In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.
The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include , where it is known as the Laplace filter, and in machine learning for clustering and semi-supervised learning on neighborhood graphs.
There are various definitions of the discrete Laplacian for graphs, differing by sign and scale factor (sometimes one averages over the neighboring vertices, other times one just sums; this makes no difference for a regular graph). The traditional definition of the graph Laplacian, given below, corresponds to the negative continuous Laplacian on a domain with a free boundary.
Let be a graph with vertices and edges . Let be a function of the vertices taking values in a ring. Then, the discrete Laplacian acting on is defined by
where is the graph distance between vertices w and v. Thus, this sum is over the nearest neighbors of the vertex v. For a graph with a finite number of edges and vertices, this definition is identical to that of the Laplacian matrix. That is, can be written as a column vector; and so is the product of the column vector and the Laplacian matrix, while is just the vth entry of the product vector.
If the graph has weighted edges, that is, a weighting function is given, then the definition can be generalized to
where is the weight value on the edge .
Closely related to the discrete Laplacian is the averaging operator:
In addition to considering the connectivity of nodes and edges in a graph, mesh Laplace operators take into account the geometry of a surface (e.g.
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Le cours étudie les concepts fondamentaux de l'analyse vectorielle et de l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.
The students gain an in-depth knowledge of several current and emerging areas of theoretical computer science. The course familiarizes them with advanced techniques, and develops an understanding of f
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph.
In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Pierre-Simon, Marquis de Laplace (ləˈplɑ:s; pjɛʁ simɔ̃ laplas; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.
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ACADEMIC PRESS INC ELSEVIER SCIENCE2022
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