to quantify the triadic contributions to the forcing and response by each pair of triplets is developed and applied to data from direct numerical simulations of a turbulent channel at Reι ≈ 550. The linear resolvent operator is incorporated to provide the missing link from energy transfer between modes to the effect on the spectral turbulent kinetic energy. The coefficients highlight the importance of interactions involving large-scale structures, providing a natural connection to the modeling assumptions in quasi-linear (QL) and generalized quasi-linear (GQL) analyses. Specifically, it is revealed that the QL and GQL reductions efficiently capture important triadic interactions in the flow, especially when including of a small number of wavenumbers into the GQL large-scale base flow. Additionally, spatio-temporal analyses of the triadic contributions to a single mode representative of the near-wall cycle demonstrate the spatio-temporal nature of the triadic interactions and the effect of the resolvent operator, which selectively amplifies certain forcing profiles. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures. © 2025, CC BY.

关键词： Navier Stokes equations

Adaptive tikhonov strategies for stochastic ensemble Kalman inversion

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arXiv 2021年

作者： Weissmann, Simon Chada, Neil K. Schillings, Claudia Tong, Xin T. Interdisciplinary Center for Scientific Computing University of Heidelberg Heidelberg69120 Germany Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal KSA 23955 Saudi Arabia Mannheim School of Computer Science and Mathematics University of Mannheim Mannheim68131 Germany Department of Mathematics National University of Singapore Singapore119077 Singapore

Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is to consider the introduction of adaptive Tikhonov strategies for EKI. This work builds upon Tikhonov EKI (TEKI) which was proposed for a fixed regularization constant. By adaptively learning the regularization parameter, this procedure is known to improve the recovery of the underlying unknown. For the analysis, we consider a continuous-time setting where we extend known results such as well-posdeness and convergence of various loss functions, but with the addition of noisy observations. Furthermore, we allow a time-varying noise and regularization covariance in our presented convergence result which mimic adaptive regularization schemes. In turn we present three adaptive regularization schemes, which are highlighted from both the deterministic and Bayesian approaches for inverse problems, which include bilevel optimization, the MAP formulation and covariance learning. We numerically test these schemes and the theory on linear and nonlinear partial differential equations, where they outperform the non-adaptive TEKI and *** Codes 65M32, 60G35, 65C35, 70F17 Copyright © 2021, The Authors. All rights reserved.

关键词： Continuous time systems

Theoretical Prediction of the Effective Dynamic Dielectric Constant of Disordered Hyperuniform Anisotropic Composites Beyond the Long-Wavelength Regime

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arXiv 2023年

作者： Kim, Jaeuk Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Torquato and Kim [Phys. Rev. X 11, 296 021002 (2021)] derived exact nonlocal strong-contrast expansions of the effective dynamic dielectric constant tensor Εe(kq, ω) that treat general three-dimensional (3D) two-phase composite microstructures, which are valid well beyond the long-wavelength regime. Here, we demonstrate that truncating this general rapidly converging expansion at the two- and three-point levels is a powerful theoretical tool from which one can extract accurate approximations suited for various microstructural symmetries. Among other results, we show that such truncations yield closed-form formulas applicable to transverse polarization in layered media and transverse magnetic polarization in transversely isotropic media, respectively. We apply these formulas to estimate Εe(kq, ω) for models of 3D disordered hyperuniform layered and transversely isotropic media: nonstealthy hyperuniform media and stealthy hyperuniform media. In particular, we show that stealthy hyperuniform layered and transversely isotropic media are perfectly transparent (trivially implying no Anderson localization, in principle) within finite wave number intervals through the third-order terms. For all models considered here, we validate that the second-order formulas, which depend on the spectral density, are already very accurate well beyond the long-wavelength regime by showing very good agreement with the finite-difference time-domain (FDTD) simulations. The high predictive power of the second-order formula is due to the fact that higher-order contributions are negligibly small, implying that it very accurately approximates multiple scattering through all orders. This implies that there can be no Anderson localization within the predicted perfect transparency interval in stealthy hyperuniform layered and transversely isotropic media in practice because the localization length (associated with only possibly negligibly small higher-order contributions) should be very large compar

关键词： Spectral density

2D Basement Relief Inversion using Sparse Regularization.

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arXiv 2024年

作者： Barboza, Francisco Márcio da Cunha Romão Silva, E. Arthur Anthony de Carvalho, Bruno Motta Department of Computing and Technology Federal University of Rio Grande do Norte Rio Grande do Norte Caicó Brazil Graduate Program on Systems and Computing Federal University of Rio Grande do Norte Rio Grande do Norte Natal Brazil Department of Informatics and Applied Mathematics Federal University of Rio Grande do Norte Rio Grande do Norte Natal Brazil

Basement relief gravimetry is a key application in geophysics, particularly important for oil exploration and mineral prospecting. It involves solving an inverse geophysical problem, where the parameters of a geological model are inferred from observed data. In this context, the geological model consists of the depths of constant-density prisms representing the basement relief, and the data correspond to the gravitational anomalies caused by these prisms. Inverse geophysical problems are typically ill-posed, as defined by Hadamard, meaning that small perturbations in the data can result in large variations in the solutions. To address this instability, regularization techniques, such as those proposed by Tikhonov, are employed to stabilize the solutions. This study presents a comparative analysis of various regularization techniques applied to the gravimetric inversion problem, including Smoothness Constraints, Total Variation, the Discrete Cosine Transform (DCT), and the Discrete Wavelet Transform (DWT) using Daubechies D4 wavelets. Optimization methods are commonly used in inverse geophysical problems because of their ability to find optimal parameters that minimize the objective function—in this case, the depths of the prisms that best explain the observed gravitational anomalies. The Genetic Algorithm (GA) was selected as the optimization technique. GA is based on Darwinian evolutionary theory, specifically the principle of natural selection, where the fittest individuals in a population are selected to pass on their traits. In optimization, this translates to selecting solutions that most effectively minimize the objective function. The results, evaluated using fit metrics and cumulative error analysis, demonstrate the effectiveness of all the regularization techniques and the Genetic Algorithm. Among the methods tested, the Smoothness constraint was briefly the most effective for the first and second synthetic models. For the third model, which was based on re

关键词： Discrete wavelet transforms

Generalization and memorization: The Bias Potential Model

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arXiv 2020年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University Department of Mathematics Princeton University

Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or *** Codes 68T07, 60-08. Copyright © 2020, The Authors. All rights reserved.

关键词： Probability distributions

Integrating machine learning with physics-based modeling

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arXiv 2020年

作者： Weinan, E. Han, Jiequn Zhang, Linfeng Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University

Machine learning is poised as a very powerful tool that can drastically improve our ability to carry out scientific research. However, many issues need to be addressed before this becomes a reality. This article focuses on one particular issue of broad interest: How can we integrate machine learning with physics-based modeling to develop new interpretable and truly reliable physical models? After introducing the general guidelines, we discuss the two most important issues for developing machine learning-based physical models: Imposing physical constraints and obtaining optimal datasets. We also provide a simple and intuitive explanation for the fundamental reasons behind the success of modern machine learning, as well as an introduction to the concurrent machine learning framework needed for integrating machine learning with physics-based modeling. Molecular dynamics and moment closure of kinetic equations are used as examples to illustrate the main issues discussed. We end with a general discussion on where this integration will lead us to, and where the new frontier will be after machine learning is successfully integrated into scientific modeling. Copyright © 2020, The Authors. All rights reserved.

关键词： Molecular dynamics

GLOBAL WELL-POSEDNESS FOR THE THERMODYNAMICALLY REFINED PASSIVELY TRANSPORTED NONLINEAR MOISTURE DYNAMICS WITH PHASE CHANGES

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arXiv 2022年

作者： Hittmeir, Sabine Klein, Rupert Li, Jinkai Titi, Edriss S. Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 Wien1090 Austria FB Mathematik & Informatik Freie Universität Berlin Arnimallee 6 Berlin14195 Germany South China Research Center for Applied Mathematics and Interdisciplinary Studies School of Mathematical Sciences South China Normal University Zhong Shan Avenue West 55 Guangzhou510631 China Department of Mathematics Texas A&M University 3368 TAMU College StationTX77843-3368 United States Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot76100 Israel

In this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies [15] take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential e.g. in the case of deep convective cloud columns in [14]. The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of *** Codes 35A01, 35B45, 35D35, 35M86, 35Q30, 35Q35, 35Q86, 76D03, 76D09, 86A10 © 2022, CC BY.

关键词： Phase transitions

Effective electromagnetic wave properties of disordered stealthy hyperuniform layered media beyond the quasistatic regime

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arXiv 2023年

Disordered stealthy hyperuniform dielectric composites exhibit novel electromagnetic wave transport properties in two and three dimensions. Here, we carry out the first study of the electromagnetic properties of one-dimensional (1D) disordered stealthy hyperuniform layered media. From an exact nonlocal theory, we derive an approximation formula for the effective dynamic dielectric constant tensor Εe(kq, ω) of general 1D media that is valid well beyond the quasistatic regime and apply it to 1D stealthy hyperuniform systems. We consider incident waves of transverse polarization, frequency ω, and wavenumber kq. Our formula for Εe(kq, ω), which is given in terms of the spectral density, leads to a closed-form relation for the transmittance T. Our theoretical predictions are in excellent agreement with finite-difference time-domain (FDTD) simulations. Stealthy hyperuniform layered media have perfect transparency intervals up to a finite wavenumber, implying no Anderson localization, but non-stealthy hyperuniform media are not perfectly transparent. Our predictive theory provides a new path for the inverse design of the wave characteristics of disordered layered media, which are readily fabricated, by engineering their spectral densities. © 2023 Optica Publishing Group © 2023, CC BY.

关键词： Spectral density