808 results on '"Balanced flow"'
Search Results
2. High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model
- Author
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Junseok Kim, Junxiang Yang, and Zhijun Tan
- Subjects
Backward differentiation formula ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Numerical analysis ,Scalar (physics) ,Applied mathematics ,Local variable ,Dissipation ,Balanced flow ,Backward Euler method ,Exponential function ,Mathematics - Abstract
In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L 2 -gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.
- Published
- 2021
3. Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow
- Author
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Alberto Abbondandolo and Pietro Majer
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Closed manifold ,stable foliation ,CW-decomposition ,gradient flow ,Morse-Smale flow ,Structure (category theory) ,Dynamical Systems (math.DS) ,Morse code ,Mathematics::Geometric Topology ,law.invention ,Flow (mathematics) ,law ,FOS: Mathematics ,Mathematics::Differential Geometry ,Geometry and Topology ,Mathematics - Dynamical Systems ,Invariant (mathematics) ,Balanced flow ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
We prove that a Morse-Smale gradient-like flow on a closed manifold has a "system of compatible invariant stable foliations" that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse-Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse-Smale gradient-like flow on a closed manifold $M$ are the open cells of a $CW$-decomposition of $M$., 57 pages, 1 figure. The list of references has been expanded and the discussion about the history of the problem and future perspectives has been improved thanks to the suggestions of some readers
- Published
- 2021
4. Stable gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints
- Author
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Christian Palus and Sören Bartels
- Subjects
65N12, 65N30, 74K20 ,Discretization ,Iterative method ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Bending of plates ,Bending ,Isometry (Riemannian geometry) ,01 natural sciences ,Computational Mathematics ,Linearization ,Mathematics - Numerical Analysis ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
Bilayer plates are compound materials that exhibit large bending deformations when exposed to environmental changes that lead to different mechanical responses in the involved materials. In this article a new numerical method that is suitable for simulating the isometric deformation induced by a given material mismatch in a bilayer plate is discussed. A dimensionally reduced formulation of the bending energy is discretized generically in an abstract setting and specified for discrete Kirchhoff triangles; convergence towards the continuous formulation is proved. A practical semi-implicit discrete gradient flow employing a linearization of the isometry constraint is proposed as an iterative method for the minimization of the bending energy; stability and a bound on the violation of the isometry constraint are proved. The incorporation of obstacles is discussed and the practical performance of the method is illustrated with numerical experiments involving the simulation of large bending deformations and investigation of contact phenomena.
- Published
- 2021
5. GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS
- Author
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Pan Zhang
- Subjects
General Mathematics ,010102 general mathematics ,Yang–Mills existence and mass gap ,Riemannian manifold ,01 natural sciences ,010101 applied mathematics ,Higgs field ,Higgs boson ,Order operator ,Gravitational singularity ,0101 mathematics ,Balanced flow ,Mathematics ,Gauge fixing ,Mathematical physics - Abstract
In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the$L^2$-bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgsk-functional with Higgs self-interaction, we show that, provided$\dim (M), for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local$L^2$-derivative estimates, energy estimates and blow-up analysis.
- Published
- 2021
6. A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion
- Author
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Antonio Russo, Sergio P. Perez, Serafim Kalliadasis, Julien Mendes, Commission of the European Communities, and Engineering & Physical Science Research Council (EPSRC)
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Discretization ,Diffusion ,Mathematics, Applied ,Numerical & Computational Mathematics ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,DENSITY-FUNCTIONAL THEORY ,Finite-volume schemes ,Thermal ,0101 mathematics ,Anisotropy ,Non-homogeneous baths ,01 Mathematical Sciences ,Mathematics ,Science & Technology ,15 Commerce, Management, Tourism and Services ,Finite volume method ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,Dynamical-density functional theory ,AGGREGATION ,MODEL ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Physical Sciences ,08 Information and Computing Sciences ,Balanced flow - Abstract
We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to study several one- and two-dimensional systems, with a general free-energy functional accounting for external fields and inter-particle potentials, and placed in non-homogeneous thermal baths characterized by anisotropic, space-dependent and time-dependent properties.
- Published
- 2021
7. Energy-production-rate preserving numerical approximations to network generating partial differential equations
- Author
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Qi Wang, Jia Zhao, and Qi Hong
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Partial differential equation ,Series (mathematics) ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference ,Computational Mathematics ,Quadratic equation ,Computational Theory and Mathematics ,Modeling and Simulation ,Dissipative system ,Applied mathematics ,Convergence tests ,Boundary value problem ,Balanced flow ,Mathematics - Abstract
We recast a network generating partial differential equation system into a singular limit of a dissipative gradient flow model, which not only identifies the consistent physical boundary conditions but also generates networks. We then develop a set of structure-preserving numerical algorithms for the gradient flow model. Using the energy quadratization (EQ) method, we reformulate the gradient flow system into an equivalent one with a quadratic energy density by introducing auxiliary variables. Subsequently, we devise a series of fully discrete, linear, second order, energy-production-rate preserving, finite difference algorithms to solve the EQ-reformulated PDE system subject to various compatible boundary conditions. We show that the numerical schemes are energy-production-rate preserving for any time steps. Numerical convergence tests are given to validate the accuracy of the fully discrete schemes. Several 2D numerical examples are given to demonstrate the capability of the schemes in predicting network generating phenomena with the gradient flow PDE system, especially, the original network generating PDE model.
- Published
- 2021
8. Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach
- Author
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Jie Ouyang and Chenhui Zhang
- Subjects
Variable time ,Scalar (mathematics) ,Selection strategy ,010103 numerical & computational mathematics ,Time step ,01 natural sciences ,010101 applied mathematics ,Auxiliary variables ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Computational Theory and Mathematics ,Fixed time ,Modeling and Simulation ,symbols ,Applied mathematics ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.
- Published
- 2021
9. Gradient Flow Analysis and Performance Comparison of CNN Models
- Author
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Seol-hyun Noh
- Subjects
Performance comparison ,Word error rate ,Balanced flow ,Algorithm ,Mathematics - Published
- 2021
10. Global Heteroclinic Rebel Dynamics Among Large 2-Clusters in Permutation Equivariant Systems
- Author
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Sindre W. Haugland, Felix P. Kemeth, Bernold Fiedler, and Katharina Krischer
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Pure mathematics ,Mathematics::Combinatorics ,Dynamics (mechanics) ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Nonlinear Sciences - Chaotic Dynamics ,37G40, 34C15, 34C23, 37B35 ,Permutation ,Symmetric group ,Modeling and Simulation ,FOS: Mathematics ,Order (group theory) ,Equivariant map ,Vector field ,Chaotic Dynamics (nlin.CD) ,Mathematics - Dynamical Systems ,Balanced flow ,Analysis ,Mathematics - Abstract
We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable \emph{rebel flow} in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart-Landau oscillators with complex linear global coupling., 46 pages, 21 figures
- Published
- 2021
11. Asymptotic behavior of gradient flows on the unit sphere with various potentials
- Author
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Hyungjin Huh and Dohyun Kim
- Subjects
Unit sphere ,Applied Mathematics ,010102 general mathematics ,Relaxation (NMR) ,State (functional analysis) ,01 natural sciences ,010101 applied mathematics ,Range (mathematics) ,Yield (chemistry) ,Statistical physics ,0101 mathematics ,Algebraic number ,Balanced flow ,Focus (optics) ,Analysis ,Mathematics - Abstract
We consider a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance | x i − x j | p + 2 for any p ≠ 0 . In this paper, we employ both attractive and repulsive couplings to study the asymptotic behavior of the system accompanied by both p > 0 (positive range) and p 0 (negative range), and this enables to yield richer dynamical phenomena. Firstly in an attractive regime, we focus on the emergence of the complete aggregation; however, the relaxation dynamics towards the aggregated state for the positive range differs from the one for the negative range. More precisely for p > 0 , the complete aggregation occurs with an algebraic rate O ( t − 1 / p ) . On the other hand for p 0 , the issue of global existence arises due to the singular interaction and is crucially related to the aggregation estimate. To this end, we show that the complete aggregation emerges in finite time and thus a solution exists until such a time. Lastly in a repulsive regime, we mainly consider the splay state for both positive and negative ranges, and several case studies are presented to obtain the qualitative insight. Finally, we compare our results with the case of p = 0 .
- Published
- 2021
12. Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions
- Author
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Chun Liu, Stefan Metzger, Kei Fong Lam, and Patrik Knopf
- Subjects
Numerical Analysis ,Applied Mathematics ,FOS: Physical sciences ,Boundary (topology) ,Numerical Analysis (math.NA) ,Mathematical Physics (math-ph) ,35A01, 35A02, 35A35, 35B40, 65M60, 65M12 ,Finite element method ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Modeling and Simulation ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,Balanced flow ,Cahn–Hilliard equation ,Conservation of mass ,Mathematical Physics ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics ,Interpolation - Abstract
The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.
- Published
- 2021
13. Regularity estimates for the gradient flow of a spinorial energy functional
- Author
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Fei He and Changliang Wang
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Spinor ,General Mathematics ,Order (ring theory) ,Ricci flow ,Covariant derivative ,Differential Geometry (math.DG) ,Flow (mathematics) ,Spinor field ,FOS: Mathematics ,Balanced flow ,Energy functional ,Mathematics - Abstract
In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow., Comment: 32 pages
- Published
- 2021
14. Normalized Gradient Flow with Lagrange Multiplier for Computing Ground States of Bose--Einstein Condensates
- Author
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Wei Liu and Yongyong Cai
- Subjects
Normalization (statistics) ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,law.invention ,Computational Mathematics ,symbols.namesake ,law ,Lagrange multiplier ,symbols ,0101 mathematics ,Balanced flow ,Ground state ,Bose–Einstein condensate ,Mathematical physics ,Mathematics - Abstract
The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evol...
- Published
- 2021
15. Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks
- Author
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Yujun Teng, Paris Perdikaris, and Sifan Wang
- Subjects
Computational Mathematics ,Conservation law ,Theoretical computer science ,Artificial neural network ,Differential equation ,business.industry ,Applied Mathematics ,Deep learning ,Domain knowledge ,Artificial intelligence ,Balanced flow ,business ,Mathematics - Abstract
The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constrai...
- Published
- 2021
16. Short retractions of CAT(1) spaces
- Author
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Alexander Lytchak and Anton Petrunin
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Tractrix ,Applied Mathematics ,General Mathematics ,Regular polygon ,Metric Geometry (math.MG) ,Construct (python library) ,Type (model theory) ,Space (mathematics) ,Lipschitz continuity ,53C20 (Primary) 53C23, 53C44 (Secondary) ,Mathematics - Metric Geometry ,Differential Geometry (math.DG) ,Flow (mathematics) ,FOS: Mathematics ,Balanced flow ,Mathematics - Abstract
We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest., Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:1903.08539
- Published
- 2020
17. Uniqueness and nonuniqueness of limits of teichmuller harmonic map flow
- Author
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Melanie Rupflin, James Kohout, and Peter M. Topping
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Surface (mathematics) ,Sequence ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Harmonic map ,01 natural sciences ,Manifold ,Flow (mathematics) ,0103 physical sciences ,Immersion (mathematics) ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Balanced flow ,QA ,Analysis ,Mathematics - Abstract
The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞ {t\to\infty} .
- Published
- 2022
18. Non-linear Morse–Bott functions on quaternionic Stiefel manifolds
- Author
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M. J. Pereira-Sáez, Daniel Tanré, and Enrique Macías-Virgós
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Pure mathematics ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Stiefel manifold ,Quadratic equation ,Product (mathematics) ,Mathematics::Differential Geometry ,0101 mathematics ,Balanced flow ,Mathematics::Symplectic Geometry ,Mathematics ,Sign (mathematics) - Abstract
In the Stiefel manifold X n , k , we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n − 2 k . The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.
- Published
- 2020
19. A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals
- Author
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Juan Pablo Borthagaray, Ricardo H. Nochetto, and Shawn W. Walker
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Applied Mathematics ,Numerical analysis ,Computation ,Mathematical analysis ,Monotonic function ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Tensor field ,010101 applied mathematics ,Computational Mathematics ,Liquid crystal ,Regularization (physics) ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
We consider the one-constant Landau-de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $$\mathbf{Q}$$ , which is constrained to be uniaxial: $$\mathbf{Q}= s (\mathbf{n}\otimes \mathbf{n}- d^{-1}\mathbf{I})$$ where $$\mathbf{n}$$ is a director field, $$s\in \mathbb {R}$$ is the degree of orientation, and $$d\ge 2$$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $$\Gamma $$ -convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method’s capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.
- Published
- 2020
20. Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries
- Author
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Herbert Egger and Anke Böttcher
- Subjects
Discretization ,General Mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Nonlinear system ,Variational principle ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Balanced flow ,010306 general physics ,Galerkin method ,Realization (systems) ,Energy functional ,Mathematics - Abstract
We study the systematic numerical approximation of a class of Allen–Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. and support the observation that their hybrid Allen–Cahn model avoids the problem of mesh-locking to a large extent.
- Published
- 2020
21. Accelerated Information Gradient Flow
- Author
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Yifei Wang and Wuchen Li
- Subjects
FOS: Computer and information sciences ,Logarithm ,Computer Science - Information Theory ,Bayesian probability ,Machine Learning (stat.ML) ,Statistics::Other Statistics ,Kernel Bandwidth ,Statistics - Computation ,Theoretical Computer Science ,symbols.namesake ,Statistics - Machine Learning ,Convergence (routing) ,FOS: Mathematics ,Mathematics - Optimization and Control ,Computation (stat.CO) ,Mathematics ,Numerical Analysis ,Information Theory (cs.IT) ,Applied Mathematics ,General Engineering ,Markov chain Monte Carlo ,Inverse problem ,Statistics::Computation ,Computational Mathematics ,Computational Theory and Mathematics ,Optimization and Control (math.OC) ,Metric (mathematics) ,symbols ,Balanced flow ,Algorithm ,Software - Abstract
We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.
- Published
- 2021
22. Local Energy Dissipation Rate Preserving Approximations to Driven Gradient Flows with Applications to Graphene Growth
- Author
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Lin Lu, Qi Wang, Yushun Wang, and Yongzhong Song
- Subjects
Numerical Analysis ,Field (physics) ,Applied Mathematics ,General Engineering ,Dissipation ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Convergence (routing) ,Benchmark (computing) ,Periodic boundary conditions ,Statistical physics ,Boundary value problem ,Balanced flow ,Convection–diffusion equation ,Software ,Mathematics - Abstract
We develop a paradigm for developing local energy dissipation rate preserving (LEDRP) approximations to general gradient flow models driven by source terms. In driven gradient flow models, the deduced energy density transport equation possesses an indefinite source. Local energy-dissipation-rate preserving algorithms are devised to respect the mathematical structure of both the driven gradient flow model and its deduced energy density transport equation. The LEDRP algorithms are also global energy-dissipation-rate preserving under proper boundary conditions such as periodic boundary conditions. However, the contrary may not be true. We then apply the paradigm to a phase field model for growth of a graphene sheet to produce a set of LEDRP algorithms. Numerical refinement tests are conducted to confirm the convergence property of the new algorithms and simulations of graphene growth are demonstrated to benchmark against existing results in the literature.
- Published
- 2021
23. On nonnegative solutions for the Functionalized Cahn–Hilliard equation with degenerate mobility
- Author
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Keith Promislow, Shibin Dai, Toai Luong, and Qiang Liu
- Subjects
Physics ,Weak convergence ,Applied Mathematics ,Weak solution ,Degenerate energy levels ,Mathematical analysis ,Dissipation ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,Surface energy ,The Functionalized Cahn–Hilliard equation ,Weak solutions ,Physics::Fluid Dynamics ,QA1-939 ,Balanced flow ,Degenerate mobility ,Cahn–Hilliard equation ,Galerkin method ,Nonnegative solutions ,Mathematics - Abstract
The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M ( u ) that is zero for u ≤ 0 . Assuming the initial data u 0 ( x ) is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.
- Published
- 2021
24. The gradient flow of the Möbius energy : 𝜀-regularity and consequences
- Author
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Simon Blatt
- Subjects
Numerical Analysis ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Möbius energy ,Scale (descriptive set theory) ,01 natural sciences ,Uniform norm ,Singularity ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Balanced flow ,Analysis ,Topology (chemistry) ,Energy (signal processing) ,Mathematics - Abstract
We study the gradient flow of the Mobius energy introduced by O’Hara (Topology 30:2 (1991), 241–247). We will show a fundamental 𝜀-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He (
- Published
- 2020
25. Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation
- Author
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Zhengguang Liu and Xiaoli Li
- Subjects
Computational Mathematics ,Numerical Analysis ,Nonlinear system ,Work (thermodynamics) ,Phase transition ,Applied Mathematics ,Crystal model ,Scalar (physics) ,Phase field models ,Statistical physics ,Balanced flow ,Diffusion (business) ,Mathematics - Abstract
The phase-field crystal equation is a sixth-order nonlinear parabolic equation and have received increasing attention in the study of the microstructural evolution of two-phase systems on atomic length and diffusive time scales. This model can be applied to simulate various phenomena such as epitaxial growth, material hardness and phase transition. Compared with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal phase-field crystal equation equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. We propose linear semi-implicit approach and scalar auxiliary variable approach with unconditional energy stability for the nonlocal phase-field crystal equation. The first contribution is that we have proved the unconditional energy stability for nonlocal phase-field crystal model and its semi-discrete schemes carefully and rigorously. Secondly, we found a fast procedure to reduce the computational work and memory requirement which the non-locality of the nonlocal diffusion term generates huge computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.
- Published
- 2020
26. A Second Order Gradient Flow of p-Elastic Planar Networks
- Author
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Paola Pozzi and Matteo Novaga
- Subjects
long-time existence ,Applied Mathematics ,Weak solution ,Mathematical analysis ,minimizing movements ,Order (ring theory) ,35K92, 53A04, 53C44 ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Planar ,Flow (mathematics) ,Mathematik ,elastic flow of networks ,FOS: Mathematics ,0101 mathematics ,Balanced flow ,Analysis ,Energy (signal processing) ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy., 27 pages
- Published
- 2020
27. A Highly Efficient and Accurate New Scalar Auxiliary Variable Approach for Gradient Flows
- Author
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Fukeng Huang, Jie Shen, and Zhiguo Yang
- Subjects
Auxiliary variables ,General Relativity and Quantum Cosmology ,Computational Mathematics ,Nonlinear system ,Applied Mathematics ,Scalar (mathematics) ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Balanced flow ,01 natural sciences ,Mathematics - Abstract
We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit l...
- Published
- 2020
28. Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation
- Author
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Bangti Jin and Manh Hong Duong
- Subjects
Discretization ,Applied Mathematics ,General Mathematics ,Convergence (routing) ,Zero (complex analysis) ,Physical system ,Applied mathematics ,Fokker–Planck equation ,Derivative ,Balanced flow ,Mathematics ,Fractional calculus - Abstract
In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.
- Published
- 2020
29. Unconditionally Bound Preserving and Energy Dissipative Schemes for a Class of Keller--Segel Equations
- Author
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Jie Shen and Jie Xu
- Subjects
Numerical Analysis ,Class (set theory) ,Discretization ,Applied Mathematics ,Mathematics::Analysis of PDEs ,Structure (category theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Quantitative Biology::Cell Behavior ,Computational Mathematics ,Energy stability ,Scheme (mathematics) ,Dissipative system ,Applied mathematics ,0101 mathematics ,Balanced flow ,Energy (signal processing) ,Mathematics - Abstract
We propose numerical schemes for a class of Keller--Segel equations. The discretization is based on the gradient flow structure. The resulting first-order scheme is mass conservative, bound preserv...
- Published
- 2020
30. The family of level sets of a harmonic function
- Author
-
Pisheng Ding
- Subjects
Algebra and Number Theory ,Mean curvature ,Applied Mathematics ,Mathematical analysis ,symbols.namesake ,Harmonic function ,Flow (mathematics) ,Special functions ,Fourier analysis ,symbols ,Geometry and Topology ,Balanced flow ,Analysis ,Mathematics ,Geometric data analysis - Abstract
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.
- Published
- 2019
31. Fast, unconditionally energy stable large time stepping method for a new Allen–Cahn type square phase-field crystal model
- Author
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Fubiao Lin, Xiaoxia Wen, and Xiaoming He
- Subjects
Applied Mathematics ,010102 general mathematics ,01 natural sciences ,Stability (probability) ,Square (algebra) ,010101 applied mathematics ,symbols.namesake ,Robustness (computer science) ,Lagrange multiplier ,Crystal model ,Benchmark (computing) ,symbols ,Applied mathematics ,0101 mathematics ,Balanced flow ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we develop a new square phase-field crystal model using the L 2 -gradient flow approach, where the total mass of atoms is conserved through a nonlocal Lagrange multiplier. We construct a fast, provably unconditionally energy stable, second-order scheme by using the recently developed SAV approach with the stabilization technique, where an extra stabilization term is added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with the classical Cahn–Hilliard type square phase-field crystal model and the non-stabilized SAV scheme for simulating some benchmark numerical examples, we demonstrate the robustness of the new model, as well as the stability and the accuracy of the developed scheme, numerically.
- Published
- 2019
32. Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour
- Author
-
Raimund Bürger, Luis Miguel Villada, Pep Mulet, and Daniel Inzunza
- Subjects
Numerical Analysis ,Applied Mathematics ,Numerical analysis ,CPU time ,Space (mathematics) ,Computer Science::Numerical Analysis ,Mathematics::Numerical Analysis ,Convolution ,Term (time) ,Computational Mathematics ,Nonlinear system ,Applied mathematics ,Balanced flow ,Reduction (mathematics) ,Mathematics - Abstract
The numerical solution of nonlinear convection-diffusion equations with nonlocal flux by explicit finite difference methods is costly due to the local spatial convolution within the convective numerical flux and the disadvantageous Courant-Friedrichs-Lewy (CFL) condition caused by the diffusion term. More efficient numerical methods are obtained by applying second-order implicit-explicit (IMEX) Runge-Kutta time discretizations to an available explicit scheme for such models in Carrillo et al. (2015) [13] . The resulting IMEX-RK methods require solving nonlinear algebraic systems in every time step. It is proven, for a general number of space dimensions, that this method is well defined. Numerical experiments for spatially two-dimensional problems motivated by models of collective behaviour are conducted with several alternative choices of the pair of Runge-Kutta schemes defining an IMEX-RK method. For fine discretizations, IMEX-RK methods turn out more efficient in terms of reduction of error versus CPU time than the original explicit method.
- Published
- 2019
33. A variant of scalar auxiliary variable approaches for gradient flows
- Author
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Mejdi Azaïez, Dianming Hou, and Chuanju Xu
- Subjects
Numerical Analysis ,Constant coefficients ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Scalar (mathematics) ,Linear system ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Modeling and Simulation ,Bounded function ,Applied mathematics ,Poisson's equation ,Balanced flow ,Spectral method ,Mathematics - Abstract
In this paper, we propose and analyze a new class of schemes based on a variant of the scalar auxiliary variable (SAV) approaches for gradient flows. Precisely, we construct more robust first and second order unconditionally stable schemes by introducing a new defined auxiliary variable to deal with nonlinear terms in gradient flows. The new approach consists in splitting the gradient flow into decoupled linear systems with constant coefficients, which can be solved using existing fast solvers for the Poisson equation. This approach can be regarded as an extension of the SAV method; see, e.g., Shen et al. (2018) [21] , in the sense that the new approach comes to be the conventional SAV method when α = 0 and removes the boundedness assumption on ∫ Ω F ( ϕ ) d x required by the SAV. The new approach only requires that the total free energy or a part of it is bounded from below, which is more realistic in physically meaningful models. The unconditional stability is established, showing that the efficiency of the new approach is less restricted to particular forms of the nonlinear terms. A series of numerical experiments is carried out to verify the theoretical claims and illustrate the efficiency of our method.
- Published
- 2019
34. Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity
- Author
-
Yuan Gao
- Subjects
Work (thermodynamics) ,Logarithm ,Applied Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Singularity ,Variational inequality ,Radon measure ,FOS: Mathematics ,0101 mathematics ,Balanced flow ,Analysis ,Analysis of PDEs (math.AP) ,Mathematical physics ,Mathematics - Abstract
In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^+$ happens., Comment: 15 pages
- Published
- 2019
35. The numerical study for the ground and excited states of fractional Bose–Einstein condensates
- Author
-
Yongyun Shao, Zhen Wang, Yu Wang, Zijian Han, and Rongpei Zhang
- Subjects
Condensed Matter::Quantum Gases ,Discretization ,Condensed Matter::Other ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,law.invention ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,law ,Modeling and Simulation ,Excited state ,Lattice (order) ,0101 mathematics ,Balanced flow ,Bose–Einstein condensate ,Harmonic oscillator ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the ground and first excited states of the fractional Bose–Einstein condensates (BEC) which is modeled by fractional Gross–Pitaevskii (GP) equation. We first introduce the normalized gradient flow method and prove its energy diminishing property. Then the weighted shifted Grunwald–Letnikov difference (WSGD) method is used to discretize the Gross–Pitaevskii equation. The corresponding normalization and energy diminishing property for the semi-discrete scheme are proved. For the time discretization, we use the implicit integration factor (IIF) method which decouples the diffusion and nonlinear terms separately. Finally the numerical methods are applied to compute the ground and first excited states of fractional BEC with harmonic oscillator, harmonic-plus-optical lattice and box potential. Our numerical results show that the ground and excited states in fractional GP equation differ from those of the standard (non-fractional) GP equation.
- Published
- 2019
36. The Energy Functional of $$G_2$$-Structures Compatible with a Background Metric
- Author
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Leonardo Bagaglini
- Subjects
Pure mathematics ,Spinor ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,Metric (mathematics) ,Torsion (algebra) ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Balanced flow ,Energy functional ,Mathematics - Abstract
The space of $$G_2$$ -structures is naturally stratified by those structures compatible with a fixed Riemannian metric. We study the restriction of the total torsion energy functional to these strata. Precisely we show the short-time existence of its negative gradient flow, we characterise the space of its critical points in terms of spinor fields and, finally, we describe the long-time behaviour of the homogeneous negative gradient flow.
- Published
- 2019
37. Energy-stable Runge–Kutta schemes for gradient flow models using the energy quadratization approach
- Author
-
Jia Zhao and Yuezheng Gong
- Subjects
Class (set theory) ,Applied Mathematics ,010102 general mathematics ,Mathematical proof ,01 natural sciences ,010101 applied mathematics ,Auxiliary variables ,Runge–Kutta methods ,Dissipative system ,Applied mathematics ,0101 mathematics ,Balanced flow ,Energy (signal processing) ,Quadratic functional ,Mathematics - Abstract
In this letter, we present a novel class of arbitrarily high-order and unconditionally energy-stable algorithms for gradient flow models by combining the energy quadratization (EQ) technique and a specific class of Runge–Kutta (RK) methods, which is named the EQRK schemes. First of all, we introduce auxiliary variables to transform the original model into an equivalent system, with the transformed free energy a quadratic functional with respect to the new variables and the modified energy dissipative law is conserved. Then a special class of RK methods is employed for the reformulated system to arrive at structure-preserving time-discrete schemes. Along with rigorous proofs, numerical experiments are presented to demonstrate the accuracy and unconditionally energy-stability of the EQRK schemes.
- Published
- 2019
38. Gradient system for the roots of the Askey-Wilson polynomial
- Author
-
J. F. van Diejen
- Subjects
Polynomial ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Gradient system ,Balanced flow ,Askey–Wilson polynomials ,Mathematics - Abstract
Recently, it was observed that the roots of the Askey-Wilson polynomial are retrieved at the unique global minimum of an associated strictly convex Morse function [J. F. van Diejen and E. Emsiz, Lett. Math. Phys. 109 (2019), pp. 89–112]. The purpose of the present note is to infer that the corresponding gradient flow converges to the roots in question at an exponential rate.
- Published
- 2019
39. Convergence of combinatorial Ricci flows to degenerate circle patterns
- Author
-
Asuka Takatsu
- Subjects
Mathematics - Differential Geometry ,Surface (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Degenerate energy levels ,Geometric Topology (math.GT) ,Ricci flow ,Mathematics - Geometric Topology ,symbols.namesake ,Differential Geometry (math.DG) ,Euler characteristic ,Convergence (routing) ,FOS: Mathematics ,symbols ,Primary 53C44, Secondary 52C26 ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Balanced flow ,Mathematics - Abstract
We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of the questions raised by B. Chow and F. Luo.
- Published
- 2019
40. Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes
- Author
-
Masato Kimura and Matteo Negri
- Subjects
Constraint (information theory) ,Quadratic equation ,Discretization ,Applied Mathematics ,Weak solution ,Convergence (routing) ,Applied mathematics ,Monotonic function ,Uniqueness ,Balanced flow ,Analysis ,Mathematics - Abstract
We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.
- Published
- 2021
41. On the Gradient Flow Formulation of the Lohe Matrix Model with High-Order Polynomial Couplings
- Author
-
Hansol Park and Seung-Yeal Ha
- Subjects
Coupling ,Polynomial ,Mathematical analysis ,Matrix norm ,FOS: Physical sciences ,Order (ring theory) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,State (functional analysis) ,82C10, 82C22, 35B37 ,Matrix (mathematics) ,Balanced flow ,Mathematical Physics ,Hamiltonian (control theory) ,Mathematics - Abstract
We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.
- Published
- 2021
42. Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation
- Author
-
Benjamin Stamm, Pascal Heid, and Thomas P. Wihler
- Subjects
Physics and Astronomy (miscellaneous) ,Degrees of freedom (statistics) ,010103 numerical & computational mathematics ,Energy minimization ,01 natural sciences ,510 Mathematics ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Numerical Analysis ,Applied Mathematics ,Numerical Analysis (math.NA) ,Finite element method ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Gross–Pitaevskii equation ,Modeling and Simulation ,Energy based ,35P30, 47J25, 49M25, 49R05, 65N25, 65N30, 65N50 ,ddc:000 ,A priori and a posteriori ,Balanced flow - Abstract
Journal of computational physics 436, 110165 (2021). doi:10.1016/j.jcp.2021.110165, Published by Elsevier, Amsterdam
- Published
- 2021
43. On the Completely Separable State for the Lohe Tensor Model
- Author
-
Dohyun Kim, Hansol Park, and Seung-Yeal Ha
- Subjects
Kuramoto model ,Mathematical analysis ,Swarm behaviour ,Statistical and Nonlinear Physics ,01 natural sciences ,010305 fluids & plasmas ,Separable state ,Tensor product ,Product (mathematics) ,0103 physical sciences ,Tensor ,Quantum information ,Balanced flow ,010306 general physics ,Mathematical Physics ,Mathematics - Abstract
We study completely separable states of the Lohe tensor model and their asymptotic collective dynamics. Here, the completely separable state means that it is a tensor product of rank-1 tensors. For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020]. We also show that the swarm double sphere model can be represented as a gradient system with an analytic potential. Using this gradient flow formulation, we provide the swarm multisphere model on the product of multiple unit spheres with possibly different dimensions, and then we construct a completely separable state of the swarm multisphere model as a tensor product of rank-1 tensors which is a solution of the proposed swarm multisphere model. This concept of separability has been introduced in the theory of quantum information. Finally, we also provide a sufficient framework leading to the complete aggregation of completely separable states.
- Published
- 2021
44. Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation
- Author
-
Mengxia Ma, Xiaodong Wang, Yong Chai, Jie Ouyang, and Chenhui Zhang
- Subjects
Numerical Analysis ,Applied Mathematics ,Computation ,General Engineering ,Regular polygon ,Function (mathematics) ,Adaptive stepsize ,01 natural sciences ,Stability (probability) ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Range (mathematics) ,Computational Theory and Mathematics ,Applied mathematics ,0101 mathematics ,Balanced flow ,Software ,Energy (signal processing) ,Mathematics - Abstract
A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.
- Published
- 2021
45. Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective
- Author
-
Max O. Souza, Léonard Monsaingeon, Fabio A. C. C. Chalub, and Ana Margarida Ribeiro
- Subjects
Partial differential equation ,Markov chain ,Applied Mathematics ,010102 general mathematics ,Evolutionary game theory ,Computer Science::Computational Geometry ,Heavy traffic approximation ,01 natural sciences ,010101 applied mathematics ,Replicator equation ,Moran process ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii), and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimisation of free energy functionals.
- Published
- 2021
46. Global existence for the p-Sobolev flow
- Author
-
Kenta Nakamura, Tuomo Kuusi, Masashi Misawa, Department of Mathematics and Statistics, and Geometric Analysis and Partial Differential Equations
- Subjects
Mathematics::Analysis of PDEs ,Nonlinear intrinsic scaling transformation ,01 natural sciences ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,CONVERGENCE ,FOS: Mathematics ,111 Mathematics ,0101 mathematics ,Expansion of positivity ,EQUATIONS ,Mathematics ,Euclidean space ,Applied Mathematics ,Yamabe flow ,010102 general mathematics ,Mathematical analysis ,LOCAL BEHAVIOR ,35B45 (35B65 35D30 35K61) ,010101 applied mathematics ,Sobolev space ,Nonlinear system ,p-Sobolev flow ,Flow (mathematics) ,Bounded function ,Balanced flow ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper, we study a doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow. In the special case p=2 our theory includes the classical Yamabe flow on a bounded domain in Euclidean space. Our main aim is to prove the global existence of the p-Sobolev flow together with its qualitative properties., arXiv admin note: text overlap with arXiv:2103.15259
- Published
- 2021
- Full Text
- View/download PDF
47. Elastic flow of networks : Short-time existence result
- Author
-
Anna Dall'Acqua, Paola Pozzi, and Chun Chi Lin
- Subjects
Banach fixed-point theorem ,010102 general mathematics ,Mathematical analysis ,Elastic energy ,01 natural sciences ,Short interval ,010101 applied mathematics ,Parabolic system ,Mathematics - Analysis of PDEs ,Mathematics (miscellaneous) ,Flow (mathematics) ,Mathematik ,FOS: Mathematics ,primary 35K52, secondary 53C44, 35K61, 35K41 ,Boundary value problem ,0101 mathematics ,Balanced flow ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study the $$L^2$$ -gradient flow of the penalized elastic energy on networks of q-curves in $$\mathbb {R}^{n}$$ for $$q \ge 3$$ . Each curve is fixed at one end-point and at the other is joint to the other curves at a movable q-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov’s theory on linear parabolic systems and Banach fixed point theorem in proper Holder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.
- Published
- 2021
48. FreezeNet: Full Performance by Reduced Storage Costs
- Author
-
Jens Mehnert, Paul Wimmer, and Alexandru Paul Condurache
- Subjects
0301 basic medicine ,business.industry ,Computation ,Random seed ,Value (computer science) ,030229 sport sciences ,Backpropagation ,03 medical and health sciences ,030104 developmental biology ,0302 clinical medicine ,Key (cryptography) ,Pruning (decision trees) ,Artificial intelligence ,Balanced flow ,business ,Algorithm ,MNIST database ,Mathematics - Abstract
Pruning generates sparse networks by setting parameters to zero. In this work we improve one-shot pruning methods, applied before training, without adding any additional storage costs while preserving the sparse gradient computations. The main difference to pruning is that we do not sparsify the network’s weights but learn just a few key parameters and keep the other ones fixed at their random initialized value. This mechanism is called freezing the parameters. Those frozen weights can be stored efficiently with a single 32bit random seed number. The parameters to be frozen are determined one-shot by a single for- and backward pass applied before training starts. We call the introduced method FreezeNet. In our experiments we show that FreezeNets achieve good results, especially for extreme freezing rates. Freezing weights preserves the gradient flow throughout the network and consequently, FreezeNets train better and have an increased capacity compared to their pruned counterparts. On the classification tasks MNIST and CIFAR-10/100 we outperform SNIP, in this setting the best reported one-shot pruning method, applied before training. On MNIST, FreezeNet achieves \(99.2\%\) performance of the baseline LeNet-5-Caffe architecture, while compressing the number of trained and stored parameters by a factor of \(\times 157\).
- Published
- 2021
49. Analysis of stochastic gradient descent in continuous time
- Author
-
Latz, Jonas, Latz, J [0000-0002-4600-0247], Apollo - University of Cambridge Repository, and Latz, Jonas [0000-0002-4600-0247]
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Computer Science - Machine Learning ,Discretization ,Markov process ,010103 numerical & computational mathematics ,Dynamical system ,01 natural sciences ,Statistics - Computation ,Article ,Machine Learning (cs.LG) ,Theoretical Computer Science ,010104 statistics & probability ,symbols.namesake ,60J25 ,FOS: Mathematics ,Applied mathematics ,Ergodic theory ,Mathematics - Numerical Analysis ,Wasserstein distance ,0101 mathematics ,Computation (stat.CO) ,Mathematics ,37A25 ,Probability (math.PR) ,Ergodicity ,90C30 ,Stochastic optimisation ,Numerical Analysis (math.NA) ,68W20 ,Stochastic gradient descent ,Computational Theory and Mathematics ,65C40 ,symbols ,90C30, 60J25, 37A25, 65C40, 68W20 ,Statistics, Probability and Uncertainty ,Balanced flow ,Convex function ,Gradient descent ,Piecewise-deterministic Markov processes ,Mathematics - Probability - Abstract
Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time representation of stochastic gradient descent. The stochastic gradient process is a dynamical system that is coupled with a continuous-time Markov process living on a finite state space. The dynamical system—a gradient flow—represents the gradient descent part, the process on the finite state space represents the random subsampling. Processes of this type are, for instance, used to model clonal populations in fluctuating environments. After introducing it, we study theoretical properties of the stochastic gradient process: We show that it converges weakly to the gradient flow with respect to the full target function, as the learning rate approaches zero. We give conditions under which the stochastic gradient process with constant learning rate is exponentially ergodic in the Wasserstein sense. Then we study the case, where the learning rate goes to zero sufficiently slowly and the single target functions are strongly convex. In this case, the process converges weakly to the point mass concentrated in the global minimum of the full target function; indicating consistency of the method. We conclude after a discussion of discretisation strategies for the stochastic gradient process and numerical experiments.
- Published
- 2021
50. Fast and Efficient Numerical Finite Difference Method for Multiphase Image Segmentation
- Author
-
Junseok Kim, Jian Wang, Chaeyoung Lee, Yibao Li, Sangkwon Kim, Hyundong Kim, Sungha Yoon, and Jintae Park
- Subjects
Article Subject ,Computer science ,General Mathematics ,General Engineering ,Process (computing) ,Stability (learning theory) ,Finite difference method ,Initialization ,Image segmentation ,Engineering (General). Civil engineering (General) ,symbols.namesake ,Goodness of fit ,Euler's formula ,symbols ,QA1-939 ,Balanced flow ,TA1-2040 ,Algorithm ,Mathematics - Abstract
We present a simple numerical solution algorithm for a gradient flow for the Modica–Mortola functional and numerically investigate its dynamics. The proposed numerical algorithm involves both the operator splitting and the explicit Euler methods. A time step formula is derived from the stability analysis, and the goodness of fit of transition width is tested. We perform various numerical experiments to investigate the property of the gradient flow equation, to verify the characteristics of our method in the image segmentation application, and to analyze the effect of parameters. In particular, we propose an initialization process based on target objects. Furthermore, we conduct comparison tests in order to check the performance of our proposed method.
- Published
- 2021
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