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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 ...
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior.
[3] [4] Presently, the two types are highly correlated and complementary and both have a wide variety of applications in, e.g., non-linear optimization, sensitivity analysis, robotics, machine learning, computer graphics, and computer vision. [5] [10] [3] [4] [11] [12] Automatic differentiation is particularly important in the field of machine ...
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). Many interesting problems can be formulated as convex optimization problems of the form
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
In particular, if a hill is defined as a height function over a plane (,), the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.