Interpolation and approximation for data living on manifold surfaces

Meshed surfaces are ubiquitous in digital geometry processing and computer graphics. The set of attributes associated with each vertex such as the vertex locations, curvature, temperature, pressure or saliency, can be recognized as data living on mani- fold surfaces. So interpolation and approximation for these data are of general interest. This paper presents two approaches for mani- fold data interpolation and approximation through the properties of Laplace-Beltrami operator （Laplace operator defined on a mani- fold surface）. The first one is to use Laplace operator minimizing the membrane energy of a scalar function defined on a manifold. The second one is to use bi-Laplace operator minimizing the thin plate energy of a scalar function defined on a manifold. These two approaches can process data living on high genus meshed surfaces. The approach based on Laplace operator is more suitable for manifold data approximation and can be applied manifold data smoothing, while the one based on bi-Laplace operator is more suit- able for manifold data interpolation and can be applied image extremal envelope computation. All the application examples demon- strate that our procedures are robust and efficient.

A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

In this article the authors propose a new approximate inertial manifold（AIM） to the Navier-Stokes equations. The solutions are in the neighborhoods of this AIM with thickness δ=o（h^2k＋1-ε）. The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solution ｜｜｜（u-uh^＊·,p-ph^＊·）｜｜｜≤C（h^2k＋1-ε＋h^＊（m＋1））,where （h, h＊） and （k, m） are co^trse and fine meshes and degree of finite element subspa^es, respectively. These results are much better them Standard G^tlerkin（SG） and nonlinear Galcrkin （NG） methods. For example, for 2D NS eqs and linear element, let uh,u^h, u^＊ be the SG, NG and their approximate solutions respectively, then ｜｜u-uh｜｜1≤Ch,｜｜u-u^h｜｜i≤Ch^2,｜｜u-u^＊｜｜1≤Ch^3,and h^＊ ≈ h^2 for NG, h^＊ ≈ h^3/2 for theirs.

LOCALIZATION AND APPROXIMATION OF ATTRACTORS FOR THE KURAMOTO-SIVASHINSKY EQUATIONS

The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood into which the orbits enter with an exponential speed and in a finite time is associated with each manifold. The thickness of these neighborhoods decreases with increasing m for a fixed order j. Besides, the neighborhoods localize the global attractor and aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.

Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms

This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.

APPROXIMATE INERTIAL MANIFOLDS UNDERNONSELFADJOINT

This paper sets up the approximate inertias manifold(AIM) in the nouselfadjoint nonlinear evolutionary equation and Ands AIMs which are explitly dafined in the weally damped forced KdV equation (WDF KdV).

CONVERGENT FAMILIES OF APPROXIMATE INERTIAL MANIFOLDS FOR NONAUTONOMOUS EVOLUTION EQUATIONS

In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is studied. A family of convergent approximate inertial manifolds for a class of evolution equations has been constructed when the spectral gap condition is satisfied.

WAVELET APPROXIMATE INERTIAL MANIFOLD AND NUMERICAL SOLUTION OF BURGERS' EQUATION

The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation has a good localization property of the numerical solution distinguishably.

EAPPROXIMATE INERTIAL MANIFOLDS FOR THE SYSTEM OF THE J-J EQUATIONS

In this paper foe Liapunov functionals has been constructed.the decay property of the high dimensional modes of the J-J equations in the Josephson junctions is obtained,and thus the approxtmate inertial manifolds are given.

Actor-Critic Reinforcement Learning and Application in Developing Computer-Vision-Based Interface Tracking

This paper synchronizes control theory with computer vision by formalizing object tracking as a sequential decision-making process.A reinforcement learning(RL)agent successfully tracks an interface between two liquids,which is often a critical variable to track in many chemical,petrochemical,metallurgical,and oil industries.This method utilizes less than 100 images for creating an environment,from which the agent generates its own data without the need for expert knowledge.Unlike supervised learning(SL)methods that rely on a huge number of parameters,this approach requires far fewer parameters,which naturally reduces its maintenance cost.Besides its frugal nature,the agent is robust to environmental uncertainties such as occlusion,intensity changes,and excessive noise.From a closed-loop control context,an interface location-based deviation is chosen as the optimization goal during training.The methodology showcases RL for real-time object-tracking applications in the oil sands industry.Along with a presentation of the interface tracking problem,this paper provides a detailed review of one of the most effective RL methodologies:actor–critic policy.

A SIMPLE POSTPROCESS PROCEDURE FOR GALERKIN METHOD

A kind simple postprocess procedure for classical Galerkin method for steady Navier_Stokes equations with stream function form was presented in this paper. The main ideal was to construct an approximate interactive rule between lower frequency components and higher frequency components by using the conception of Approximate Inertial Manifold(AIM) and a kind of new decomposition of the true solution. It is demonstrated in this paper that this kind of postprocess Galerkin method could derive a higher accuracy solution with lower computing efforts.

THE RESEARCH OF LONGTIME DYNAMIC BEHAVIOR IN WEAKLY DAMPED FORCED KdV EQUATION

It is presented that there exists approximate inertial manifolds in weakly damped forced Kdv equation with with periodic boundary conditionsIIbns. The approximate inertial manifolds provide approximant of the attractror by finite dimensional smooth manifolds which are exphcitly defined And the concepl leads to new numerical schemes which are well adapted to the longtime behavior of dynamical system.

NUMERICAL ANALYSIS OF LONGTIME DYNAMIC BEHAVIOR IN WEAKLY DAMPED FORCED KdV EQUATION

The numerical analysis of the approximate inertial manifold in,weakly damped forced KdV equation is given. The results of numerical analysis under five models is the same as that of nonlinear spectral analysis.

WAVELET BASIS ANALYSIS IN PERTURBEDPERIODIC KdV EQUATION

In the paper by using the spline wavelet basis to constructr the approximate inertial manifold, we study the longtime behavior of perturbed perodic KdV equation.

FULL DISCRETE NONLINEAR GALERKIN METHOD FOR THE NAVIER-STOKES EQUATIONS

This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.

A class of epidemic virus transmission population dynamic system

A class of epidemic virus transmission population dynamic system is considered. Firstly, using the functional homotopic analysis method, an initial approximate function is selected. Then, the arbitrary order approximate analytic solutions are obtained successively. Finally, the accuracy of the obtained approximate analytic solutions is described. The influence of the various physical parameters for the epidemic virus transmission population dynamic system is discussed.

APPROXIMATE INERTIAL MANIFOLDS TO THE NAVIER-STOKES EQUATIONS

In this paper,we construct an infinite family of approximate inertial manifolds for the Navier-Stokes equations.These manifolds provide higher and higher order approximations to the attractor.Our manifolds are constructed by contraction principle and therefore can be easily approximated by simple explicit functions in real computations.

Approximate Inertial Manifolds to the Generalized Symmetric Regularized Long Wave Equations with Damping Term

Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.

GEVREY CLASS REGULARITY AND APPROXIMATE INERTIAL MANIFOLDS FOR THE NEWTON-BOUSSINESQ EQUATIONS

The authors show the Gevrey class regularity of the solutions for the two-dimensional Newton-Boussinesq Equations. Based on this fact, an approximate inertial manifold for the system is constructed, which attracts all solutions to an exponentially thin neighborhood of it in a finite time.

Long-Time Behaviour of the Solutions for the Multidimensional Kolmogorov-Spieqel-Sivashinsky Equation

In this paper, we study the existence and long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation. We first show the existence of the global solution for this equation, and then prove the existence of the global attractor and establish the esti- mates of the upper bounds of Hausdorff and fractal dimensions for the attractor. We also obtain the Gevrey class regularity for the solutions and construct an approximate inertial manifold for the system.