Article
Mathematics
Jin Wang, Zhengyuan Shi
Summary: The multi-reconstruction algorithm proposed in this study, based on a modified vector-valued Allen-Cahn equation, is able to reconstruct multi-component surfaces without overlapping or self-intersections, producing smooth surfaces and preserving the original data effectively. The algorithm involves one linear equation and two nonlinear equations, with the linear equation discretized using implicit methods and solved using fast Fourier transform. The ability to apply the algorithm directly to graphics processing units allows for faster implementation compared to traditional central processing units.
Article
Mathematics, Applied
Dongsun Lee
Summary: This paper presents the mathematical formulation and numerical construction of minimal surfaces, along with two classical examples: Scherk's minimal surface and catenoid.
Article
Mathematics, Applied
Liang Wu, Mejdi Azaieza, Tomas Chacon Rebollo, Chuanju Xu
Summary: In this paper, an efficient reduced-order finite element method is proposed and analyzed for the parametrized Allen-Cahn equation. The equation is first discretized using a stabilized semi-implicit scheme in time and a finite element method in space for a given parameter. Then, a reduced basis is constructed using proper orthogonal decomposition (POD) based on a set of snapshots. The main contribution of this work lies in the error analysis of the reduced-order model, where an error estimate is derived for the first time for the parametrized Allen-Cahn equation, considering the impact of the diffusion parameter.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Narcisse Batangouna
Summary: In this article, we study the time semidiscretization of the Ginzburg-Landau equation using the backward Euler scheme. We construct an exponential attractor for the dynamical system associated with each time step tau. We prove that as tau approaches 0, this attractor converges to the exponential attractor of the dynamical system associated with the Allen-Cahn equation, and also show that the fractal dimension of the attractor and the global attractor is bounded by a constant independent of tau.
Article
Computer Science, Artificial Intelligence
Jeremy M. Budd, Yves van Gennip, Jonas Latz, Simone Parisotto, Carola-Bibiane Schoenlieb
Summary: This paper presents a method for jointly reconstructing and segmenting images using graph-based segmentation methods. The convergence properties of the scheme are analyzed, and the performance in image reconstruction is evaluated through experiments on distorted images. The results show that the proposed method achieves highly accurate segmentations and outperforms sequential reconstruction-segmentation approaches in terms of reconstruction and segmentation.
SIAM JOURNAL ON IMAGING SCIENCES
(2023)
Article
Optics
Salim B. Ivars, Muriel Botey, Ramon Herrero, Kestutis Staliunas
Summary: We propose a method to control turbulence by modifying the excitation cascade. The method is based on the asymmetric coupling between spatiotemporal excitation modes using non-Hermitian potentials. We demonstrate that unidirectional coupling towards larger or smaller wave numbers can increase or reduce the energy flow in turbulent states, thereby influencing the character of turbulence. The study uses the complex Ginzburg-Landau equation, a universal model for pattern formation and turbulence in various systems.
Article
Mathematics, Interdisciplinary Applications
Orazio Descalzi, Carlos Cartes
Summary: This article investigates the formation of localized spatiotemporal chaos in the complex cubic Ginzburg-Landau equation with nonlinear gradient terms and reviews the influence of multiplicative noise on stationary pulses stabilized by nonlinear gradients. Surprising results are obtained through numerical simulations and explained analytically, including the induction of velocity change in propagating dissipative solitons.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Tiemo Pedergnana, Nicolas Noiray
Summary: This study presents a detailed analysis of the transformation rules for Langevin equations under general nonlinear mappings, and shows how to identify systems with exact potentials by understanding their differential-geometric properties. The results imply a broad class of exactly solvable stochastic models for nonlinear oscillations.
Article
Engineering, Multidisciplinary
Revanth Mattey, Susanta Ghosh
Summary: A physics informed neural network (PINN) is a method that incorporates the physics of a system into a neural network's loss function by satisfying the system's boundary value problem. To address the accuracy issue for highly non-linear and higher-order time-varying partial differential equations, a novel backward compatible PINN (bc-PINN) scheme is proposed, which solves the PDE sequentially over successive time segments using a single neural network and re-trains the network to satisfy the already obtained solutions for previous time segments. Two techniques, using initial conditions and transfer learning, are introduced to improve the accuracy and efficiency of the bc-PINN scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics
Yasuhito Miyamoto, Tatsuki Mori, Tohru Tsujikawa, Shoji Yotsutani
Summary: This study investigates the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions and determines the stability/instability of all symmetric solutions and a part of asymmetric solutions, with numerical simulations supporting the results.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Enyu Fan, Changpin Li
Summary: This paper numerically studies the Allen-Cahn equations with different kinds of time fractional derivatives and investigates the influences of time derivatives on the solutions of the considered models.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Engineering, Mechanical
Ahmed H. Arnous, Taher A. Nofal, Anjan Biswas, Yakup Yildirim, Asim Asiri
Summary: This paper presents a method for extracting cubic-quartic optical soliton solutions for the complex Ginzburg-Landau equation with five distinct forms of nonlinear refractive index. By utilizing the proposed algorithm, a diverse range of optical solitons, including hybrid types, that satisfy the specified parameter restrictions can be obtained.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Qinghua Chen, Yutian Lei
Summary: This paper investigates the asymptotic behavior of solutions of an integral equation of the Allen-Cahn type as |x| approaches infinity, with certain conditions on uniform continuity and integrability. The study reveals the limits of the solutions under various scenarios.
Article
Mathematics, Applied
Azer Khanmamedov
Summary: This paper considers the hyperbolic relaxation of the 2D Cahn-Hilliard/Allen-Cahn equation with cubic nonlinearity of class C-2, proving that the semigroup generated by weak solutions possesses a global attractor. It also improves previously obtained results regarding global attractors for the 2D Cahn-Hilliard equation with the inertial term.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Materials Science, Multidisciplinary
Anjan Biswas, Abdul H. Kara, Yunzhou Sun, Qin Zhou, Yakup Yildirim, Hashim M. Alshehri, Milivoj R. Belic
Summary: This paper discusses the conserved densities and quantities for the perturbed complex Ginzburg-Landau model using Lie symmetry analysis, and finds that for certain nonlinear forms, the Hamiltonian ceases to exist due to divergent integrals.
RESULTS IN PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Dong Wang, Haohan Li, Xiaoyu Wei, Xiao-Ping Wang
JOURNAL OF COMPUTATIONAL PHYSICS
(2017)
Article
Computer Science, Interdisciplinary Applications
Xianmin Xu, Dong Wang, Xiao-Ping Wang
JOURNAL OF COMPUTATIONAL PHYSICS
(2017)
Article
Mathematics, Applied
Shidong Jiang, Dong Wang, Xiao-Ping Wang
JOURNAL OF SCIENTIFIC COMPUTING
(2018)
Article
Mathematics, Applied
Dong Wang, Xiao-Ping Wang, Ya-Guang Wang
SIAM JOURNAL ON APPLIED MATHEMATICS
(2017)
Article
Mathematics, Applied
Dong Wang, Braxton Osting
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Multidisciplinary Sciences
Dong Wang, Andrej Cherkaev, Braxton Osting
Article
Computer Science, Interdisciplinary Applications
Dong Wang, Xiao-Ping Wang, Xianmin Xu
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Mathematics, Applied
Dong Wang, Shidong Jiang, Xiao-Ping Wang
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Braxton Osting, Dong Wang, Yiming Xu, Dominique Zosso
Summary: Archetypal analysis is an unsupervised learning method that utilizes convex polytopes to summarize multivariate data, with archetype points being the key components. Consistency results are proven, showing convergence of archetype points under certain assumptions, along with convergence rates for optimal objective values. Experiments with various distributions support the analysis and demonstrate the effectiveness of the method for summarizing data.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2021)
Article
Mathematics, Interdisciplinary Applications
Dong Wang, Braxton Osting, Xiao-Ping Wang
MULTISCALE MODELING & SIMULATION
(2019)