A consistent variational formulation of Bishop nonlocal rods

A Nonlocal Variational Formulation for the Improvement of Tone Mapped Images

Due to technical limitations, common display devices can only reproduce images having a low range of intensity values (dynamic range). As a consequence, the dynamic range of images encoding real world scenes, which is large, has to be compressed in order for them to be reproduced on a common display, and this technique is called tone mapping. Because there is no ground truth to compare with, ev...

A Nonlocal Metric Formulation of MOND

We study a class of nonlocal, but causal, covariant and conserved field equations for the metric. Although nonlocal, these equations do not seem to possess extra graviton solutions in weak field perturbation theory. Indeed, the equations reduce to those of general relativity when the Ricci scalar vanishes throughout spacetime. When a static matter source is present we show how these equations c...

A variational formulation for PDEs

Definition 1. A classic solution (sometimes called strong solution) to the problem is a function u ∈ C2([a, b]) such that it vanishes at the endpoints and pointwise it satisfies the equation −u′′(x)+ u(x) = f(x). Let’s change approach. Suppose that we formally multiply the differential equation by an arbitrary function φ ∈ C1([a, b]) with φ(a) = φ(b) = 0 and we integrate over the interval [a, b...

Primal interface formulation for coupling multiple PDEs: A consistent derivation via the Variational Multiscale method

This paper presents a primal interface formulation that is derived in a systematic manner from a Lagrange multiplier method to provide a consistent framework to couple different partial differential equations (PDE) as well as to tie together nonconforming meshes. The derivation relies crucially on concepts from the Variational Multiscale (VMS) approach wherein an additive multiscale decompositi...

Variational Formulation of Problems and Variational Methods