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adaptivity and A Posteriori Error Control for Bifurcation Problems Ⅰ:The Bratu Problem

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Communications in Computational Physics 2010年第9期8卷 845-865页

作者： K.Andrew Cliffe Edward J.C.Hall Paul Houston Eric T.Phipps Andrew G.Salinger School of Mathematical Sciences University of NottinghamUniversity ParkNottingham NG72RDUK Computer Science Research Institute Sandia National LaboratoriesAlbuquerqueNew MexicoUSA

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is *** particular,we study in detail the numerical approximation of the Bratu problem,based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method.A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual(DWR)*** experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.

关键词： Bifurcation theory Bratu problem a posteriori error estimation adaptivity discontinuous Galerkin methods

Bio-organic adaptive photonic crystals enable supramolecular solvatochromism

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Nano Research 2023年第10期16卷 12092-12097页

作者： Jiahao Zhang Yan Zhang Yancheng Wang Sigal Rencus-Lazar Deqing Mei Ehud Gazit Kai Tao State Key Laboratory of Fluid Power and Mechatronic Systems School of Mechanical EngineeringZhejiang UniversityHangzhou 310027China Zhejiang-Israel Joint Laboratory of Self-Assembling Functional Materials Hangzhou Global Scientific and Technological Innovation CentreZhejiang UniversityHangzhou 311200China Key Laboratory of Advanced Manufacturing Technology of Zhejiang Province School of Mechanical EngineeringZhejiang UniversityHangzhou 310027China The Shmunis School of Biomedicine and Cancer Research George S.Wise Faculty of Life SciencesTel Aviv UniversityTel Aviv 6997801Israel Department of Materials Science and Engineering Iby and Aladar Fleischman Faculty of EngineeringTel Aviv UniversityTel Aviv 6997801Israel

Photonic crystals(PCs)exhibit promising structural coloration properties and possess extensive application prospects in diverse optical ***,state-of-the-art inorganic or polymeric PCs show limited adaptivity as their configurations are fixed once ***,bio-organic adaptive PCs are fabricated via drop-casting of amphiphilic guanine-based peptide nucleic acid selfassembled *** high formation activation energy of up to 81.8 kJ·mol−1 suggests that the self-assembly step dominates the entire ***,the configurations along with the structural coloration of the supramolecular PCs are sensitive to self-assembly influencing parameters,showing temperature-encoded structural color evolution and solvent polaritydependent *** findings demonstrate that the supramolecular PCs are adaptive,thus showing promising potential for detection of organic solvents of different polarities in a visual and real-time manner for environmental protection or optical applications.

关键词： peptide nucleic acids molecular manufacturing supramolecular photonic crystals solvatochromic effects adaptivity

Discontinuous Galerkin Methods for a Class of Nonvariational Problems

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Communications on Applied Mathematics and Computation 2022年第2期4卷 634-656页

作者： Andreas Dedner Tristan Pryer Mathematics Institute University of WarwickCoventry CV47ALUK Department of Mathematical Sciences University of BathBath BA27AYUK

We extend the fnite element method introduced by Lakkis and Pryer(SIAM ***.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)*** is done by viewing the“fnite element Hessian”as an auxiliary variable in the *** the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic ***,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous *** conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM ***.39(5):1749–1779,2001/2002).We also give an a posteriori analysis of the method in the case where the problem has a strong *** analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin *** evidence is presented showing that the method works well also in a more general setting.

关键词： Nonvariational problems Discontinuous Galerkin Error estimates adaptivity

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

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Communications on Applied Mathematics and Computation 2022年第1期4卷 60-83页

作者： Zhanjing Tao Juntao Huang Yuan Liu Wei Guo Yingda Cheng School of Mathematics Jilin UniversityJilin 130012China Department of Mathematics Michigan State UniversityEast LansingMI 48824USA Department of Mathematics Statistics and PhysicsWichita State UniversityWichitaKS 67260USA Department of Mathematics and Statistics Texas Tech UniversityLubbockTX 70409USA Department of Mathematics Department of Computational MathematicsScience and EngineeringMichigan State UniversityEast LansingMI 48824USA

This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger *** solutions to such equations often exhibit solitary wave and local structures,which make adaptivity essential in improving the simulation *** scheme uses the ultra-weak discontinuous Galerkin(DG)formulation and belongs to the framework of adaptive multiresolution *** numerical experiments are presented to demon-strate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

关键词： Multiresolution Sparse grid Ultra-weak discontinuous Galerkin method Schrödinger equation adaptivity

A Physarum-inspired approach to supply chain network design

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Science China(Information Sciences) 2016年第5期59卷 191-204页

作者： Xiaoge ZHANG Andrew ADAMATZKY Xin-She YANG Hai YANG Sankaran MAHADEVAN Yong DENG School of Computer and Information Science Southwest University School of Engineering Vanderbilt University Unconventional Computing Center University of the West of England School of Science and Technology Middlesex University Department of Civil and Environmental Engineering the Hong Kong University of Science and Technology

A supply chain is a system which moves products from a supplier to customers, which plays a very important role in all economic activities. This paper proposes a novel algorithm for a supply chain network design inspired by biological principles of nutrients’ distribution in protoplasmic networks of slime mould Physarum polycephalum. The algorithm handles supply networks where capacity investments and product flows are decision variables, and the networks are required to satisfy product demands. Two features of the slime mould are adopted in our algorithm. The first is the continuity of flux during the iterative process, which is used in real-time updating of the costs associated with the supply links. The second feature is adaptivity. The supply chain can converge to an equilibrium state when costs are changed. Numerical examples are provided to illustrate the practicality and flexibility of the proposed method algorithm.

关键词： supply chain design Physarum capacity investments network optimization adaptivity

Robust Reconstruction of Block Sparse Signals from Adaptively One-Bit Measurements

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Chinese Journal of Electronics 2020年第5期29卷 937-944页

作者： HOU Jingyao WANG Jianjun ZHANG Feng HUANG Jianwen School of Mathematics and Statistics Southwest University College of Artificial Intelligence Southwest University School of Mathematics and Statistics Ningxia University School of Mathematics and Statistics Tianshui Normal University

Though various theoretical results and algorithms have been proposed in one-bit Compressed sensing(1-bit CS), there are few studies on more structured signals, such as block sparse signals. We address the problem of recovering block sparse signals from one-bit measurements. We first propose two recovery schemes, one based on second-order cone programming and the other based on hard thresholding, for common non-adaptively thresholded one-bit measurements. Note that the worst-case error in recovering sparse signals from non-adaptively thresholded one-bit measurements is bounded below by a polynomial of oversampling *** break the limit, we introduce a recursive strategy that allows the thresholds in quantization to be adaptive to previous measurements at each iteration. Using the scheme, we propose two iterative algorithms and show that corresponding recovery errors are both exponential functions of the oversampling factor. Several simulations are conducted to reveal the superiority of our methods to existing approaches.

关键词： One-bit compressed sensing Convex programming Hard thresholding adaptivity Iterative algorithms

Continuous adjoint-based error estimation and its application to adaptive discontinuous Galerkin method

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Applied Mathematics and Mechanics(English Edition) 2016年第11期37卷 1419-1430页

作者： Huiqiang YUE Tiegang LIU V.SHAYDUROV Key Laboratory of Mathematics Informatics and Behavioral Semantics Ministry of EducationSchool of Mathematics and Systems Science Beihang University

An adaptive mesh refinement algorithm based on a continuous adjoint ap- proach is developed. Both the primal equation and the adjoint equation are approximated with the discontinuous Galerkin （DG） method. The proposed adaptive algorithm is used in compressible Euler equations. Numerical tests are made to show the superiority of the proposed adaptive algorithm.

关键词： adaptivity discontinuous Galerkin (DG) adjoint

Selected Recent Applications of Sparse Grids

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Numerical Mathematics(Theory,Methods and Applications) 2015年第1期8卷 47-77页

作者： Benjamin Peherstorfer Christoph Kowitz Dirk Pfluger Hans-Joachim Bungartz Scientific Computing Group Dept.of Computer ScienceBoltzmannstr.385748 GarchingGermany Institutefor Advanced Study Technische UniversitätMünchenLichtenbergstr.2a85748 GarchingGermany Institute for Parallel and Distributed Systems University of StuttgartUni-versitätsstr.3870569 StuttgartGermany

Sparse grids have become a versatile tool for a vast range of applications reaching from interpolation and numerical quadrature to data-driven problems and uncertainty *** review four selected real-world applications of sparse grids:financial product pricing with the Black-Scholes model,interactive explo-ration of simulation data with sparse-grid-based surrogate models,analysis of simu-lation data through sparse grid data mining methods,and stability investigations of plasma turbulence simulations.

关键词： Sparse grids hierarchical decomposition adaptivity combination technique

Enhancing accessibility to web mapping systems with technology-aligned adaptive profiles

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International Journal of Digital Earth 2014年第4期7卷 256-278页

作者： Jacob Delfos Bert Veenendaal Tele Tan Department of Spatial Sciences Curtin UniversityPerthWAAustralia Department of Computing Curtin UniversityPerthWAAustralia

Web-based geographic information systems have advanced rapidly on the back of web-based technologies,increased bandwidths and access to Digital Earth imagery and ***,these advances are causing its capabilities to slowly overtake those of ***,the introduction of non-desktop devices such as smartphones,tablets and netbooks is starting to undo progress made towards standardisation of web-based *** varia-tions in screen sizes,computational power,bandwidth,and operating environ-ments are once again introducing the need to ensure software remains functional across different platforms,standards-compliant or *** two issues highlight the need for a mechanism to tune content and capability to end-users and their environment,to prevent information and complexity overload in a field already troubled by poor usability,while promoting cross-platform *** paper proposes the use of adaptivity to accommodate for users from different backgrounds accessing web mapping systems in different technical *** describes adaptive profiles aligned to the finite number of states a system can adopt,rather than the limitless range of user or environment characteristics that cannot be adapted *** profile consists of a combination of adaptive states comprising functionality,information detail,or technical demands to optimise for individual users or technical environments.

关键词： web mapping digital earth adaptivity usability compatibility user modelling