Motivated by the interior tomography problem, we propose a method for exact reconstruction of a region of interest of a function from its local Radon transform in any number of dimensions. Our aim is to verify the feasibility of a one-dimensional reconstruction procedure that can provide the foundation for an efficient algorithm. For a broad class of functions, including piecewise polynomials a...

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It known that solutions to the Dirichlet problem involving such an operator exhibit algebraic boundary singularity regardless domain regularity. This, in turn, deteriorates global regularity and as result convergence rate numerical solutions. For finite element discretizations, we derive local error estimates $H^s$...

We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a seminorm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f . We determine a decomposition formula for this Laurent norm; an expression for the Laurent norm for f in terms of the Laurent norms for each of the irreducibl...

A new multidimensional optimization problem is considered in the tropical mathematics setting. The problem is to minimize a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield and given by a conjugate transposition operator. A special case of the problem, which arises in just-in-time scheduling, serves as a motivation for the study. To solve the general pr...

This paper derives analytical estimates for the rates of convergence of numerical first and second derivative operators involved in flux reconstruction (FR). These estimates yield the rate of convergence for steady-state advection-diffusion problems when error is measured in the vector seminorm induced by the advection-diffusion operator. This serves to rigorously quantify the effect of polynom...

When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill–posed operator equations, assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it is clear from works of Hegland, Engl a...

We establish theoretical convergence results for an Iteratively Regularized Gauss Newton (IRGN) algorithm with a specific Tikhonov regularization. This Tikhnov regularization, which uses a seminorm generated by a linear operator, is motivated by mapping of the minimization variables to physical space which exposes the different scales of the parameters and therefore also suggests appropriate we...

We extend some previous results of ours [1] on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates in terms of the Euclidean norm. For example, one can use as a separating family of seminorms the absolute values of the components of the actions: with ...

In this papers we analyze the minimization of seminorms ‖L · ‖ on R under the constraint of a bounded I-divergence D(b,H ·) for rather general linear operators H and L. The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Often H represents, e.g., a linear blur operator and L is some discrete d...

In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show equivalence function space vector field characterized by Gagliardo-type seminorm with 'projected difference' that corresponding Sobolev space. As an application, will use obtain Caccioppoli-type for nonlinear system nonlocal equations, which in turn is key ingredient a...