Search results
Results from the WOW.Com Content Network
Backward finite difference [ edit ] To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same.
The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method. [4]
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
Proximal gradient methods are a generalized form of projection used to solve non-differentiable convex optimization problems.. A comparison between the iterates of the projected gradient method (in red) and the Frank-Wolfe method (in green).
Figure 1.Comparison of different schemes. In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and provides numerical solutions to differential equations. [1]
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable.