456 results on '"Balanced flow"'
Search Results
2. High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model
- Author
-
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 gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints
- Author
-
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
4. Energy-production-rate preserving numerical approximations to network generating partial differential equations
- Author
-
Qi Wang, Jia Zhao, and Qi Hong
- Subjects
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
5. Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach
- Author
-
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
6. Asymptotic behavior of gradient flows on the unit sphere with various potentials
- Author
-
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
7. Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions
- Author
-
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
8. Normalized Gradient Flow with Lagrange Multiplier for Computing Ground States of Bose--Einstein Condensates
- Author
-
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
9. Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks
- Author
-
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
10. Short retractions of CAT(1) spaces
- Author
-
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
11. Uniqueness and nonuniqueness of limits of teichmuller harmonic map flow
- Author
-
Melanie Rupflin, James Kohout, and Peter M. Topping
- Subjects
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
12. A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals
- Author
-
Juan Pablo Borthagaray, Ricardo H. Nochetto, and Shawn W. Walker
- Subjects
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
13. Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries
- Author
-
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
14. The gradient flow of the Möbius energy : 𝜀-regularity and consequences
- Author
-
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
15. Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation
- Author
-
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
16. A Second Order Gradient Flow of p-Elastic Planar Networks
- Author
-
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
17. A Highly Efficient and Accurate New Scalar Auxiliary Variable Approach for Gradient Flows
- Author
-
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
18. Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation
- Author
-
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
19. Unconditionally Bound Preserving and Energy Dissipative Schemes for a Class of Keller--Segel Equations
- Author
-
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
20. 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
21. Fast, unconditionally energy stable large time stepping method for a new Allen–Cahn type square phase-field crystal model
- Author
-
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
22. 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
23. A variant of scalar auxiliary variable approaches for gradient flows
- Author
-
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
24. Accelerated Information Gradient Flow
- Author
-
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
25. Local Energy Dissipation Rate Preserving Approximations to Driven Gradient Flows with Applications to Graphene Growth
- Author
-
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
26. On nonnegative solutions for the Functionalized Cahn–Hilliard equation with degenerate mobility
- Author
-
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
27. 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
28. 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
29. 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
30. 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
31. 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
32. 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
33. Nonlinear systems coupled through multi-marginal transport problems
- Author
-
Maxime Laborde, Department of Mathematics and Statistics [Montréal], and McGill University = Université McGill [Montréal, Canada]
- Subjects
Applied Mathematics ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,01 natural sciences ,Convexity ,010101 applied mathematics ,Nonlinear system ,Mathematics - Analysis of PDEs ,Dimension (vector space) ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Applied mathematics ,Uniqueness ,0101 mathematics ,Balanced flow ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.
- Published
- 2019
34. Biomembrane modeling with isogeometric analysis
- Author
-
Luca Dedè, Alfio Quarteroni, and Andrea Bartezzaghi
- Subjects
Backward differentiation formulas ,Canham–Helfrich energy ,Geometric partial differential equation ,Isogeometric analysis ,Lagrange multiplier ,Lipid biomembrane ,Computational Mechanics ,Mechanics of Materials ,Mechanical Engineering ,Physics and Astronomy (all) ,Computer Science Applications1707 Computer Vision and Pattern Recognition ,Discretization ,erythrocyte cytoskeleton ,finite-element-method ,General Physics and Astronomy ,010103 numerical & computational mathematics ,shape ,Energy minimization ,01 natural sciences ,canham-helfrich energy ,large-deformation ,Quantitative Biology::Subcellular Processes ,symbols.namesake ,Computational mechanics ,Applied mathematics ,0101 mathematics ,bending energy ,Mathematics ,Physics::Biological Physics ,Partial differential equation ,Computer Science Applications ,010101 applied mathematics ,Quantitative Biology::Quantitative Methods ,Nonlinear system ,membranes ,flow ,partial-differential-equations ,symbols ,simulations ,Balanced flow ,bilayers - Abstract
We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.
- Published
- 2019
35. Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics
- Author
-
Stefano Almi, Sandro Belz, and Matteo Negri
- Subjects
Pointwise ,Numerical Analysis ,Discretization ,Truncation ,Applied Mathematics ,Finite element method ,Sobolev space ,Computational Mathematics ,Modeling and Simulation ,Convergence (routing) ,Applied mathematics ,Balanced flow ,Analysis ,Mathematics ,Variable (mathematics) - Abstract
We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovered a posteriori by a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateral L2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.
- Published
- 2019
36. Convergence of Riemannian 4-manifolds with L2L^{2}-curvature bounds
- Author
-
Norman Zergänge
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Einstein manifold ,Curvature ,01 natural sciences ,0103 physical sciences ,Convergence (routing) ,010307 mathematical physics ,0101 mathematics ,Balanced flow ,Critical exponent ,Analysis ,Geometry and topology ,Mathematics - Abstract
In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L 2 {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L 2 {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called L 2 {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the L 2 {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.
- Published
- 2019
37. Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension
- Author
-
Luis Miguel Villada, Raimund Bürger, Pep Mulet, and Daniel Inzunza
- Subjects
Computational Mathematics ,Numerical Analysis ,Nonlinear system ,Implicit explicit ,Applied Mathematics ,Mathematical analysis ,Space dimension ,Structure (category theory) ,Balanced flow ,Analysis ,Mathematics - Published
- 2019
38. Optimal $L^1$-type Relaxation Rates for the Cahn--Hilliard Equation on the Line
- Author
-
Sebastian Scholtes, Maria G. Westdickenberg, and Felix Otto
- Subjects
Applied Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Type (model theory) ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Computational Mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Relaxation (physics) ,0101 mathematics ,Balanced flow ,Cahn–Hilliard equation ,Nonlinear Sciences::Pattern Formation and Solitons ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics ,Line (formation) - Abstract
In this paper we derive optimal algebraic-in-time relaxation rates to the kink for the Cahn-Hilliard equation on the line. We assume that the initial data have a finite distance---in terms of either a first moment or the excess mass---to a kink profile and capture the decay rate of the energy and the perturbation. Our tools include Nash-type inequalities, duality arguments, and Schauder estimates.
- Published
- 2019
39. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem
- Author
-
Mikaela Iacobelli
- Subjects
Asymptotic analysis ,Diffusion equation ,Exponential convergence ,Applied Mathematics ,Quantization (signal processing) ,Stability (probability) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Periodic boundary conditions ,Balanced flow ,Diffusion (business) ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in [ 3 ]. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for \begin{document}$ W_2 $\end{document} -stability of solutions.
- Published
- 2019
40. Asymptotic stability of local Helfrich minimizers
- Author
-
Daniel Lengeler
- Subjects
35Q92, 35B35, 35Q74, 35K25, 76D27 ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Geometric flow ,01 natural sciences ,Stability (probability) ,Condensed Matter::Soft Condensed Matter ,Quantitative Biology::Subcellular Processes ,Quantitative Biology::Quantitative Methods ,Willmore energy ,Mathematics - Analysis of PDEs ,Exponential stability ,Stability theory ,0103 physical sciences ,FOS: Mathematics ,Fluid dynamics ,010307 mathematical physics ,0101 mathematics ,Balanced flow ,Well posedness ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([12, 11]). The proof is based on a Lojasiewicz-Simon inequality.
- Published
- 2018
41. Convergences of the squareroot approximation scheme to the Fokker–Planck operator
- Author
-
Martin Heida
- Subjects
Finite volume method ,Applied Mathematics ,Operator (physics) ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Modeling and Simulation ,Scheme (mathematics) ,Convergence (routing) ,Applied mathematics ,Fokker–Planck equation ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.
- Published
- 2018
42. Entropy method for generalized Poisson–Nernst–Planck equations
- Author
-
Victor A. Kovtunenko and José Rodrigo González Granada
- Subjects
Algebra and Number Theory ,Simplex ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Entropy (classical thermodynamics) ,Nonlinear system ,Variational principle ,symbols ,Applied mathematics ,Nernst equation ,0101 mathematics ,Balanced flow ,Conservation of mass ,Mathematical Physics ,Analysis ,Mathematics - Abstract
A proper mathematical model given by nonlinear Poisson–Nernst–Planck (PNP) equations which describe electrokinetics of charged species is considered. The model is generalized with entropy variables associating the pressure and quasi-Fermi electro-chemical potentials in order to adhere to the law of conservation of mass. Based on a variational principle for suitable free energy, the generalized PNP system is endowed with the structure of a gradient flow. The well-posedness theorems for the mixed formulation (using the entropy variables) of the gradient-flow problem are provided within the Gibbs simplex and supported by a-priori estimates of the solution.
- Published
- 2018
43. 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
44. 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
45. 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
46. 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
47. Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system
- Author
-
Virginie Ehrlacher, Greta Marino, Jan-Frederik Pietschmann, MATHematics for MatERIALS (MATHERIALS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC), Chemnitz University of Technology / Technische Universität Chemnitz, VE acknowledges support from the ANR JCJC project COMODO (ANR-19-CE46-0002) and from the PHC PROCOPE project Number 42632VA. GM and JFP were supported by the DAADvia the PPP Grant No. 57447206, ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- Subjects
Cross-diffusion ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Global existence ,Entropy (arrow of time) ,Mathematics ,Cahn-Hilliard ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Degenerate energy levels ,Zero (complex analysis) ,Degenerate Ginzburg-Landau ,Extension (predicate logic) ,Weak solutions ,010101 applied mathematics ,35D30, 35G31, 35G50 ,1991Mathematics Subject Classification.35D30, 35G31, 35G50 ,Balanced flow ,Analysis ,Analysis of PDEs (math.AP) - Abstract
The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment., 33 pages, comments are welcome
- Published
- 2021
48. On the nonlocal Cahn–Hilliard equation with nonlocal dynamic boundary condition and boundary penalization
- Author
-
Patrik Knopf and Andrea Signori
- Subjects
Surface (mathematics) ,Dynamic boundary conditions ,Reaction rates ,Asymptotic analysis ,Boundary (topology) ,FOS: Physical sciences ,Robin boundary conditions ,Type (model theory) ,01 natural sciences ,Mathematics - Analysis of PDEs ,Phase (matter) ,Gradient flow ,FOS: Mathematics ,Boundary value problem ,0101 mathematics ,Cahn–Hilliard equation ,Mathematical Physics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,35A01, 35A02, 35A15, 35K61, 35B40 ,Mathematical Physics (math-ph) ,Nonlocal Cahn–Hilliard equation ,010101 applied mathematics ,Relaxation (approximation) ,Balanced flow ,Analysis ,Analysis of PDEs (math.AP) - Abstract
The Cahn--Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. In recent times, various dynamic boundary conditions have been introduced to model interactions of the materials with the boundary more precisely. To take long-range interactions of the materials into account, we propose a new model consisting of a nonlocal Cahn--Hilliard equation subject to a nonlocal dynamic boundary condition that is also of Cahn--Hilliard type and contains an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal total free energy with respect to a suitable inner product of order $H^{-1}$ which contains both bulk and surface contributions. The total free energy is considered as nonlocal since it comprises convolutions in the bulk and on the surface of the phase-field variables with certain interaction kernels. The main difficulties arise from defining a suitable kernel on the surface and from handling the resulting boundary convolution. In the main model, the chemical potentials in the bulk and on the surface are coupled by a Robin type boundary condition depending on a specific relaxation parameter related to the rate of chemical reactions. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when the relaxation parameter tends to zero or infinity. By this approach, we also obtain weak and strong well-posedness of the corresponding limit systems.
- Published
- 2021
49. Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
- Author
-
Harald Garcke and Robert Nürnberg
- Subjects
Mathematics - Differential Geometry ,Mean curvature flow ,Geodesic ,Curve-shortening flow ,Applied Mathematics ,Mathematical analysis ,Numerical Analysis (math.NA) ,Weak formulation ,Curvature ,Computational Mathematics ,Differential Geometry (math.DG) ,Flow (mathematics) ,FOS: Mathematics ,Boundary value problem ,Mathematics::Differential Geometry ,Mathematics - Numerical Analysis ,Balanced flow ,Mathematics - Abstract
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetric self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models., 42 pages, 21 figures
- Published
- 2020
50. Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations
- Author
-
Jiang Yang, Xiao Li, Zhonghua Qiao, and Lili Ju
- Subjects
Pointwise ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Quantitative Biology::Genomics ,01 natural sciences ,Parabolic partial differential equation ,Computer Science Applications ,Integrating factor ,010101 applied mathematics ,Computational Mathematics ,Runge–Kutta methods ,Modeling and Simulation ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Boundary value problem ,Mathematics - Numerical Analysis ,0101 mathematics ,Balanced flow ,Mathematics - Abstract
A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. The MBP plays a crucial role in understanding the physical meaning and the wellposedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge–Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen–Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen–Cahn type of equations.
- Published
- 2020
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.