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Multigrid Methods
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called Multiresolution analysis, multiresolution methods, very useful in problems exhibiting Multiscale modeling, multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners. The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a ''global'' correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to solve, is similar to the fine grid problem in tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental assumptions of linear elasticity are infinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce yielding. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. Mathematical formulation Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krylov Subspace
In linear algebra, the order-''r'' Krylov subspace generated by an ''n''-by-''n'' matrix ''A'' and a vector ''b'' of dimension ''n'' is the linear subspace spanned by the images of ''b'' under the first ''r'' powers of ''A'' (starting from A^0=I), that is, :\mathcal_r(A,b) = \operatorname \, \. Background The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931. Properties * \mathcal_r(A,b), A\,\mathcal_r(A,b)\subset \mathcal_(A,b). * Let r_0 = \operatorname \operatorname \, \. Then \ are linearly independent unless r>r_0, \mathcal_r(A,b) \subset \mathcal_(A,b) for all r, and \operatorname \mathcal_(A,b) = r_0. So r_0 is the maximal dimension of the Krylov subspaces \mathcal_r(A,b). * The maximal dimension satisfies r_0\leq 1 + \operatorname A and r_0 \leq n. * Consider \dim \operatorname \, \ = \deg\,p(A), where p(A) is the minimal polynomial of A. We have r_0\leq \deg\,p(A). Moreover, for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Péclet Number
In continuum mechanics, the Péclet number (, after Jean Claude Eugène Péclet) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate Potential gradient, gradient. In the context of species or mass transfer, the Péclet number is the product of the Reynolds number and the Schmidt number (). In the context of the thermal fluids, the thermal Péclet number is equivalent to the product of the Reynolds number and the Prandtl number (). The Péclet number is defined as : \mathrm = \dfrac. For mass transfer, it is defined as : \mathrm_L = \frac = \mathrm_L \, \mathrm, where is the characteristic length, the local flow velocity, the Fick's law, mass diffusion coefficient, the Reynolds number, the Schmidt number. Such ratio can also be re-written in terms of times, as a ratio be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convection–diffusion Equation
The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion equation, diffusion and convection (advection equation, advection) equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift velocity, drift–diffusion equation, or (generic) scalar transport equation. Equation The general equation in conservative form is \frac = \mathbf \cdot (D \mathbf c - \mathbf c) + R where * is the variable of interest (species concentration for mass transfer, temperature for heat transfer), * is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, * is the velocity field that the quantity is moving with. It is a function of time and space. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overhead (computing)
Overhead in computer systems consists of shared functions that benefit all users or processes but are not directly attributable to any specific task. It is thus similar to overhead in organizations. Computer system overhead shows up as slower processing, less memory, less storage capacity, less network bandwidth, or bigger latency than would be expected from reading the system specifications. It is a special case of engineering overhead. Overhead can be a deciding factor in software design, with regard to structure, error correction, and feature inclusion. Examples of computing overhead may be found in object-oriented programming (OOP), functional programming, data transfer, data structures, and file systems on data storage devices. Software design Choice of implementation A programmer/software engineer may have a choice of several algorithms, encodings, data types or data structures, each of which have known characteristics. When choosing among them, their respective overhead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Poisson Equation
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. On a two-dimensional rectangular grid Using the finite difference numerical method to discretize the 2-dimensional Poisson equation (assuming a uniform spatial discretization, \Delta x=\Delta y) on an grid gives the following formula: ( ^2 u )_ = \frac (u_ + u_ + u_ + u_ - 4 u_) = g_ where 2 \le i \le m-1 and 2 \le j \le n-1 . The preferred arrangement of the solution vector is to use Enumeration, natural ordering which, prior to removing boundary elements, would look like: \mathbf = \begin u_ , u_ , \ldots , u_ , u_ , u_ , \ldots , u_ , \ldots , u_ \end^\mathsf This will result in an linear syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling (statistics), sampling or experimentation, which represent the values of a function for a limited number of values of the Dependent and independent variables, independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the function approximation, approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residual (numerical Analysis)
Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find ''x'' such that : f(x)=b. Given an approximation ''x''0 of ''x'', the residual is : b - f(x_0) that is, "what is left of the right hand side" after subtracting ''f''(''x''0)" (thus, the name "residual": what is left, the rest). On the other hand, the error is : x - x_0 If the exact value of ''x'' is not known, the residual can be computed, whereas the error cannot. Residual of the approximation of a function Similar terminology is used dealing with differential, integral and functional equations. For the approximation f_\text of the solution f of the equation : T(f)(x)=g(x) \, , the residual can either be the function : ~g(x)~ - ~T(f_\text)(x), or can be said to be the maximum of the norm of this difference : \max_ , g(x)-T(f_\text)(x), over the domain \mathcal X, where the function f_\text is expected to approximate the solution f , or some integral of a function of the diffe ... [...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. 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 Let \mathbf A\mathbf x = \mathbf b be a square system of linear equations, where: \mathbf A = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_ \end, \qquad \mathbf = \begin x_ \\ x_2 \\ \vdots \\ x_n \end , \qquad \mathbf = \begin b_ \\ b_2 \\ \vdots \\ b_n \end. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
