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Direct Multiple Shooting Method
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods. Single shooting methods Shooting methods can be used to solve boundary value problems (BVP) like : y''(t) = f(t, y(t), y'(t)), \quad y(t_a) = y_a, \quad y(t_b) = y_b, in which the time points ''t''a and ''t''b are known and we seek :y(t),\quad t \in (t_a,t_b). Single shooting methods proceed as follows. Let ''y''(''t''; ''t''0, ''y''0) denote the solution of the initial value problem (IVP) : y''(t) = f(t, y(t), y'(t)) ... [...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] |
Numerical Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A first-order differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathematical definition Let F(x,y)=0 be a well-posed problem, i.e. F:X \times Y \rightarrow \mathbb is a real or complex functional relationship, defined on the cross-product of an input data set X and an output data set Y, such that exists a locally lipschitz function g:X \rightarrow Y called resolvent, which has the property that for every root (x,y) of F, y=g(x). We define numerical method for the approximation of F(x,y)=0, the sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ... of problems : \left \_ = \left \_, with F_n:X_n \times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Value Problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shooting Method
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem. In layman's terms, one "shoots" out trajectories in different directions from one boundary until one finds the trajectory that "hits" the other boundary condition. Mathematical description Suppose one wants to solve the boundary-value problem y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1. Let y(t; a) solve the initial-value problem y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a. If y(t_1; a) = y_1 , then y(t; a) is also a solution of the boundary-value problem. The shooting method is the process of solving the initial value problem for many different values of a until one finds the solution y(t; a) that sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Method
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Computing
Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: bit-level, instruction-level, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance increases must largely come from increasing the number of processors (or cores) on a die, rather than m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Value Problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. Definition An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb \times \mathbb^n \to \mathbb^n where \Omega is an open set of \mathbb \times \mathbb^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In higher dimensions, the differential equation is replaced with a family of equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parareal
Parareal is a parallel algorithm from numerical analysis and used for the solution of initial value problems. It was introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of the most widely studied parallel-in-time integration methods. Parallel-in-time integration methods In contrast to e.g. Runge-Kutta or multi-step methods, some of the computations in Parareal can be performed in parallel and Parareal is therefore one example of a parallel-in-time integration method. While historically most efforts to parallelize the numerical solution of partial differential equations focussed on the spatial discretization, in view of the challenges from exascale computing, parallel methods for temporal discretization have been identified as a possible way to increase concurrency in numerical software. Because Parareal computes the numerical solution for multiple time steps in parallel, it is categorized as a ''parallel across the steps'' method. This is in cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
