|
Leimkuhler–Matthews Method
In mathematics, the Leimkuhler-Matthews method (or LM method in its original paper ) is an algorithm for finding discretized solutions to the Brownian dynamics :\mathrm X = -\nabla V(X ) \, \mathrm t + \sigma \, \mathrm W, where \sigma>0 is a constant, V(X) is an energy function and W(t) is a Wiener process. This stochastic differential equation has solutions (denoted X(t) \in \mathbb^N at time t ) distributed according to \pi(X) \propto \exp(-V(x)) in the limit of large-time, making solving these dynamics relevant in sampling-focused applications such as classical molecular dynamics and machine learning. Given a time step \Delta t>0, the Leimkuhler-Matthews update scheme is compactly written as :X_ = X_t -\nabla V(X_t) \Delta t + \sigma\frac2 \, (R_t+R_), with initial condition X_0 := X(0) , and where X_t \approx X(t) . The vector R_t is a vector of independent normal random numbers redrawn at each step so \text R_t \cdot R_N\delta_ (where \textbullet/math> denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milstein Method
In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published it in 1974. Description Consider the autonomous Itō stochastic differential equation: \mathrm X_t = a(X_t) \, \mathrm t + b(X_t) \, \mathrm W_t with initial condition X_ = x_, where W_ stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time ,T/math>. Then the Milstein approximation to the true solution X is the Markov chain Y defined as follows: * partition the interval ,T/math> into N equal subintervals of width \Delta t>0: 0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text\tau_n:=n\Delta t\text\Delta t = \frac * set * recursively define for by: |
Bayesian Inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sampling (statistics)
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population in question. Sampling has lower costs and faster data collection than measuring the entire population and can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heun's Method
In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods. The procedure for calculating the numerical solution to the initial value problem: :y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0, by way of Heun's method, is to first calculate the intermediate value \tilde_ and then the final approximation y_ at the next integration point. :\tilde_ = y_i + h f(t_i,y_i) :y_ = y_i + \frac (t_i, y_i) + f(t_,\tilde_) : where h is the step size and t_=t_i+h. Description Euler's method is used as the foundation for Heun's method. Euler's method uses the line tangent to the function at the beginning of the interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Method (SDE)
In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs. Most basic scheme Consider the Itō diffusion X satisfying the following Itō stochastic differential equation : = a(X_) \, t + b(X_) \, W_, with initial condition X_0=x_0, where W_t stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time ,T/math>. Then the basic Runge–Kutta approximation to the true solution X is the Markov chain Y defined as follows: * partition the interval ,T/math> into N subintervals of width \delta=T/N > 0: 0 = \tau_ < \tau_ < \dots < \tau_ = T; * set ; * recursively compute for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Dynamics
Brownian dynamics (BD) can be used to describe the motion of molecules for example in molecular simulations or in reality. It is a simplified version of Langevin dynamics and corresponds to the limit where no average acceleration takes place. This approximation can also be described as 'overdamped' Langevin dynamics, or as Langevin dynamics without inertia. In Langevin dynamics, the equation of motion is :M\ddot = - \nabla U(X) - \gamma \dot + \sqrt R(t) where *\gamma is a friction coefficient, *U(X) is the particle interaction potential, *\nabla is the gradient operator such that - \nabla U(X) is the force calculated from the particle interaction potentials *the dot is a time derivative such that \dot is the velocity, and \ddot is the acceleration *T is the temperature *k_B is Boltzmann's constant *R(t) is a delta-correlated stationary Gaussian process with zero-mean, satisfying :\left\langle R(t) \right\rangle =0 :\left\langle R(t)R(t') \right\rangle = \delta(t-t'). In Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Probability Distribution")
