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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

Sharp concentration of simple random tensors

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arXiv 2025年

作者： Al-Ghattas, Omar Chen, Jiaheng Sanz-Alonso, Daniel Department of Statistics University of Chicago IL60637 United States Committee on Computational and Applied Mathematics University of Chicago IL60637 United States

This paper establishes sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings. © 2025, CC BY.

关键词： Tensors

Turn-off/turn-on biosensing of tetracycline and ciprofloxacin antibiotics using fluorescent iron oxide quantum dots

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Analytical Methods 2024年第8期16卷 1261-1271页

作者： Sudewi, Sri Li, Chien-Hung Chabib, Lutfi Rasool, Akhtar Arputharaj, Emmanuvel Zulfajri, Muhammad Huang, Genin Gary Department of Medicinal and Applied Chemistry Kaohsiung Medical University Kaohsiung80708 Taiwan Department of Pharmacy Faculty of Mathematics and Natural Science Universitas Sam Ratulangi Manado95115 Indonesia Pharmacy Study Program Faculty of Mathematics and Science Universitas Islam Indonesia Yogyakarta55584 Indonesia KST BJ Habibie Setu Tangerang Selatan15314 Indonesia Department of Medical Laboratory Science and Biotechnology Kaohsiung Medical University Kaohsiung80708 Taiwan Department of Chemistry Education Universitas Serambi Mekkah Banda Aceh23245 Indonesia Department of Medical Research Kaohsiung Medical University Hospital Kaohsiung80708 Taiwan Department of Chemistry National Sun Yat-sen University Kaohsiung80424 Taiwan

A fluorescence probe based on iron oxide quantum dots (IO-QDs) was synthesized using the hydrothermal method for the determination of tetracycline (TCy) and ciprofloxacin (CPx) in aqueous solution. The IO-QDs were characterized using high-resolution transmission electron microscopy (HR-TEM), powder X-ray diffraction (P-XRD), vibrating sample magnetometry (VSM), and Fourier-transform infrared spectroscopy (FTIR). The as-prepared IO-QDs are fluorescent, stable, and with a fluorescence quantum yield (QY) of 9.8 ± 0.12%. The fluorescence of IO-QDs was observed to be quenched and enhanced in the presence of TCy and CPx, respectively. The fluorescence intensity ratio shows linearity at concentrations from 1-100 μM and 5-100 μM for TCy and CPx, respectively;the detection limit for TCy and CPx was estimated to be 0.71 μM and 1.56 μM, respectively. The proposed method was also successfully utilized in the spiked samples of drinking water and honey with good recoveries. The method offered convenience, rapid detection, high sensitivity, selectivity, and cost-efficient alternative options for the determination of TCy and CPx in real samples. © 2024 The Royal Society of Chemistry.

关键词： Iron oxides

Multiresolution Approximation of a Bayesian Inverse Problem using Second-Generation Wavelets

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arXiv 2023年

作者： Shervani-Tabar, Navid Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. It is common practice to decompose the unknown field into some basis and infer the decomposition parameters instead of directly inferring the unknown. Given the multiscale nature of permeability fields, wavelets are a natural choice for parameterizing them. This study uses a Bayesian approach to incorporate the statistical sparsity that characterizes discrete wavelet coefficients. First, we impose a prior distribution incorporating the hierarchical structure of the wavelet coefficient and smoothness of reconstruction via scale-dependent hyperparameters. Then, Sequential Monte Carlo (SMC) method adaptively explores the posterior density on different scales, followed by model selection based on Bayes Factors. Finally, the permeability field is reconstructed from the coefficients using a multiresolution approach based on second-generation wavelets. Here, observations from the pressure sensor grid network are computed via Multilevel Adaptive Wavelet Collocation Method (AWCM). Results highlight the importance of prior modeling on parameter estimation in the inverse problem. Copyright © 2023, The Authors. All rights reserved.

关键词： Inverse problems

Solving Fractional Non-Linear Differential Equations by Laplace Homotopy Analysis Method

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Solving Fractional Non-Linear Differential Equations by Lapl...

2021 International Conference on Advancements in Electrical, Electronics, Communication, Computing and Automation, ICAECA 2021

作者： Malathy, R. Saratha, S.R. Krishnan, G. Sai Sundara SNS College of Engineering Department of Mathematics Coimbatore India Kumaraguru College of Technology Department of Mathematics Coimbatore India Applied Mathematics and Computational Sciences PSG College of Technology Coimbatore India

ISBN: (纸本)9781665428293

This study addresses solving fractional non-linear differential equations by the LHAM (Laplace Homotopy analysis method). We present a methodology for designing Modified Riemann-Liouville Integral to the fractional non-linear differential equation. Based on the union of the homotopy analysis approach and the suggested integral transform is used to find the convergence of the Inviscid Burger's equation. The obtained criteria can be verified by numerically using Mathematica software. Finally, some simulation studies are stated to demonstrated the usefulness of the derived results. © 2021 IEEE.

关键词： Laplace transforms