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Polynomial Chaos
Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms of a polynomial function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability distribution of these random variables. PCE can be used, e.g., to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. Note that despite its name, PCE has no immediate connections to chaos theory. PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables. It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991 and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002. Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may 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'' 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. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Quantification
Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense. Many problems in the natural sciences and engineering are also rife with sources of uncertainty. Computer experiments on computer simulations are the most common approach to study problems in uncertainty quantification. Sources Uncertainty can enter mathematical models and experimental measurements in various contexts. One way to categorize the sources of uncertainty is to consider: ; Paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis
Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of the purchase of a security and the sale of a similar security **Basis of futures, the value differential between a future and the spot price **Basis (options), the value differential between a call option and a put option **Basis swap, an interest rate swap *Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation *Tax basis, cost of an asset and technology *Basis function *Basis (linear algebra) **Dual basis **Orthonormal basis **Schauder basis *Basis (universal algebra) *Basis of a matroid *Generating set of an ideal: **Gröbner basis ** Hilbert's basis theorem *Generating set of a group *Base (topology) *Change of basis *Greedoid *Normal basis *Polynomial basis *Radial basis function *Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Increments
In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. Lévy processes) or by construction (e.g. random walks) Definition A stochastic process X=(X_t)_ has stationary increments if for all t \geq 0 and h > 0 , the distribution of the random variables : Y_:=X_ -X_t depends only on h and not on t . Examples Having stationary increments is a defining property for many large families of stochastic processes such as the Lévy processes. Being special Lévy processes, both the Wiener process and the Poisson processes have stationary increments. Other families of stochastic processes such as random walks have stationary increments by construction. An example of a stochastic process with stationary increments that is not a Lévy process is gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional (mathematics)
In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with linear forms, which are linear mapping from a vector space V into its Field (mathematics), field of scalars (that is, an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers more generally to a mapping from a space X into the field of Real numbers, real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte-Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects result in displacements, or changes of an object's position relative to its environment. Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, Huygens, and Newton laid the foundation for what is now known as classical mechanics. As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Askey-scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic orthogonal polynomials. Askey scheme for hypergeometric orthogonal polynomials give the following version of the Askey scheme: ;_4F_3(4): Wilson , Racah ;_3F_2(3): Continuous dual Hahn , Continuous Hahn , Hahn , dual Hahn ;_2F_1(2): Meixner–Pollaczek , Jacobi , Pseudo Jacobi , Meixner , Krawtchouk ;_2F_0(1)\ \ / \ \ _1F_1(1): Laguerre , Bessel , Charlier ;_2F_0(0): Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ... Here _pF_q(n) indicates a hypergeometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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