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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 also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Approximation
In general, a function approximation problem asks us to select a function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of applied mathematics, and computer science in particular , such as predicting the growth of microbes in microbiology. Function approximations are used where theoretical models are unavailable or hard to compute. One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.). Second, the target function, call it ''g'', may be unknown; instead of an explicit formula, only a set of poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-index Notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. Definition and basic properties An ''n''-dimensional multi-index is an ''n''-tuple :\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of non-negative integers (i.e. an element of the ''n''-dimensional set of natural numbers, denoted \mathbb^n_0). For multi-indices \alpha, \beta \in \mathbb^n_0 and x = (x_1, x_2, \ldots, x_n) \in \mathbb^n one defines: ;Componentwise sum and difference :\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n) ;Partial order :\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\ ;Sum of components (absolute value) :, \alpha , = \alpha_1 + \alpha_2 + \cdots + \alpha_n ;Factorial :\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n! ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splines (mathematics)
Spline may refer to: Mathematics * Spline (mathematics), a mathematical function used for interpolation or smoothing * Smoothing spline, a method of smoothing using a spline function Devices * Spline (mechanical), a mating feature for rotating elements * Flat spline, a device to draw curves * Spline drive, a type of screw drive * Spline cord, a type of thin rubber cord used to secure a window screen to its frame * Spline (or star filler A star filler (also known as cross filler, splines, separators and crossweb fillers) is a type of plastic insert in Cat 5 and Cat 6 cable which separates the individual stranded pair sets from each other while inside of the cable. It increases the ...), a type of plastic cable filler for CAT cable Other * Spline (alien beings), in Stephen Baxter's Xeelee Sequence novels See also * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spline (mathematics)
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term ''spline'' more frequently refers to a piecewise polynomial ( parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. Introduction The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subdivision Surface
In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface) is a curved surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface, the underlying ''inner mesh'', can be calculated from the coarse mesh, known as the ''control cage'' or ''outer mesh'', as the functional limit of an iterative process of subdividing each polygonal face into smaller faces that better approximate the final underlying curved surface. Less commonly, a simple algorithm is used to add geometry to a mesh by subdividing the faces into smaller ones without changing the overall shape or volume. Overview A subdivision surface algorithm is recursive in nature. The process starts with a base level polygonal mesh. A refinement scheme is then applied to this mesh. This process takes that mesh and subdivides it, creating new vertices and new faces. The positions of the new vertices in the mesh are compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radial Basis Function
A radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed point \mathbf, called a ''center'', so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf-\mathbf\right\, ). Any function \varphi that satisfies the property \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ) is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection \_k which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Distance Weighting
Inverse distance weighting (IDW) is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the known points. This method can also be used to create spatial weights matrices in spatial autocorrelation analyses (e.g. Moran's ''I''). The name given to this type of method was motivated by the weighted average applied, since it resorts to the inverse of the distance to each known point ("amount of proximity") when assigning weights. Definition of the problem The expected result is a discrete assignment of the unknown function u in a study region: :u(x): x \to \mathbb, \quad x \in \mathbf \sub \mathbb^n, where \mathbf is the study region. The set of N known data points can be described as a list of tuples: : x_1, u_1), (x_2, u_2), ..., (x_N, u_N) The function is to be "smooth" (continuous and once differentiable), to be exact (u(x_i) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Lagrange Equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thin Plate Energy Functional
The exact thin plate energy functional (TPEF) for a function f(x,y) is :\int_^ \int_^ (\kappa_1^2 + \kappa_2^2) \sqrt \,dx \,dy where \kappa_1 and \kappa_2 are the principal curvatures of the surface mapping f at the point (x,y). This is the surface integral of \kappa_1^2 + \kappa_2^2, hence the \sqrt in the integrand. Minimizing the exact thin plate energy functional would result in a system of non-linear equations. So in practice, an approximation that results in linear systems of equations is often used. The approximation is derived by assuming that the gradient of f is 0. At any point where f_x = f_y =0, the first fundamental form g_ of the surface mapping f is the identity matrix and the second fundamental form b_ is :\begin f_ & f_ \\ f_ & f_ \end. We can use the formula for mean curvature H=b_g^/2 to determine that H = (f_+f_)/2 and the formula for Gaussian curvature K=b/g (where b and g are the determinants of the second and first fundamental forms, respectively) t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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