Search results for "mathematical analysis"

Mappings of finite distortion: Reverse inequalities for the Jacobian

2007

Let f be a nonconstant mapping of finite distortion. We establish integrability results on 1/Jf by studying weights that satisfy a weak reverse Holder inequality where the associated constant can depend on the ball in question. Here Jf is the Jacobian determinant of f.

Riesz-Fischer Maps, Semi-frames and Frames in Rigged Hilbert Spaces

2021

In this note we present a review, some considerations and new results about maps with values in a distribution space and domain in a σ-finite measure space X. Namely, this is a survey about Bessel maps, frames and bases (in particular Riesz and Gel’fand bases) in a distribution space. In this setting, the Riesz-Fischer maps and semi-frames are defined and new results about them are obtained. Some examples in tempered distributions space are examined.

On the fractional integral of Weyl inL p

1994

Carleson's counterexample and a scale of Lorentz-BMO spaces on the bitorus

2005

We introduce a full scale of Lorentz-BMO spaces BMO L p,q on the bidisk, and show that these spaces do not coincide for different values ofp andq. Our main tool is a detailed analysis of Carleson's construction in [C].

Computing Difficulties for Deriving Poverty Indices from Some Functional Forms of Lorenz Curves

2014

We examine three families of classical one-parameter functional forms for estimating a Lorenz curve: the power form (Pareto, elementary form), the exponential form (Gupta, elementary form) and fractional form (Rohde). For the first time, we systematically study these functions not for their ability to be estimated but on the point of view of the possibility of deriving poverty indices, which implies first determining the headcount ratio (i.e., the percentage of poor). We show that computing difficulties have been largely underestimated. Two forms, the most simple ones, pose no problem: the elementary power and exponential forms. However, the Pareto functional form poses problem with a restr…

Numerical Recovery of Source Singularities via the Radiative Transfer Equation with Partial Data

2013

The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator $X_{V}^{*}X_{V}$ where $X_{V}$ is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by F. Monard and G. Bal. We will see that one can detect quite well the visible singularities of an internal optical source $f$ for generic anisotropic $k$ and $\sigma$, with or without noise added to the accessible data $X_{V}f$. In particular, we use a truncated Neumann series to estimate $X_{V}$ and $X_{V}^{*}$, which provides a good approximation of $X_{V}^{*…

Shape Sensitivity Analysis and Gradient-Based Optimization of Large Structures Using MLFMA

2014

A fast method for computing the action of shape-differentiated electric field integral equation (EFIE) system matrix to a vector is derived exploiting the multilevel fast multipole algorithm (MLFMA). The proposed method is used in conjunction with the adjoint-variable method (AVM) to compute the shape gradient of arbitrary objective functions depending on shape of a metallic scatterer. The method is demonstrated numerically by optimizing the shape of a parabolic reflector illuminated with a half-wave dipole.

On shape differentiation of discretized electric field integral equation

2013

Abstract This work presents shape derivatives of the system matrix representing electric field integral equation discretized with Raviart–Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries of the original system matrix. The results are compared to derivatives computed with automatic differentiation technique and finite differences, and are found to be in an excellent agreement. Furthermore, the derived formulas are employed to analyze shape sensitivity of the input impedance of a planar inverted F-antenna, and the results are compared to those obtained using a finite difference approximation.

Justification of point electrode models in electrical impedance tomography

2011

The most accurate model for real-life electrical impedance tomography is the complete electrode model, which takes into account electrode shapes and (usually unknown) contact impedances at electrode-object interfaces. When the electrodes are small, however, it is tempting to formally replace them by point sources. This simplifies the model considerably and completely eliminates the effect of contact impedance. In this work we rigorously justify such a point electrode model for the important case of having difference measurements ("relative data") as data for the reconstruction problem. We do this by deriving the asymptotic limit of the complete model for vanishing electrode size. This is s…

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

2015

We consider the computationally efficient time integration of Maxwell’s equations using discrete exterior calculus (DEC) as the computational framework. With the theory of DEC, we associate the degrees of freedom of the electric and magnetic fields with primal and dual mesh structures, respectively. We concentrate on mesh constructions that imitate the geometry of the close packing in crystal lattices that is typical of elemental metals and intermetallic compounds. This class of computational grids has not been used previously in electromagnetics. For the simulation of wave propagation driven by time-harmonic source terms, we provide an optimized Hodge operator and a novel time discretizati…