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About 10,000 papers for “probabilistic methods”

Probabilistic Methods

Remco van der Hofstad · Cambridge University Press eBooks · 2016

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Cited by 0

Probabilistic Methods

Arindam Chakraborty · Encyclopedia of Maritime and Offshore Engineering · 2017

Abstract The article provides a high‐level introduction to the probability methods. Various types of randomness were discussed including several numerical methods to generate pseudo random numbers that can be simulated using computer. The article also discusses Bayesian statistics and time series methods. The application of such methods is abundant in the Marine and Offshore engineering.

Cited by 3

Probabilistic Methods

Asafu-Adjei, Joseph · VCU Scholars Compass (Virginia Commonwealth University) · 2007

The Probabilistic Method was primarily used in Combinatorics and pioneered by Erdös Pai, better known to Westerners as Paul Erdos in the 1950s. The probabilistic method is a powerful tool for solving many problems in discrete mathematics, combinatorics and also in graph .theory. It is also very useful to solve problems in number theory, combinatorial geometry, linear algebra and real analysis. More recently, it has been applied in the development of efficient algorithms and in the study of various computational problems.Broadly, the probabilistic method is somewhat opposite of the extremal graph theory. Instead of considering how a graph can behave in the extreme, we consider how a collection of graphs behave on 'average' where by we can formulate a probability space. The method allows one to prove the existence of a structure with particular properties by defining an appropriate probability space of structures and show that the desired properties hold in the space with positive probability.(please see PDF for complete abstract)

Cited by 0Open access

Probabilistic Methods

Alexander Lubotzky, Dan Segal · Birkhäuser Basel eBooks · 2003

Probability has entered into group theory along several paths. One, initiated by Erdos and Turan in the 1960s, is the investigation of probabilistic properties of finite groups; a typical example of this is the classic theorem of Dixon that a random pair of elements generates the alternating group Alt(n) with probability that tends to 1 as n → ∞, and its recent extension by Kantor, Lubotzky, Liebeck and Shalev to the definitive result: a random pair of elements generates a finite simple group with probability that tends to 1 as the group order tends to ∞. Another path is based on the fact that a profinite group G, being a compact topological group, has a finite Haar measure μ. Normalizing this so that μ(G) = 1, we may consider G as a probability space: this means that the measure of a subset X of G is construed as the probability that a random element of G lies in X. It is now natural to ask questions such as: what is the probability that a random k-tuple of elements generates G? Formally, this probability is defined as: P(G,k)= $$\mu \left\{ {({x_1}, \ldots ,{x_k}) \in {G^{(k)}}|\left\langle {{x_1}, \ldots ,{x_k}} \right\rangle = G} \right\},$$ (1) (11.1) where μ denotes also the product measure on G(k).

Cited by 1

Probabilistic Inversion Methods

Alexandrine Gesret · 2024

Model parameterization becomes an issue in itself when it comes to solving structural imaging inverse problems. Indeed, in probabilistic inversion, it is necessary to use the most sparse parametrization possible to enable sampling, which involves the forward problem a number of times. At the same time, this parameterization must represent reality as accurately as possible. The best-known Markov chain Monte Carlo (MCMC) algorithm for sampling a probability distribution is the Metroplis-Hastings algorithm. Its implementation is simple, involving two steps: a new state is proposed followed by the computation of the acceptance probability. One approach that enables escaping a local minimum consists of having several interacting chains, as proposed by Geyer, which also speeds up convergence. The reversible-jump MCMC algorithm is an extension of the Metropolis-Hastings algorithm, which takes into account the variable dimension of the parameter space.

Cited by 0

Probabilistic Forecast Methods

Stephen Haben, Marcus Voß, William Holderbaum · 2023

Abstract The previous two chapters were concerned with point forecasts which only produce a single estimate for each time step in the forecast horizon, i.e. one value $$L_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> for each of the time periods $$t=N+1, N+2, \ldots , N+k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> (assuming a forecast horizon of length k steps ahead starting at forecast origin N ). Point estimates are limited in their description of the future demand, especially when the underlying data has a large degree of uncertainty. A more detailed picture of the possible values of the demand can be produced by estimating the distribution of the demand for each period in the forecast horizon. Forecasts which estimate the spread of the distribution are often called probabilistic forecasts . That is the subject of this chapter.

Cited by 0Open access

Probabilistic methods: entropy compression

Kardoš, František · Centre International de Rencontres Mathématiques · 2024

We will overview two probabilistic methods used to prove the existence of diverse combinatorial objects with desired properties: First, we will introduce Lovász Local Lemma as a classical tool to prove that, informally speaking, with non-zero probability none of the “bad” events occur if those bad events are of low probability and somewhat essentially independent from each other. Next, we will move on to its algorithmic counterpart, the Entropy Compression Method, used to ensure that a randomized algorithm eventually finds a solution with desired properties. The two methods will be illustrated by applying them to various settings in graph theory, combinatorics, and geometry. https://www.labri.fr/perso/fkardos/

Cited by 0Open access

Probabilistic Methods in Schrödinger Equations

Zhi-Ming Ma, Renming Song · Birkhäuser Boston eBooks · 1990

It is well known that for the time homogeneous Schrödinger equation $$\left( {\frac{\Delta }{2} + q} \right)u = 0$$ where Δ is the Laplacian on a domain D, there are typically four types of boundary value problems:

Cited by 11