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Monte Carlo Integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals. There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods. Overview In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bias Of An Estimator
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In statistics, "bias" is an property of an estimator. Bias is a distinct concept from consistency: consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased (see bias versus consistency for more). All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in unbiased estimation of standard deviation); because a biased esti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Numerica
''Acta Numerica'' is a mathematics journal publishing research on numerical analysis. It was established in 1992 to publish widely accessible summaries of recent advances in the field. One volume is published each year, consisting of review and survey articles from authors invited by the journal's editorial board. The journal is indexed by '' Mathematical Reviews'' and '' Zentralblatt MATH''. During the period of 2004–2009, it had an MCQ of 3.43, the highest of all journals indexed by ''Mathematical Reviews'' over that period of time."Top Journal MCQs cited in the MR Citation Database", MathSciNet, Retrieved 2011-1-13. References Similar Journals * Mathematics of Computation (published by the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance Reduction
In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main variance reduction methods are * common random numbers * antithetic variates * control variates * importance sampling * stratified sampling * moment matching * conditional Monte Carlo * and quasi random variables (in Quasi-Monte Carlo method) For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of specialized techniques; for example, parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method In Statistical Physics
Monte Carlo in statistical physics refers to the application of the Monte Carlo method to problems in statistical physics, or statistical mechanics. Overview The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution: :\langle A\rangle=\int_ A_ \frac d\vec. where E(\vec)=E_ is the energy of the system for a given state defined by \vec - a vector with all the degrees of freedom (for instance, for a mechanical system, \vec = \left(\vec, \vec \right) ), \beta\equiv 1/k_bT and :Z= \int_ e^d\vec is the partition function. One possible approach to solve this multivariable integral i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auxiliary Field Monte Carlo
Auxiliary-field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical (Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, Baeurle 2002a). Reweighting procedure and numerical sign problem The distinctive ingredient of "auxiliary-field Monte Carlo" is the fact that the interactions are decoupled by means of the application of the Hubbard–Stratonovich transformation, which permits the reformulation of many-body theory in terms of a scalar auxiliary-field representation. This reduces the many-body problem to the calculation of a sum or integral over all possible auxiliary-field configurations. In this sense, there is a trade-off: instead of dealing with one very complicated many-body problem, one faces the calculation of an infinite number of simple external-field problems. It is here, as in other related methods, that Monte Carlo enters the game in the guis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Monte Carlo Method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences) to achieve variance reduction. This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is to approximate the integral of a function ''f'' as the average of the function evaluated at a set of points ''x''1, ..., ''x''''N'': : \int_ f(u)\,u \approx \frac\,\sum_^N f(x_i). Since we are integrating over the ''s''-dimensional unit cube, each ''x''''i'' is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the ''x''''i'' are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Mont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis–Hastings Algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. New samples are added to the sequence in two steps: first a new sample is proposed based on the previous sample, then the proposed sample is either added to the sequence or rejected depending on the value of the probability distribution at that point. The resulting sequence can be used to approximate the distribution (e.g. to generate a histogram) or to compute an integral (e.g. an expected value). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, there are usually other methods (e.g. adaptive rejection sampling) that can directly return independent samples from the distribution, and these are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theory), intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to family of sets, families of sets and to indexed family, indexed families of sets. By definition, a collection of sets is called a ''family of sets'' (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An \left(A_i\right)_, is by definition a set-valued Function (mathematics), function (that is, it is a function that assigns a set A_i to every ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Quadrature
Adaptive quadrature is a numerical integration method in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional algorithms may fail. General scheme Adaptive quadrature follows the general scheme 1. procedure integrate ( f, a, b, τ ) 2. Q \approx \int_a^b f(x)\,\mathrmx 3. \varepsilon \approx \left, Q - \int_a^b f(x)\,\mathrmx\ 4. if ''ε'' > ''τ'' then 5. m = (a + b) / 2 6. Q = integrate(f, a, m, τ/2) + integrate(f, m, b, τ/2) 7. endif 8. return Q An approximation Q to the integral of f(x) over the interval ,b/math> is computed (line 2), as well as an error estimate \varepsilon (line 3). If the estimated error is larger than the required tol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strata
In geology and related fields, a stratum (: strata) is a layer of Rock (geology), rock or sediment characterized by certain Lithology, lithologic properties or attributes that distinguish it from adjacent layers from which it is separated by visible surfaces known as either ''Bed (geology), bedding surfaces'' or ''bedding planes''.Salvador, A. ed., 1994. ''International stratigraphic guide: a guide to stratigraphic classification, terminology, and procedure. 2nd ed.'' Boulder, Colorado, The Geological Society of America, Inc., 215 pp. . Prior to the publication of the International Stratigraphic Guide, older publications have defined a stratum as being either equivalent to a single Bed (geology), bed or composed of a number of beds; as a layer greater than 1 cm in thickness and constituting a part of a bed; or a general term that includes both ''bed'' and ''Lamination (geology), lamina''.Neuendorf, K.K.E., Mehl, Jr., J.P., and Jackson, J.A. , eds., 2005. ''Glossary of Geolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. Mathematica's Wolfram Language is fundamentally based on Lisp; for example, the Mathematica command Most is identically equal to the Lisp command butlast. There is a substantial literature on the development of computer algebra systems (CAS). __TOC_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
