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Design and Adaptive Finite Element simulation of Conformal Transformation Optics Devices with Isotropic Materials

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SSRN

SSRN 2022年

作者： Yang, Wei He, Bin Wang, Hao Department of Mathematics Lanzhou Jiaotong University Gansu Lanzhou730070 China School of Mathematics and Statistics Anhui Normal University Anhui Wuhu241000 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Traditional transform optical devices are designed from anisotropic magnetic materials which are difficult to fabricate. In this paper, we design and simulate a conformal carpet invisible cloak and bending waveguide device with isotropic material. According to the existing conformal coordinate transformation, we deduce the perfect parameters required by the conformal device. And the finite element method and error analysis of the model equation and the basic process of adaptive algorithm are given. We first simulated a carpet cloak and bending waveguide device with perfect parameters;then we used the idea of parameter segmentation to design a carpet cloak that requires 35 isotropic materials and use the idea of layering to design the conformal bending waveguide device, and relevant numerical results are presented;finally, we designed the adaptive finite element algorithm to realize the numerical simulation of Maxwell's equations in isotropic material isotropic material. © 2022, The Authors. All rights reserved.

关键词： Error analysis

A GENERAL ALTERNATING-DIRECTION IMPLICIT FRAMEWORK WITH GAUSSIAN PROCESS REGRESSION PARAMETER PREDICTION FOR LARGE SPARSE LINEAR SYSTEMS

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arXiv 2021年

作者： Jiang, Kai Su, Xuehong Zhang, Juan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of existing ADI methods can be unified in the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework’s relatively optimal parameters. The GPR method only requires a small training data set to learn the regression prediction mapping, can predict accurate splitting parameters, and has high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations. Finally, we use the three-dimensional convection-diffusion equation, two-dimensional parabolic equation, and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. As a result, our proposed methods can solve much larger linear systems which these existing ADI methods have not reached. Copyright © 2021, The Authors. All rights reserved.

关键词： Linear systems

NUMERICAL METHODS AND ANALYSIS OF COMPUTING QUASIPERIODIC SYSTEMS

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arXiv 2022年

作者： Jiang, Kai Li, Shifeng Zhang, Pingwen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China

Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation. Copyright © 2022, The Authors. All rights reserved.

关键词： Fast Fourier transforms

GLOBAL OPTIMIZATION FOR THE PORTFOLIO SELECTION MODEL WITH HIGH-ORDER MOMENTS

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arXiv 2022年

作者： Yang, Liu Yang, Yi Zhong, Suhan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University School of Mathematics and Computational Sciences Xiangtan University Hunan Xiangtan411105 China School of Mathematics Hunan City University Hunan Yiyang413000 China Department of Mathematics Texas A&M University College StationTX77843-3368 United States

In this paper, we study the global optimality of polynomial portfolio optimization (PPO). The PPO is a kind of portfolio selection model with high-order moments and flexible risk preference parameters. We introduce a perturbation sample average approximation method, which can give a robust approximation of the PPO in form of linear conic optimization. The approximated problem can be solved globally with Moment-SOS relaxations. We summarize a semidefinite algorithm, which can be used to find reliable approximations of the optimal value and optimizer set of the PPO. Numerical examples are given to show the efficiency of the algorithm. Copyright © 2022, The Authors. All rights reserved.

关键词： Global optimization

The Mean Time to Absorption on Horizontal Partitioned Sierpinski Gasket Networks

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arXiv 2022年

作者： Zhang, Zhizhuo Wu, Bo Yu, Zuguo School of Mathematics Southeast University Nanjing210023 China School of Applied Mathematics Nanjing University of Finance and Economics Nanjing210023 China Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China

The random walk is one of the most basic dynamic properties of complex networks, which has gradually become a research hotspot in recent years due to its many applications in actual networks. An important characteristic of the random walk is the mean time to absorption, which plays an extremely important role in the study of topology, dynamics and practical application of complex networks. Analyzing the mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random walks on the network. The existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network, but they have failed to prove whether these effects are related to the scale of these self-similar structures. In this article, we construct and study a class of Horizontal Partitioned Sierpinski Gasket network model based on the classic Sierpinski gasket network, which is composed of local self-similar structures, and the scale of these structures will be controlled by the partition coefficient k. Then, the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained, which prove that the size of the self-similar structure in the network will directly restrict the influence of the self-similar structure on the properties of random walks on the network. Finally, we also analyzed the mean time to absorption of different absorption nodes on the network to find the location of the node with the highest absorption efficiency. Copyright © 2022, The Authors. All rights reserved.

关键词： Complex networks

Nest-DGIL: Nesterov-optimized Deep Geometric Incremental Learning for CS Image Reconstruction

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arXiv 2023年

作者： Fan, Xiaohong Yang, Yin Chen, Ke Feng, Yujie Zhang, Jianping The School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Xiangtan411105 China The Centre for Mathematical ImagingTechniques Department of Mathematical Sciences The University of Liverpool Department of Mathematics and Statistics University of Strathclyde Glasgow United Kingdom The School of Mathematics and Computational Science Xiangtan University The Key Laboratory of Intelligent Computing & Information Processing The Ministry of Education Xiangtan411105 China

Proximal gradient-based optimization is one of the most common strategies to solve inverse problem of images, and it is easy to implement. However, these techniques often generate heavy artifacts in image reconstruction. One of the most popular refinement methods is to fine-tune the regularization parameter to alleviate such artifacts, but it may not always be sufficient or applicable due to increased computational costs. In this work, we propose a deep geometric incremental learning framework based on the second Nesterov proximal gradient optimization. The proposed end-to-end network not only has the powerful learning ability for high-/low-frequency image features, but also can theoretically guarantee that geometric texture details will be reconstructed from preliminary linear reconstruction. Furthermore, it can avoid the risk of intermediate reconstruction results falling outside the geometric decomposition domains and achieve fast convergence. Our reconstruction framework is decomposed into four modules including general linear reconstruction, cascade geometric incremental restoration, Nesterov acceleration, and post-processing. In the image restoration step, a cascade geometric incremental learning module is designed to compensate for missing texture information from different geometric spectral decomposition domains. Inspired by the overlap-tile strategy, we also develop a post-processing module to remove the block effect in patch-wise-based natural image reconstruction. All parameters in the proposed model are learnable, an adaptive initialization technique of physical parameters is also employed to make model flexibility and ensure converging smoothly. We compare the reconstruction performance of the proposed method with existing state-of-the-art methods to demonstrate its superiority. Our source codes are available at https://***/fanxiaohong/Nest-DGIL. Copyright © 2023, The Authors. All rights reserved.

关键词： Image reconstruction

Nucleation and phase transition of decagonal quasicrystals

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arXiv 2024年

作者： Zhou, Tiejun Zhang, Lei Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonL8S 4M1 Canada

In this work, we study the nucleation of quasicrystals from liquid or periodic crystals by developing an efficient order-order phase transition algorithm, namely the nullspace-preserving saddle search method. Specifically, we focus on nucleation and phase transitions of the decagonal quasicrystal (DQC) based on the Lifshitz-Petrich model. We present the nucleation path of DQC from the liquid and demonstrate one- and two-stage transition paths between DQC and periodic crystals. We provide a perspective of the group-subgroup phase transition and nucleation rates to understand the nucleation and phase transition mechanisms involving DQC. These results reveal the one-step and stepwise modes of symmetry breaking or recovery in the phase transition from DQC, where the stepwise modes are more probable. Copyright © 2024, The Authors. All rights reserved.

关键词： Quasicrystals

Theory of polygonal phases self-assembled from T-shaped liquid crystalline molecules

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arXiv 2023年

作者： He, Zhijuan Wang, Xin Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases. Copyright © 2023, The Authors. All rights reserved.

关键词： Free energy

Adaptive finite element method for phase field fracture models based on recovery error estimates

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Journal of computational and Applied Mathematics 2026年 472卷

作者： Tian Tian Chunyu Chen Liang He Huayi Wei School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 China

The phase-field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase-field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our implementation.

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