RiemannSobolevNewton

The Sobolev gradient related to F and the natural Riemannian metric is the Newton quotient. A solid mathematical basis for that statement , focusing on a complex-valued function F can be formed by noting that Riemannian geometry, which has roots in Gauss's ideas, is essential to this discussion.

Consider a positive integer n and F as a function mapping an open subset G of complex n-dimensional space to itself. The Jacobian derivative F'(z) and its inverse should exist for all z in G. The image of G under F is denoted by M.

A metric is defined for each z in G and g, h in complex n-dimensional space, based on the tangent space to M. The standard inner product (dot product) on the complex n-dimensional space is used to define this metric. The norm associated with the inner product is denoted by || · ||z, while the standard norm in complex n-dimensional space is abbreviated as || · ||. For a fixed z, the norm of a linear transformation T is defined accordingly.

A geometric interpretation of the second inner product of two elements g, h in complex n-dimensional space is that due to the explicit appearance of F', a variable metric is introduced, which changes from point to point in M, as is common in Riemannian geometry.

The concept of a SobolevGradient refers to the gradient of a function with respect to a varying metric. The derivative of the function ϕ(z) is given by a specific formula and the goal is to find a gradient of ϕ at z, i.e., a vector h in complex n-dimensional space that maximizes a particular function Φ(h).

To solve this maximization problem, an upper bound for Φ needs to be provided, followed by the specific choice of h for which this bound is assumed. The Cauchy-Schwarz inequality can be applied to achieve this. Moreover, an identity is found using the definition of the second scalar product.

With the choice of hS, the Newton quotient is identified as the required gradient. This confirms that the Sobolev gradient of ϕ relative to the natural Riemannian metric is the Newton quotient, leading to the "N" in RSN. For complex numbers with n=1, the continuous Newton method offers a flow concerning the natural Riemannian metric.

It should come as no surprise that flows derived from the continuous Newton method possess impressive properties.