Level-set method: Difference between revisions

Revision as of 12:54, 16 January 2024

Revision as of 18:03, 18 January 2024

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[[Image:level set method.png|thumb|right|400px|An illustration of the level-set method]]

The figure on the right illustrates several ideas about the level-set method. In the upper-left corner we see a shape; that is, a [[bounded region]] with a well-behaved boundary. Below it, the red surface is the graph of a level set function <math>\varphi</math> determining this shape, and the flat blue region represents the ''X-Y'' plane. The boundary of the shape is then the zero-level set of <math>\varphi</math>, while the shape itself is the set of points in the plane for which <math>\varphi</math> is positive (interior of the shape) or zero (at the boundary).

In the top row, the shape can be seen changing its topology by splitting in two. It would be difficult to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. One would need an algorithm able to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves. On the bottom row, however, the level set function accomplishes this change by translating downward. This is an example of when it can be easier to work with a shape through its level-set function than with the shape directly, where the method would need to consider and handle all the possible deformations the shape might undergo.

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Here, <math>|\cdot|</math> is the [[Euclidean norm]] (denoted customarily by single bars in PDEs), and <math>t</math> is time. This is a [[partial differential equation]], in particular a [[Hamilton–Jacobi equation]], and can be solved numerically, for example, by using [[finite difference]]s on a Cartesian grid.<ref name=osher>{{cite book |last=Osher |first=Stanley J. |authorlink = Stanley Osher |author2=Fedkiw, Ronald P. |authorlink2=Ronald Fedkiw |title=Level Set Methods and Dynamic Implicit Surfaces|publisher=[[Springer-Verlag]] |year=2002 |isbn= 978-0-387-95482-0}}</ref><ref name=sethian>{{cite book |last=Sethian |first=James A. |authorlink = James Sethian |title= Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science|publisher=[[Cambridge University Press]] |year=1999 |isbn= 978-0-521-64557-7}}</ref>

However, numerical solution of the level set equation may require complex techniques. Simple finite difference methods fail quickly. [[Upwind scheme|Upwinding]] methods such as the [[Godunov method]] are considered better; however, the level set method does not guarantee preservation of the volume and shape of the set level in an advection field that maintains shape and size, for example a uniform or [[rotational velocity]] field. Instead, the shape of the level set may become distorted and the level set may disappear over a few time steps. Therefore, high-order finite difference schemes, such as high-order essentially non-oscillatory (ENO) schemes, are often required, and even then the feasibility of long-term simulations is questionable. More complex methods have been developed to overcome this; for example, combinations of the leveling method with tracking marker particles suggested by the velocity field.<ref>{{Citation |last1 = Enright |first1 = D. |last2 = Fedkiw |first2 = R. P.| last3 = Ferziger |first3 = J. H. |authorlink3 = Joel H. Ferziger| last4 = Mitchell |first4 = I.| title = A hybrid particle level set method for improved interface capturing| journal = J. Comput. Phys.| volume = 183 |issue = 1 |year = 2002 |pages = 83&ndash;116| url = http://www.cs.ubc.ca/~mitchell/Papers/myJCP02.pdf |doi=10.1006/jcph.2002.7166|bibcode = 2002JCoPh.183...83E |citeseerx = 10.1.1.15.910}}</ref>

==Example==

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* [[Computational fluid dynamics]]

* [[Combustion]]

* [[Trajectory|Trajectory planning]]

* [[Mathematical optimization|Optimization]]

* [[Image processing]]

* [[Computational biophysics]]

* Discrete [[complex dynamics]]: visulalisationvisualization of [[b:Fractals/Iterations in the complex plane/Mandelbrot set|parameter plane]] and [[b:Fractals/Iterations in the complex plane/Julia set|dynamic plane]]

==See also==