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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 Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 cross-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 calle ... of problems : \left \_ = \left \_, with F_n:X_n \times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Sethian
James Albert Sethian is a professor of mathematics at the University of California, Berkeley and the head of the Mathematics Grouat the United States Department of Energy, United States Department of Energy's Lawrence Berkeley National Laboratory. Sethian was born in Washington, D.C. on May 10, 1954. He received a B.A. (1976) from Princeton and a M.A. (1978) and Ph.D (1982) from Berkeley under the direction of Alexandre Chorin. Beginning in 1983, he was a National Science Foundation postdoctoral fellow, lastly at the Courant Institute under Peter Lax. In 1985, he returned to Berkeley to join the mathematics faculty, where he is currently a full professor. Sethian was elected member of the National Academy of Engineering in 2008 as well as the National Academy of Sciences in 2013. Sethian has acted as Interim Director Research at Thinking Machines Corporation and held visiting positions at the National Center for Atmospheric Research and the National Institute of Standards and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Value Problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eikonal Equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation. The classical eikonal equation in geometric optics is a differential equation of the form where x lies in an open subset of \mathbb^n, n(x) is a positive function, \nabla denotes the gradient, and , \cdot , is the Euclidean norm. The function n is given and one seeks solutions u . In the context of geometric optics, the function n is the refractive index of the medium. More generally, an eikonal equation is an equation of the form where H is a function of 2n variables. Here the function H is given, and u is the solution. If H(x,y)= , y, - n(x) , then equation () becomes (). Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics. One fast computational algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dijkstra's Algorithm
Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. The algorithm exists in many variants. Dijkstra's original algorithm found the shortest path between two given nodes, but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree. For a given source node in the graph, the algorithm finds the shortest path between that node and every other. It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level-set Method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the ''Eulerian approach''). Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water. The figure on the right illustrates several important 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 \varphi determining this shape, and the flat blue region r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ron Kimmel
Ron Kimmel ( he, רון קימל, b. 1963) is a professor of Computer Science and Electrical and Computer Engineering (by courtesy) at the Technion Israel Institute of Technology. He holds a D.Sc. degree in electrical engineering (1995) from the Technion, and was a post-doc at UC Berkeley and Berkeley Labs, and a visiting professor at Stanford University. He has worked in various areas of image and shape analysis in computer vision, image processing, and computer graphics. Kimmel's interest in recent years has been non-rigid shape processing and analysis, medical imaging, computational biometry, deep learning, numerical optimization of problems with a geometric flavor, and applications of metric and differential geometry. Kimmel is an author of two books, an editor of one, and an author of numerous articles. He is the founder of the Geometric Image Processing La and a founder and advisor of several successful image processing and analysis companies. Kimmel's contributions includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level-set Method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the ''Eulerian approach''). Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water. The figure on the right illustrates several important 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 \varphi determining this shape, and the flat blue region r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Sweeping Method
In applied mathematics, the fast sweeping method is a numerical method for solving boundary value problems of the Eikonal equation. : , \nabla u(\mathbf), = \frac 1 \text \mathbf \in \Omega : u(\mathbf) = 0 \text \mathbf \in \partial \Omega where \Omega is an open set in \mathbb^n, f(\mathbf) is a function with positive values, \partial \Omega is a well-behaved boundary of the open set and , \cdot, is the Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s .... The fast sweeping method is an iterative method which uses upwind difference for discretization and uses Gauss–Seidel iterations with alternating sweeping ordering to solve the discretized Eikonal equation on a rectangular grid. The origins of this approach lie in control theory. Although fast swe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bellman–Ford Algorithm
The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. The algorithm was first proposed by , but is instead named after Richard Bellman and Lester Ford Jr., who published it in 1958 and 1956, respectively. Edward F. Moore also published a variation of the algorithm in 1959, and for this reason it is also sometimes called the Bellman–Ford–Moore algorithm. Negative edge weights are found in various applications of graphs, hence the usefulness of this algorithm. If a graph contains a "negative cycle" (i.e. a cycle whose edges sum to a negative value) that is reachable from the source, then there is no ''cheapest'' path: any path that has a point on the negative cycle can be made cheaper by one more walk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
