scholarly journals Mass-conserving diffusion-based dynamics on graphs

2021 ◽  
pp. 1-49
Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP
Keyword(s):  
Multiscale Modeling ◽  
Classification Problems ◽  
Theoretical Justification ◽  
Finite Graphs ◽  
Discrete Scheme ◽  
Cahn Equation ◽  
Separating Flows ◽  
Special Case ◽  
Appl Math ◽  
Convergence Of The Scheme

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

The Adjustment of Incomplete Direction Observations

1965 ◽  
Vol 19 (5) ◽  
pp. 462-471
Author(s):  
G. G. Bennett
Keyword(s):  
Iterative Method ◽  
Approximate Method ◽  
Theoretical Justification ◽  
Special Case

An iterative method for the solution of the problem of adjusting incomplete direction observations has been known and practised for many years, apparently without theoretical justification. The method is shown to be theoretically sound and convergent in all cases. In fact, the approximate method may be classed as a general case, which includes the special case of the adjustment of complete direction observations.

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

On the Rates of Convergence of Chlodovsky–Durrmeyer Operators and their Bézier Variant

2009 ◽  
Vol 16 (4) ◽  
pp. 693-704
Author(s):  
Harun Karsli ◽  
Paulina Pych-Taberska
Keyword(s):  
Recent Result ◽  
Pointwise Convergence ◽  
Rates Of Convergence ◽  
Durrmeyer Operators ◽  
The One ◽  
Special Case ◽  
Appl Math ◽  
Bézier Variant

Abstract We consider the Bézier variant of Chlodovsky–Durrmeyer operators 𝐷𝑛,α for functions 𝑓 measurable and locally bounded on the interval [0,∞). By using the Chanturia modulus of variation we estimate the rate of pointwise convergence of (𝐷𝑛,α 𝑓) (𝑥) at those 𝑥 > 0 at which the one-sided limits 𝑓(𝑥+), 𝑓(𝑥–) exist. In the special case α = 1 the recent result of [Ibikli, Karsli, J. Inequal. Pure Appl. Math. 6: 12, 2005] concerning the Chlodovsky–Durrmeyer operators 𝐷𝑛 is essentially improved and extended to more general classes of functions.

scholarly journals Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 1-26 ◽  
Author(s):  
Raffaele Folino
Keyword(s):  
Initial Velocity ◽  
Numerical Experiments ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Slow Motion ◽  
Cahn Equation ◽  
Parabolic Case ◽  
Appl Math ◽  
One Space Dimension

The aim of this paper is to prove that, for specific initial data [Formula: see text] and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval [Formula: see text] shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990) 983–997], if [Formula: see text] is the diffusion coefficient, we show that in a time scale of order [Formula: see text] nothing happens and the solution maintains the same number of transitions of its initial datum [Formula: see text]. The novelty consists mainly in the role of the initial velocity [Formula: see text], which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.

scholarly journals GLOBAL ASYMPTOTIC STABILITY FOR GENERAL SYMMETRIC RATIONAL DIFFERENCE EQUATIONS

2009 ◽  
Vol 81 (2) ◽  
pp. 251-259 ◽  
Author(s):  
CONG ZHANG ◽  
HONG-XU LI ◽  
NAN-JING HUANG
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Global Attractivity ◽  
Positive Equilibrium ◽  
Globally Asymptotically Stable ◽  
Rational Difference Equations ◽  
Rational Difference Equation ◽  
Special Case ◽  
Appl Math

AbstractWe investigate the global asymptotic stability for positive solutions to a class of general symmetric rational difference equations and prove that the unique positive equilibrium 1 of the general symmetric rational difference equations is globally asymptotically stable. As a special case of our result, we solve the conjecture raised by Berenhaut, Foley and Stević [‘The global attractivity of the rational difference equationyn=(yn−k+yn−m)/(1+yn−kyn−m)’,Appl. Math. Lett.20(2007), 54–58].

Influence of corner layers in the variational determination of bubble solutions of the constrained Allen–Cahn equation

2008 ◽  
Vol 19 (5) ◽  
pp. 561-574
Author(s):  
M. C. JORGE ◽  
A. A. MINZONI ◽  
C. A. VARGAS
Keyword(s):  
Variational Approach ◽  
Bubble Radius ◽  
Mass Conservation ◽  
Normal Derivative ◽  
Leading Order ◽  
Cahn Equation ◽  
Level Lines ◽  
Corner Layer ◽  
Appl Math

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

scholarly journals On a discrete scheme for time fractional fully nonlinear evolution equations

2020 ◽  
Vol 120 (1-2) ◽  
pp. 151-162 ◽  
Author(s):  
Yoshikazu Giga ◽  
Qing Liu ◽  
Hiroyoshi Mitake
Keyword(s):  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Discrete Scheme ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Convergence Of The Scheme

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

Fractional-Order Extreme Learning Machine With Mittag-Leffler Distribution

2019 ◽  
Author(s):  
Haoyu Niu ◽  
Yuquan Chen ◽  
YangQuan Chen
Keyword(s):  
Extreme Learning Machine ◽  
Experimental Result ◽  
Classification Problems ◽  
Lévy Distribution ◽  
Levy Distribution ◽  
Heavy Tailed Distribution ◽  
Heavy Tailed ◽  
Learning Machine ◽  
Hidden Neurons ◽  
Special Case

Abstract Extreme Learning Machine (ELM) has a powerful capability to approximate the regression and classification problems for a lot of data. ELM does not need to learn parameters in hidden neurons, which enables ELM to learn a thousand times faster than conventional popular learning algorithms. Since the parameters in the hidden layers are randomly generated, what is the optimal randomness? Lévy distribution, a heavy-tailed distribution, has been shown to be the optimal randomness in an unknown environment for finding some targets. Thus, Lévy distribution is used to generate the parameters in the hidden layers (more likely to reach the optimal parameters) and better computational results are then derived. Since Lévy distribution is a special case of Mittag-Leffler distribution, in this paper, the Mittag-Leffler distribution is used in order to get better performance. We show the procedure of generating the Mittag-Leffler distribution and then the training algorithm using Mittag-Leffler distribution is given. The experimental result shows that the Mittag-Leffler distribution performs similarly as the Lévy distribution, both can reach better performance than the conventional method. Some detailed discussions are finally presented to explain the experimental results.

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