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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. 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 sweeping methods have existed in control theory, it was first proposed for Eikonal equations by Hongkai Zhao, an applied mathematician at the University of California, Irvine The University of California, Irvine (UCI or UC Irvine) is a public land-grant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variati ... [...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 called ... of problems : \left \_ = \left \_, with F_n:X_n \ti ... [...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 differentia ... [...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] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 spaces of any positive integer dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Seidel Method
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. Description The Gauss–Seidel method is an iterative technique for solving a square system of linear equations with unknown : A\mathbf x = \mathbf b . It is defined by the iteration L_* \mathbf^ = \mathbf - U \mathbf^, where \mathbf^ is the -th approximation or iteration of \mathbf,\,\mathbf^ is the next or -th i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hongkai Zhao
Hongkai Zhao is a Chinese mathematician and Ruth F. DeVarney Distinguished Professor of Mathematics at Duke University. He was formerly the Chancellor's Professor in the Department of Mathematics at the University of California, Irvine. He is known for his work in scientific computing, imaging and numerical analysis, such as the fast sweeping method for Hamilton-Jacobi equation and numerical methods for moving interface problems. Zhao had obtained his Bachelor of Science degree in the applied mathematics from the Peking University in 1990 and two years later got his Master's in the same field from the University of Southern California. From 1992 to 1996 he attended University of California, Los Angeles where he got his Ph.D. in mathematics. From 1996 to 1998 Zhao was a Gábor Szegő Assistant Professor at the Department of Mathematics of Stanford University and then got promoted to Research Associate which he kept till 1999. He has been at the University of California, Irvine si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of California, Irvine
The University of California, Irvine (UCI or UC Irvine) is a public land-grant research university in Irvine, California. One of the ten campuses of the University of California system, UCI offers 87 undergraduate degrees and 129 graduate and professional degrees, and roughly 30,000 undergraduates and 6,000 graduate students are enrolled at UCI as of Fall 2019. The university is classified among " R1: Doctoral Universities – Very high research activity", and had $436.6 million in research and development expenditures in 2018. UCI became a member of the Association of American Universities in 1996. The university was rated as one of the "Public Ivies” in 1985 and 2001 surveys comparing publicly funded universities the authors claimed provide an education comparable to the Ivy League. The university also administers the UC Irvine Medical Center, a large teaching hospital in Orange, and its affiliated health sciences system; the University of California, Irvine, Arboretum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
