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Walk-on-spheres Method
In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation, and was since then generalized to other problems. It relies on probabilistic interpretations of PDEs, and simulates paths of Brownian motion (or for some more general variants, diffusion processes), by sampling only the exit-points out of successive spheres, rather than simulating in detail the path of the process. This often makes it less costly than "grid-based" algorithms, and it is today one of the most widely used "grid-free" algorithms for generating Brownian paths. Informal description Let \Omega be a bounded Domain (mathematical analysis), domain in \mathbb^d with a sufficiently regular boundary \Gamma, let ''h'' be a function on \Gamma, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 ones are common random numbers, antithetic variates, control variates, importance sampling, stratified sampling, moment matching, conditional Monte Carlo and quasi random variables. 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, particle transport simulations make extensive use of "weight windows" and "splitti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variants Of Random Walks
Variant may refer to: In arts and entertainment * ''Variant'' (magazine), a former British cultural magazine * Variant cover, an issue of comic books with varying cover art * ''Variant'' (novel), a novel by Robison Wells * " The Variant", 2021 episode of the TV series ''Loki'' **Sylvie (Marvel Cinematic Universe), a character who was originally referred to as the Variant In gaming * Chess variant, a game derived from, related to or similar to chess in at least one respect *List of poker variants * List of ''Tetris'' variants In mathematics and computing *Variant (logic), a term or formula obtained from another one by consistently renaming all variables * Variant symlinks, a symbolic link to a file that has a variable name embedded in it *Variant type, in programming languages *Z-variant, unicode characters that share the same etymology but have slightly different appearances Computer security * In network security, varieties of computer worms are called variants. In biolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Maruyama Method
In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately, the same generalization cannot be done for any arbitrary deterministic method. Consider the stochastic differential equation (see Itô calculus) :\mathrm X_t = a(X_t, t) \, \mathrm t + b(X_t, t) \, \mathrm W_t, 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'' Then the Euler–Maruyama approximation to the true solution ''X'' is the Markov chain ''Y'' defined as follows: * partition the interval , ''T''into ''N'' equal subintervals of width \Delta t>0: ::0 = \tau_ float: """ Sample a ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes And Boundary Value Problems
In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation. Introduction: Kakutani's solution to the classical Dirichlet problem Let D be a domain (an open and connected set) in \mathbb^. Let \Delta be the Laplace operator, let g be a bounded function on the boundary \partial D, and consider the problem: :\begin - \Delta u(x) = 0, & x \in D \\ \displaystyle = g(x), & x \in \partial D \end It can be shown that if a solution u exists, then u(x) is the expected value of g(x) at the (random) first exit point from D for a canonical Brownian motion starting a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman–Kac Formula
The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven. It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Theorem Consider the partial differential equation :\frac(x,t) + \mu(x,t) \frac(x,t) + \tfrac \sigma^2(x,t) \frac(x,t) -V(x,t) u(x,t) + f(x,t) = 0, defined for all x \in \mathbb and t \in , T/math>, subject to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Processes
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. Formal definition The Bessel process of order ''n'' is the real-valued process ''X'' given (when ''n'' ≥ 2) by :X_t = \, W_t \, , where , , ·, , denotes the Euclidean norm in R''n'' and ''W'' is an ''n''-dimensional Wiener process (Brownian motion). For any ''n'', the ''n''-dimensional Bessel process is the solution to the stochastic differential equation (SDE) :dX_t = dW_t + \frac\frac where W is a 1-dimensional Wiener process (Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...). Note that this SDE makes sense for any real parameter n (although the drift term is singular at zero). Notation A notation for the Bessel process of dimension started at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Partial Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson–Boltzmann Equation
The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions. Origins Background and derivation The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively. In the Gouy-Chapman model, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or double layer. Due to thermal motio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson. Statement of the equation Poisson's equation is \Delta\varphi = f where \Delta is the Laplace operator, and f and \varphi are real or complex-valued functions on a manifold. Usually, f is given and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three-dimensional Cartesian coordinates, it takes the form \left( \frac + \frac + \frac \right)\varph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
