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  2. Energy minimization - Wikipedia

    en.wikipedia.org/wiki/Energy_minimization

    As described above, some method such as quantum mechanics can be used to calculate the energy, E(r) , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r and the second derivative matrix of the system, ∂∂E/∂r i ∂r j, also known as the Hessian matrix, which describes the curvature of the PES at r.

  3. Gradient method - Wikipedia

    en.wikipedia.org/wiki/Gradient_method

    In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.

  4. Conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_method

    The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b

  5. LOBPCG - Wikipedia

    en.wikipedia.org/wiki/LOBPCG

    Kantorovich in 1948 proposed calculating the smallest eigenvalue of a symmetric matrix by steepest descent using a direction = of a scaled gradient of a Rayleigh quotient = (,) / (,) in a scalar product (,) = ′, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors and , i.e. in a locally optimal manner.

  6. Darken's equations - Wikipedia

    en.wikipedia.org/wiki/Darken's_equations

    To derive Darken's second equation the gradient in Gibb's chemical potential is analyzed. The gradient in potential energy, denoted by F 2 , is the force which causes atoms to diffuse. [ 1 ] To begin, the flux J is equated to the product of the differential of the gradient and the mobility B , which is defined as the diffusing atom's velocity ...

  7. Line search - Wikipedia

    en.wikipedia.org/wiki/Line_search

    It can also be solved loosely, by asking for a sufficient decrease in h that does not necessarily approximate the optimum. One example of the former is conjugate gradient method. The latter is called inexact line search and may be performed in a number of ways, such as a backtracking line search or using the Wolfe conditions.

  8. Barzilai-Borwein method - Wikipedia

    en.wikipedia.org/wiki/Barzilai-Borwein_method

    The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...

  9. Derivation of the conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Derivation_of_the...

    In particular, the gradient descent method would be slow. This can be seen in the diagram, where the green line is the result of always picking the local gradient direction. It zig-zags towards the minimum, but repeatedly overshoots.