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Performance of the multigrid method with time-stepping to solve 1D and 2D wave equations

International Journal for Computational Methods in engineering Science and Mechanics 2022年第1期23卷 45-56页

作者： Malacarne, Maicon F. Pinto, Marcio A. V. Franco, Sebastião R. Graduate Program on Numerical Methods in Engineering–PPGMNE Federal University of Parana–UFPR PR Curitiba Brazil Department of Mechanical Engineering Federal University of Parana–UFPR PR Curitiba Brazil Department of Mathematics State University of Centro-Oeste–UNICENTRO PR Irati Brazil

This work aims to discuss a proposed solution for wave equations that utilize discretization by means of the finite difference method, weighted by a parameter η, with sweeping done according to the time-stepping method. The multigrid method is employed to speed up the convergence in obtaining the solution of the system of equations resulting from the discretization. To validate the proposed model, the discretization errors, effective and apparent orders, convergence factor, orders of complexity, and the computational time were assessed. A comparison between the singlegrid and multigrid methods was carried out in order to determine the most advantageous one. © 2021 Taylor & Francis Group, LLC.

关键词： Wave equations

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Optimal treatment strategies to control acute HIV infection

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Infectious Disease Modelling 2021年第1期6卷 1202-1219页

作者： Shohel Ahmed Sumaiya Rahman Md Kamrujjaman Department of Mathematics Bangladesh University of Engineering and TechnologyDhaka1000Bangladesh Department of Statistics University of DhakaDhaka1000Bangladesh Department of Mathematics University of DhakaDhaka1000Bangladesh

Various antiretroviral therapies(ART)are administered to symptomatic human immunodeficiency virus(HIV)infected individuals to improve their *** treatment effectiveness may depend on suppressing development of drug resistance,reduce evolution of new viral strains,minimize serious side effects and the costs of *** paper deals with some results concerning optimal drug administration scheme successful in improving patients'health especially in poorly resourced *** model under consideration describes the interaction between the uninfected cells,the latently infected cells,the productively infected cells,and the free ***,in viral infection,the drug strategy aspects either the virus infectivity or reduce the virion *** mathematical model proposed here,deals with both situations with the objective function based on a combination of maximizing benefit relied on T cells count(the white cells that coordinate activities of the immune system)and minimizing the systemic *** existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal *** optimality system is derived and solved numerically using the forward and backward sweep method(FBSM).Our results indicate that early initiation of treatment makes a profound impact in both improving the quality of life and reducing the economic costs of therapy.

关键词： HIV Mathematical model Critical efficacy Optimal control Combined therapy

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Regularised Canonical Correlation Analysis: graphical lasso, biplots and beyond

arXiv

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arXiv 2024年

作者： Wells, Lennie Thurimella, Kumar Bacallado, Sergio Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom Department of Chemical Engineering and Biotechnology University of Cambridge United Kingdom

Recent developments in regularized Canonical Correlation Analysis (CCA) promise powerful methods for high-dimensional, multiview data analysis. However, justifying the structural assumptions behind many popular approaches remains a challenge, and features of realistic biological datasets pose practical difficulties that are seldom discussed. We propose a novel CCA estimator rooted in an assumption of conditional independencies and based on the Graphical Lasso. Our method has desirable theoretical guarantees and good empirical performance, demonstrated through extensive simulations and real-world biological datasets. Recognizing the difficulties of model selection in high dimensions and other practical challenges of applying CCA in real-world settings, we introduce a novel framework for evaluating and interpreting regularized CCA models in the context of Exploratory Data Analysis (EDA), which we hope will empower researchers and pave the way for wider *** Codes 62H20 (Primary) 62H12, 62P10 (Secondary) © 2024, CC BY.

关键词： Correlation methods

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Local minima of the empirical risk in high dimension: General theorems and convex examples

arXiv

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arXiv 2025年

作者： Asgari, Kiana Montanari, Andrea Saeed, Basil Department of Management Science and Engineering Stanford University United States Department of Statistics Department of Mathematics Stanford University United States Department of Electrical Engineering Stanford University United States

We consider a general model for high-dimensional empirical risk minimization whereby the data xi are d-dimensional isotropic Gaussian vectors, the model is parametrized by Θ ∈ Rd×k, and the loss depends on the data via the projection ΘTxi. This setting covers as special cases classical statistics methods (e.g. multinomial regression and other generalized linear models), but also two-layer fully connected neural networks with k hidden neurons. We use the Kac-Rice formula from Gaussian process theory to derive a bound on the expected number of local minima of this empirical risk, under the proportional asymptotics in which n, d → ∞, with n d. Via Markov’s inequality, this bound allows to determine the positions of these minimizers (with exponential deviation bounds) and hence derive sharp asymptotics on the estimation and prediction error. In this paper, we apply our characterization to convex losses, where high-dimensional asymptotics were not (in general) rigorously established for k ≥ 2. We show that our approach is tight and allows to prove previously conjectured results. In addition, we characterize the spectrum of the Hessian at the minimizer. A companion paper applies our general result to non-convex examples. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

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HRSR-SVM: Hybrid Reptile Search Remora-based Support Vector Machine for forecasting stock price movement

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International Journal of Information Technology (Singapore) 2023年第6期15卷 3127-3134页

作者： Shanthini, P. Matilda Parthasarathy, S. Venkatesan, P. Nandhini, S. Department of Mathematics SRM Institute of Science and Technology Ramapuram Tamil Nadu Chennai 600089 India Research and Statistics Sri Ramachandra Medical College & RI Sri Ramachandra University Chennai India Department of Computer Science and Engineering Sathyabama Institute of Science and Technology Chennai India

The prediction of stock movements remains an arduous process because of the dynamic and volatile nature of the stock market. In recent times, numerous stock prediction approaches are developed to forecast future stock movements. However, they have not achieved satisfactory results in predicting future stock prices. The objective of the paper is to enhance the performance of stock prediction by the Support Vector Machine-based Hybrid Reptile Search Remora (SVM-HRSR) approach. The data gathered from the stock market are utilized to examine the performance of the proposed prediction model which is collected from the National Association of Securities Dealers Automated Quotations (NASDAQ). The unstructured stock market data is structured to increase the prediction performance using the preprocessing steps namely data minimization and min–max normalization. The proposed SVM-HRSR approach accurately determines stock market movements based on the predicted patterns. The performance influencing factors of the Support Vector Machine (SVM) such as kernel parameter and penalty parameter is tuned mechanically using the Hybrid Reptile Search Remora (HRSR) algorithm to improve the prediction accuracy. Some of the indicators such as Mean Absolute Percentage Error (MAPE), Correlation Coefficient (C_c) , Root Mean Square Error (RMSE), Hit Rate (HR) and Symmetric Mean Absolute Percentage Error (SMAPE) estimated the performance of the SVM-HRSR method. The experimental results inherit that the proposed method achieved less error rates as compared to existing methods. © 2023, The Author(s), under exclusive licence to Bharati Vidyapeeth's Institute of Computer Applications and Management.

关键词： Hybrid Reptile Search Remora algorithm NASDAQ Stock prediction Stock price movement Support Vector Machine

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GRAPH FOURIER NEURAL KERNELS (G-FUNK): LEARNING SOLUTIONS OF NONLINEAR DIFFUSIVE PARAMETRIC PDES ON MULTIPLE DOMAINS

arXiv

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arXiv 2024年

作者： Loeffler, Shane E. Ahmad, Zan Ali, Syed Yusuf Yamamoto, Carolyna Popescu, Dan M. Yee, Alana Lal, Yash Trayanova, Natalia Maggioni, Mauro Johns Hopkins University Department of Biomedical Engineering United States Johns Hopkins University Department of Applied Mathematics and Statistics United States

Predicting the time-dependent dynamics of complex systems governed by nonlinear partial differential equations (PDEs), with varying parameters and domains, is a difficult problem that is motivated by applications in many fields. We introduce a novel family of neural operators based on a Graph Fourier Neural Kernel (G-FuNK), for learning solution generators of nonlinear PDEs with varying coefficients, across multiple domains, for which the highest-order term in the PDE is diffusive. G-FuNKs are constructed by combining components that are parameter- and domain-adapted, with others that are not. The latter components are learned from training data, using a variation of Fourier Neural Operators, and are transferred directly across parameters and domains. The former, parameter- and domain-adapted components are constructed as soon as a parameter and a domain on which the PDE needs to be solved are given. They are obtained by constructing a weighted graph on the (discretized) domain, with weights chosen so that the Laplacian on that weighted graph approximates the highest order, diffusive term in the generator of the PDE, which is parameter- and domain-specific, and satisfies the boundary conditions. This approach proves to be a natural way to embed geometric and directionally-dependent information about the domains, allowing for improved generalization to new test domains without need for retraining. Finally, we equip G-FuNK with an integrated ordinary differential equation (ODE) solver to enable the temporal evolution of the system’s state. Our experiments demonstrate G-FuNK’s ability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations on multiple geometries and varying anisotropic diffusivity fields. We achieve low relative errors on unseen domains and fiber fields, significantly speeding up prediction capabilities compared to traditional finite-element solvers. © 2024, CC BY-NC-SA.

关键词： Ordinary differential equations

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On the Use of Multivariate Medians for Nearest Neighbour Imputation 1

引用

4th International Conference on Deep Learning, Artificial Intelligence and Robotics, ICDLAIR 2022

作者： Gissi, Francesco Benedetto, Vincenzo Bhandari, Parth Pérez-Fernández, Raúl Department of Engineering University of Sannio Viale Traiano 1 Benevento82100 Italy Kebula srl Via della Biblioteca 2 Fisciano84084 Italy Department of Statistics and O.R. and Mathematics Didactics University of Oviedo Oviedo Spain

ISBN: (数字)9783031303968

ISBN: (纸本)9783031303951

Most data analysis techniques, e.g. those faced in [15–18] usually require datasets without missing data, however, in the current age of large datasets it is becoming more and more common to deal with huge datasets for which some values are missing. For this reason, several techniques for the imputation of missing values have been developed. In the present work, we present one such technique based on the use of multivariate medians within the context of nearest neighbour imputation. Several experiments are carried out proving a better performance than some state-of-the-art methods in presence of noise. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Large dataset

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Exploring 3D Reconstruction with Drone Images: Advances and Challenges in Urban Environments 50

Exploring 3D Reconstruction with Drone Images: Advances and ...

引用

50th Annual Conference of the IEEE Industrial Electronics Society, IECON 2024

作者： Miranda Baldivieso, Thiago João da Silva Batista, Taise Grazielle Chagas, Fabio Suim Rodrigues Velho, Luiz Carlos Pacheco Ferreira Rosa, Paulo Fernando Defense Engineering Graduate Program Military Institute of Engineering Rio de Janeiro Brazil State Institute of the Environment Inea RJ Rio de Janeiro Brazil Department of Science and Industry systems University of South-Eastern Norway Kongsberg Norway Department of Computer Graphics Institute for Pure and Applied Mathematics Rio de Janeiro Brazil Electrical Engineering Graduate Program Military Institute of Engineering Rio de Janeiro Brazil

ISBN: (纸本)9781665464543

This working-in-progress paper aims to present a three-dimensional reconstruction using aerial images in different environments. The experiments were conducted with aircraft in both external and internal settings, starting with image acquisition, followed by the application of specific photogrammetry software-both commercial and open-source-and concluding with a qualitative evaluation of the results. © 2024 IEEE.

关键词： Aerial photography

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Rician Channel Modelling for Super Wideband MIMO Communications

Rician Channel Modelling for Super Wideband MIMO Communicati...

引用

IEEE Conference on Wireless Communications and Networking

作者： Peter J. Smith Sachitha C. Bandara Erfan Khordad Robin Evans Rajitha Senanayake Department of Electrical and Electronic Engineering University of Melbourne Melbourne Australia School of Mathematics and Statistics Victoria University of Wellington Wellington New Zealand

ISBN: (数字)9798350368369

ISBN: (纸本)9798350368376

Recent developments in Multiple-Input-Multiple-Output (MIMO) technology include packing a large number of antenna elements in a compact array to access the bandwidth benefits provided by higher mutual coupling (MC). The resulting super-wideband (SW) systems require a circuit-theoretic framework to handle the MC and channel models which span extremely large bands. Hence, in this paper, we make two key contributions. First, we develop a physically-consistent Rician channel model for use with SW systems. Secondly, we express the circuit-theoretic models in terms of a standard MIMO model, so that insights into the effects of antenna layouts, MC, and bandwidth can be made using standard communication theory. For example, we show the bandwidth widening resulting from the new channel model. In addition, we show that MC distorts line-of-sight paths which has beamforming implications. We also highlight the interaction between spatial correlation and MC and show that tight coupling reduces spatial correlations at low frequencies.

关键词： Mutual coupling Correlation Computational modeling Frequency-domain analysis Rician channels Channel models Integrated circuit modeling Wideband Standards Antenna arrays

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On causality in domain adaptation and semi-supervised learning: an information-theoretic analysis for parametric models

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2024年第1期25卷 12614-12670页

作者： Xuetong Wu Mingming Gong Jonathan H. Manton Uwe Aickelin Jingge Zhu Department of Electrical and Electronic Engineering University of Melbourne Parkville Australia School of Mathematics and Statistics University of Melbourne Parkville Australia Department of Computing and Information Systems University of Melbourne Parkville Australia

Recent advancements in unsupervised domain adaptation (UDA) and semi-supervised learning (SSL), particularly incorporating causality, have led to significant methodological improvements in these learning problems. However, a formal theory that explains the role of causality in the generalization performance of UDA/SSL is still lacking. In this paper, we consider the UDA/SSL scenarios where we access m labelled source data and n unlabelled target data as training instances under different causal settings with a parametric probabilistic model. We study the learning performance (e.g., excess risk) of prediction in the target domain from an information-theoretic perspective. Specifically, we distinguish two scenarios: the learning problem is called causal learning if the feature is the cause and the label is the effect, and is called anti-causal learning otherwise. We show that in causal learning, the excess risk depends on the size of the source sample at a rate of O( 1/m) only if the labelling distribution between the source and target domains remains unchanged. In anti-causal learning, we show that the unlabelled data dominate the performance at a rate of typically O( 1/n). These results bring out the relationship between the data sample size and the hardness of the learning problem with different causal mechanisms.

关键词： causality domain adaptation semi-supervised learning parametric models generalization error

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