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Radial Basis Function Interpolation
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing Order of accuracy, high-order accurate interpolation, interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions. RBF interpolation is a Meshfree method, mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid, and does not require the formation of a Types of mesh, mesh. It is often spectrally accurate and stable for large numbers of nodes even in high dimensions. Many interpolation methods can be used as the theoretical foundation of algorithms for approximating linear operators, and RBF interpolation is no exception. RBF interpolation has been used to approximate differential operators, integral operators, and Differential geometry of surfaces, surface differential operators. Examples Let f(x) = \exp(x \cos(3 \pi x)) and let x_k = \frac, k=0, 1, \dots, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characterizing the approximation error, errors introduced thereby. What is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or Rational function, rational (ratio of polynomials) approximations. The objective is to make the approxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a Set (mathematics), set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension (vector space), dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frame of reference, frames of reference. De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov. The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few borehol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyharmonic Spline
In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension. Definition A polyharmonic spline is a linear combination of polyharmonic radial basis functions (RBFs) denoted by \varphi plus a polynomial term: where * \mathbf = _1 \ x_2 \ \cdots \ x_ (\textrm denotes matrix transpose, meaning \mathbf is a column vector) is a real-valued vector of d independent variables, * \mathbf_i = _ \ c_ \ \cdots \ c_ are N vectors of the same size as \mathbf (often called centers) that the curve or surface must interpolate, * \mathbf = _1 \ w_2 \ \cdots \ w_N are the N weights of the RBFs, * \mathbf = _1 \ v_2 \ \cdots \ v_ are the d+1 weights of the polynomial. The polynomial with the coefficients \mathbf improves fitting accuracy for polyharmonic smoothing splines and al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Matrix
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., ''m'' × ''n'' for an ''m'' × ''n'' matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system, as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. The notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double-precision Floating-point Format
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754 standard, the 64-bit base-2 format is officially referred to as binary64; it was called double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 ''single precision'' and, more recently, base-10 representations (decimal floating point). One of the first programming languages to provide floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Of Ill-conditioning
Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by Microsoft * Microsoft Edge Legacy, a discontinued web browser developed by Microsoft * EdgeHTML, the layout engine used in Microsoft Edge Legacy * ThinkPad Edge, a Lenovo laptop computer series marketed from 2010 * Silhouette edge, in computer graphics, a feature of a 3D body projected onto a 2D plane * Explicit data graph execution, a computer instruction set architecture Telecommunication(s) * EDGE (telecommunication), a 2G digital cellular communications technology * Edge Wireless, an American mobile phone provider * Motorola Edge series, a series of smartphones made by Motorola * Samsung Galaxy Note Edge, a phablet made by Samsung * Samsung Galaxy S7 Edge or Samsung Galaxy S6 Edge, smartphones made by Samsung * Ubuntu Edge, a protot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bed Of Nails
A bed of nails is an rectangle, oblong piece of wood, the size of a bed, with Nail (fastener), nails pointing upwards out of it. While it appears at first glance that anyone lying on such a "bed" would be injured by the nails, if the nails are numerous enough, the weight is distributed among them so that the pressure exerted by each nail is not enough to puncture the person's skin. Uses One use of such a device is for magic (illusion), magic tricks or physics demonstrations. For example, the bed of nails was used in vaudeville in the United States, as well as in sideshows of circuses and carnivals. A famous example requires a volunteer to lie on a bed of several thousand nails, with a board on top of him. Cinder blocks are placed on the board and then smashed with a sledgehammer. Despite the seemingly unavoidable force, the volunteer is not harmed: the force from the blow is spread among the thousands of nails, resulting in reduced pressure; the breaking of the blocks also di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bump Function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain \Reals^n forms a vector space, denoted \mathrm^\infty_0(\Reals^n) or \mathrm^\infty_\mathrm(\Reals^n). The dual space of this space endowed with a suitable topology is the space of distributions. Examples The function \Psi : \mathbb \to \mathbb given by \Psi(x) = \begin \exp\left( \frac\right), & \text , x, . In fact, by definition of support, we have that \operatorname(\Psi):=\overline =\overline, where the closure is taken with respect the Euclidean topology of the real line. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function \exp\le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
