|
Fast Marching
The fast marching methodJ.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591--1595, 1996/ref> is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation: : , \nabla u(x), =1/f(x) \text x \in \Omega : u(x) = 0 \text x \in \partial \Omega Typically, such a problem describes the evolution of a closed surface as a function of time u with speed f in the normal direction at a point x on the propagating surface. The speed function is specified, and the time at which the contour crosses a point x is obtained by solving the equation. Alternatively, u(x) can be thought of as the minimum amount of time it would take to reach \partial\Omega starting from the point x. The fast marching method takes advantage of this optimal control interpretation of the problem in order to build a solution outwards starting from the "known information", i.e. the boundary values. The algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Numerical Method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathematical definition Let F(x,y)=0 be a well-posed problem, i.e. F:X \times Y \rightarrow \mathbb is a real or complex functional relationship, defined on the Cartesian product of an input data set X and an output data set Y, such that exists a locally lipschitz function g:X \rightarrow Y called resolvent, which has the property that for every root (x,y) of F, y=g(x). We define numerical method for the approximation of F(x,y)=0, the sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ... of problems : \left \_ = \left \_, with F_n:X_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |