Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations $AX\approx B$ and $AXC\approx B$. This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.     

High-order multigrid strategies for hybrid high-order discretizations of elliptic equations

This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high-order method. Combinations of $h$-, $p$-, and $hp$-coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non-nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.     

The inverse conductivity problem via the calculus of functions of bounded variation

In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of $BV$ functions. The space of the functions of bounded variation is recommended here as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical investigation of the inversion of the inclusion problem, we propose and implement a suitable minimization scheme of an enriched—constructed herein—functional, by exploiting the inner structure of $BV$ space. Finally, we validate and illustrate our theoretical results with numerical experiments.     

Computability of graphs

We consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type, which means that they have the following property: if X is a computable topological space and $f:A\to X$ a topological imbedding such that $f(A)$ and $f(B)$ are semicomputable sets in X, then $f(A)$ is a computable set in X. It is known, e.g., that $(M,\partial M)$ has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that $(G,E)$ has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.