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Iterative solvers in adaptive FEM

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arXiv 2024年

作者： Bringmann, Philipp Miraçi, Ani Praetorius, Dirk

MSC Codes 41A25, 65N15, 65N30, 65N50, 65Y20This chapter provides an overview of state-of-the-art adaptive finite element methods (AFEMs) for the numerical solution of second-order elliptic partial differential equations (PDEs), where the primary focus is on the optimal interplay of local mesh refinement and iterative solution of the arising discrete systems. Particular emphasis is placed on the thorough description of the essential ingredients necessary to design adaptive algorithms of optimal complexity, i.e., algorithms that mathematically guarantee the optimal rate of convergence with respect to the overall computational cost and, hence, time. Crucially, adaptivity induces reliability of the computed numerical approximations by means of a-posteriori error control. This ensures that the error committed by the numerical scheme is bounded from above by computable quantities. The analysis of the adaptive algorithms is based on the study of appropriate quasi-error quantities that include and balance different components of the overall error. Importantly, the quasi-errors stemming from an adaptive algorithm with contractive iterative solver satisfy a centerpiece concept, namely, full R-linear convergence. This guarantees that the adaptive algorithm is essentially contracting this quasi-error at each step and it turns out to be the cornerstone for the optimal complexity of AFEM. The unified analysis of the adaptive algorithms is presented in the context of symmetric linear PDEs. Extensions to goal-oriented, non-symmetric, as well as non-linear PDEs are presented with suitable nested iterative solvers fitting into the general analytical framework of the linear symmetric case. Numerical experiments highlight the theoretical results and emphasize the practical relevance and gain of adaptivity with iterative solvers for numerical simulations with optimal complexity. Copyright © 2024, The Authors. All rights reserved.

关键词： adaptive algorithms

AGSQTL: adaptive Green Space Quality Transfer Learning for Urban Environmental Monitoring 8

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AGSQTL: Adaptive Green Space Quality Transfer Learning for U...

8th IET Smart Cities Symposium, SCS 2024

作者： Chundru, Swathi Balantrapu, Siva Subrahmanyam Maroju, Praveen Kumar Alam, Naved Dutta, Pushan Kumar Whig, Pawan Motivity labs INC QA Automation Engineer United States United States TX United States Department of computer science and engineering Jamia hamdard New Delhi India School of Engineering and Technology Amity India Threws VIPS-TC New Delhi India

ISBN: (纸本)9781837242672

This study presents a novel machine learning approach, adaptive Green Space Quality Transfer Learning (AGSQTL), for assessing urban green space quality. Unlike traditional transfer learning methods, AGSQTL incorporates domain-specific adaptations and dynamic feature analysis tailored to green space monitoring. The algorithm utilizes a pre-trained source model and adapts it to the target domain of green space images, categorizing them as Healthy, Dried, or Contaminated. AGSQTL introduces several innovative components: domain-specific layers for capturing unique green space features, an adaptive domain discrepancy adjustment mechanism, dynamic feature pruning based on Grad-CAM importance analysis, and a self-attention mechanism for spatial context understanding. The model is trained on a dataset of 986 green space images, augmented to balance class distribution. Performance evaluation includes accuracy, precision, recall, F1-score, Cohen's Kappa, and ROC-AUC, as well as practical considerations such as model file size and inference time. The algorithm demonstrates superior adaptability to green space features compared to standard transfer learning approaches, achieving 99.2% accuracy on the test set. Grad-CAM visualizations confirm the model's ability to identify relevant areas for classification. AGSQTL was deployed as an interactive web application using Streamlit, allowing real-time green space quality assessment. This research contributes a more adaptive and efficient methodology for urban green space monitoring, with potential applications in environmental management and urban planning. © The Institution of Engineering & Technology 2024.

关键词： adaptive algorithms

An investigation on the use of Large Language Models for hyperparameter tuning in Evolutionary algorithms

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arXiv 2024年

作者： Custode, Leonardo Lucio Caraffini, Fabio Yaman, Anil Iacca, Giovanni University of Trento Trento Italy Swansea University Swansea United Kingdom Vrije Universiteit Amsterdam Amsterdam Netherlands

Hyperparameter optimization is a crucial problem in Evolutionary Computation. In fact, the values of the hyperparameters directly impact the trajectory taken by the optimization process, and their choice requires extensive reasoning by human operators. Although a variety of self-adaptive Evolutionary algorithms have been proposed in the literature, no definitive solution has been found. In this work, we perform a preliminary investigation to automate the reasoning process that leads to the choice of hyperparameter values. We employ two open-source Large Language Models (LLMs), namely Llama2-70b and Mixtral, to analyze the optimization logs online and provide novel real-time hyperparameter recommendations. We study our approach in the context of step-size adaptation for (1 + 1)-ES. The results suggest that LLMs can be an effective method for optimizing hyperparameters in Evolution Strategies, encouraging further research in this direction. Copyright © 2024, The Authors. All rights reserved.

关键词： adaptive algorithms

adaptive Cloaking Region Obfuscation in Road Networks

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Adaptive Cloaking Region Obfuscation in Road Networks

Fully Actuated System Theory and Applications (CFASTA), Conference on

作者： Baihe Ma Xu Wang Yanna Jiang Xiaojie Lin The Global Big Data Technologies Centre University of Technology Sydney Australia

ISBN: (数字)9798350373691

ISBN: (纸本)9798350373707

The development of location-based services (LBS) promotes the rapid growth of location data, which also leads to an increasing threat to location privacy. Existing location obfuscation techniques focus on two-dimensional planes that may generate off-road locations. In this paper, we propose an adaptive Cloaking Region Obfuscation (ACRO) mechanism to protect vehicles’ location privacy in road networks. The proposed ACRO mechanism employs route distances to measure the indistinguishability of vehicles in road networks. The ACRO mechanism perturbs vehicle locations to obfuscated cloaking regions based on the route distance between two cloaking regions. The ACRO mechanism adaptively tunes location privacy budgets according to drivers’ privacy requirements. Comprehensive experiments confirm that the proposed ACRO mechanism effectively protects location privacy in road networks.

关键词： Privacy Data privacy adaptive systems Roads adaptive algorithms Vehicles

NON-INTRUSIVE LEAST-SQUARES FUNCTIONAL A POSTERIORI ERROR ESTIMATOR: LINEAR AND NONLINEAR PROBLEMS WITH PLAIN CONVERGENCE

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arXiv 2024年

作者： Li, Ziyan Zhang, Shun Department of Mathematics City University of Hong Kong Kowloon Tong Hong Kong

The a posteriori error estimator using the least-squares functional can be used for adaptive mesh refinement and error control even if the numerical approximations are not obtained from the corresponding least-squares method. This suggests the development of a versatile non-intrusive a posteriori error estimator. In this paper, we present a systematic approach for applying the least-squares functional error estimator to linear and nonlinear problems that are not solved by the least-squares finite element methods. For the case of an elliptic PDE solved by the standard conforming finite element method, we minimize the least-squares functional with conforming approximation inserted to recover the other physical meaningful variable. By combining the numerical approximation from the original method with the auxiliary recovery approximation, we construct the least-squares functional a posteriori error estimator. Furthermore, we introduce a new interpretation that views the non-intrusive least-squares functional error estimator as an estimator for the combined solve-recover process. This simplifies the reliability and efficiency analysis. We extend the idea to a model nonlinear problem. Plain convergence results are proved for adaptive algorithms of the general second order elliptic equation and a model nonlinear problem with the non-intrusive least-squares functional a posteriori error estimators. Copyright © 2024, The Authors. All rights reserved.

关键词： adaptive algorithms

On Multilevel Energy-Based Fragmentation Methods

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arXiv 2024年

作者： Barker, James Griebel, Michael Hamaekers, Jan Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Fraunhofer Institute for Algorithms and Scientific Computing SCAI Schloss Birlinghoven 1 Sankt Augustin53757 Germany

Energy-based fragmentation methods approximate the potential energy of a molecular system as a sum of contribution terms built from the energies of particular subsystems. Some such methods reduce to truncations of the many-body expansion (MBE);others combine subsystem energies in a manner inspired by the principle of inclusion/exclusion (PIE). The combinatorial technique of Möbius inversion of sums over partially ordered sets, which generalizes the PIE, is known to provide a non-recursive expression for the MBE contribution terms, and has also been connected to related cluster expansion methods. We build from these ideas a very general framework for decomposing potential functions into energetic contribution terms associated with elements of particular partially ordered sets (posets) and direct products thereof. Specific choices immediately reproduce not only the MBE, but also a number of other existing decomposition forms, including, e.g., the multilevel ML-BOSSANOVA schema. Furthermore, a different choice of poset product leads to a setup familiar from the combination technique for high-dimensional approximation, which has a known connection to quantum-chemical composite methods. We present the ML-SUPANOVA decomposition form, which allows the further refinement of the terms of an MBE-like expansion of the Born-Oppenheimer potential according to systematic hierarchies of ab initio methods and of basis sets. We outline an adaptive algorithm for the a posteori construction of quasi-optimal truncations of this decomposition. Some initial experiments are reported and *** Codes 35J10, 65D40, 65Z05, 92E10, 06A07 © 2024, CC BY.

关键词： adaptive algorithms

Local adaptive 2D Recursive Sliding Mean Filtration of Halftone Images

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Local Adaptive 2D Recursive Sliding Mean Filtration of Halft...

Information, Communication and Energy Systems and Technologies (ICEST), International Scientific Conference on

作者： Rumen P. Mironov Faculty of Telecommunications Technical University of Sofia Sofia Bulgaria

ISBN: (数字)9798350386196

ISBN: (纸本)9798350386202

An analysis of locally-adaptive filtration of 2D halftone images, based on the recursive approach is made. The local averaging operation is used, implemented as a sliding mean 2D filter with a central symmetric kernel. The basic characteristics of the presented filter are analysed, together with the main features of the adaptive algorithm for filtration of Gaussian noises. The computational complexity of the developed algorithm is evaluated for the recursive and non-recursive mode and the experimental results for test images are given. The given experimental results from the simulation in GNU Octave environment for operating systems Windows 10 of the developed filtration algorithm, based on a set of real halftone images.

关键词： Visualization adaptive systems Filtration Two-dimensional displays Signal processing algorithms adaptive filters adaptive algorithms

Efficient adaptive Solvers for Doubly Degenerate Elliptic Equations Using a Posteriori Error Estimates

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SSRN

SSRN 2024年

作者： Ahmed, Elyes Amdouni, Saber SINTEF Digital Oslo Norway Norway Laboratory For Mathematical And Numerical Modeling In Engineering Science LAMSIN LR99ES20 University of Tunis El Manar National Engineering School of Tunis Tunis Tunisia

In this work, we derive a posteriori error estimates for a class of doubly nonlinear and degenerate elliptic equations, including the Stefan problem and fast and slow diffusion in porous media. Our approach employs equilibrated flux reconstructions, providing guaranteed and fully computable upper bounds on an energy-type norm and local efficiency. These bounds remain independent of the strength of nonlinearity and degeneracy rates. These estimators drive an adaptive solver, dynamically switching between nonlinear solvers to achieve optimal iterations. The adaptive algorithm accounts for discretization, regularization, quadrature, and linearization error components. When Newton's method encounters challenges in achieving convergence, the adaptive algorithm transitions to the $L$-scheme solver. This solver optimally precomputes the stabilization (or tuning) parameter $L>0$ during an offline phase, mirroring the behavior of the Jacobian. The adaptive algorithm is exemplified through four prototypical examples, showcasing its effective error control and notable computational savings. © 2024, The Authors. All rights reserved.

关键词： adaptive algorithms

FPGA/ARM Design of Multibeam adaptive Array for GNSS Receiver

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FPGA/ARM Design of Multibeam Adaptive Array for GNSS Receive...

East-West Design & Test Symposium (EWDTS)

作者： Vladimir I. Tsarik Dmitrii A. Bitsulia Victor I. Djigan Amungo Navigation Yerevan Armenia

ISBN: (数字)9798331515768

ISBN: (纸本)9798331515775

A multibeam adaptive Antenna Array (AAA) for the receivers of the Global Navigation Satellite Systems' (GNSS) signals is presented. The AAA channel signals are preprocessed by correlators. This increases the Signal-to-Noise Ratio (SNR) and allows to use the adaptive algorithms that employ the desired signals. The paper describes the details of the correlator implementation in a Field Programmable Gate Array (FPGA) as well as the implementation of the Matrix Inversion Lemma (MIL) Recursive Least Squares (RLS) algorithm for the AAA weights calculation that uses a core of the Advanced RISC Machine (ARM) processor which is a part of the Xilinx XCZU5EG System-on-Chip (SoC). The ARM processor computes the AAA weights using complex-valued double-precision arithmetic. One beam of the resulting 16 channel AAA occupies less than 1 % of all possible resources of the XCZU5EG FPGA which is a part of the SoC. One iteration of the adaptive algorithm (weights calculation) per beam takes about 0.015 ms. This allows to easily implement up to 16 beams of the considered AAA, which is a maximum possible number of the Global Positioning System (GPS) satellites visible at a certain point on the Earth surface at an arbitrary time moment. The designed adaptive algorithm also requires about 8 kB of external memory per AAA beam. The rest of the SoC resources and the external memory is used for other digital elements of the AAA/GNSS receiver, such as synchronization, data processing, etc., which are not considered in this paper.

关键词： Global navigation satellite system Satellites Reduced instruction set computing Array signal processing adaptive arrays adaptive algorithms Correlators Field programmable gate arrays Signal to noise ratio Global Positioning System