Anna Aboud | Searcharxiv

arXiv2025

A new graph-directed construction of nonlocal energies on the unit interval

We present an analytic construction of nonlocal energies on the unit interval. The energies are defined using a new graph-directed construction of discrete energies on dyadic approximations of the interval. When the discrete jump kernels are comparable to the kernel of the fractional discrete Laplacian, we prove that the discrete energies Mosco converge and the limiting energy is equivalent to the fractional Gagliardo seminorm.

math.AP↗

arXiv2018

The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products $\{\langle x, ϕ_{n} \rangle\}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\{\langle x, ϕ_{n} \rangle\}$; these approximations only converge to $x$ when $\{ϕ_{n}\}$ is ${effective}$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\{\langle x, ϕ_{n} \rangle\}$ by using a second sequence $\{ψ_{n}\}$ in the reconstruction. This allows for the recovery of $x$ even when the sequence $\{ϕ_{n}\}$ is not effective; in particular, our dualization yields a reconstruction when the sequence $\{ϕ_{n}\}$ is $almost$ $effective$. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle x, ϕ_{n} \rangle\}$ using $\{ψ_{n}\}$ converges to $x$, in which case $\{(ϕ_n, ψ_n)\}$ is called an $effective$ $pair$.

math.FA↗