Spatio-temporal motion segmentation via level set partial differential equations

2003 ◽  
Author(s):  
A.-R. Mansouri ◽  
A. Mitiche ◽  
R. El-Feghali
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Level Set ◽  
Motion Segmentation ◽  
Partial Differential ◽  
Spatio Temporal

Invariant solutions of fractional-order spatio-temporal partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nkosingiphile Mnguni ◽  
Sameerah Jamal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Growth Models ◽  
Spatial Patterning ◽  
Partial Differential ◽  
Symmetry Approach ◽  
Time Component ◽  
Spatio Temporal ◽  
Lie Symmetry Approach

Abstract This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

Computation of geometric partial differential equations and mean curvature flow

Acta Numerica ◽  
2005 ◽  
Vol 14 ◽  
pp. 139-232 ◽  
Author(s):  
Klaus Deckelnick ◽  
Gerhard Dziuk ◽  
Charles M. Elliott
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Level Set ◽  
Phase Field ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Partial Differential ◽  
Geometric Partial Differential Equations ◽  
Geometric Pdes

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Spatio-Temporal Estimation of Wildfire Growth

2013 ◽  
Author(s):  
Balaji R. Sharma ◽  
Manish Kumar ◽  
Kelly Cohen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Growth Model ◽  
Wildland Fire ◽  
Time Estimation ◽  
Fire Growth ◽  
Growth Data ◽  
Partial Differential ◽  
Spatio Temporal

This work presents a methodology for real-time estimation of wildland fire growth, utilizing afire growth model based on a set of partial differential equations for prediction, and harnessing concepts of space-time Kalman filtering and Proper Orthogonal Decomposition techniques towards low dimensional estimation of potentially large spatio-temporal states. The estimation framework is discussed in its criticality towards potential applications such as forest fire surveillance with unmanned systems equipped with onboard sensor suites. The effectiveness of the estimation process is evaluated numerically over fire growth data simulated using a well-established fire growth model described by coupled partial differential equations. The methodology is shown to be fairly accurate in estimating spatio-temporal process states through noise-ridden measurements for real-time deploy ability.

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