Variational and optimal control approaches for the second-order Herglotz problem on spheres

The present paper extends the classical second–order variational problem of Herglotz type to the more general context of the Euclidean sphere Sn following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold Sn such as the problem of finding cubic polynomials on S^n. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.publishe

Fractional variational principle of Herglotz for a new class of problems with dependence on the boundaries and a real parameter

The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied. This type of problem generalizes several problems recently studied in the literature. Moreover, it allows us to unify conservative and non-conservative dynamical processes in the same model. The dependence of the Lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard Herglotz’s variational problem into a problem of a different nature.publishe

Dispersion Relations in Scattering and Antenna Problems

This dissertation deals with physical bounds on scattering and absorption of acoustic and electromagnetic waves. A general dispersion relation or sum rule for the extinction cross section of such waves is derived from the holomorphic properties of the scattering amplitude in the forward direction. The derivation is based on the forward scattering theorem via certain Herglotz functions and their asymptotic expansions in the low-frequency and high-frequency regimes. The result states that, for a given interacting target, there is only a limited amount of scattering and absorption available in the entire frequency range. The forward dispersion relation is shown to be valuable for a broad range of frequency domain problems involving acoustic and electromagnetic interaction with matter on a macroscopic scale. In the modeling of a metamaterial, i.e., an engineered composite material that gains its properties by its structure rather than its composition, it is demonstrated that for a narrow frequency band, such a material may possess extraordinary characteristics, but that tradeoffs are necessary to increase its usefulness over a larger bandwidth. The dispersion relation for electromagnetic waves is also applied to a large class of causal and reciprocal antennas to establish a priori estimates on the input impedance, partial realized gain, and bandwidth of electrically small and wideband antennas. The results are compared to the classical antenna bounds based on eigenfunction expansions, and it is demonstrated that the estimates presented in this dissertation offer sharper inequalities, and, more importantly, a new understanding of antenna dynamics in terms of low-frequency considerations. The dissertation consists of 11 scientific papers of which several have been published in peer-reviewed international journals. Both experimental results and numerical illustrations are included. The General Introduction addresses closely related subjects in theoretical physics and classical dispersion theory, e.g., the origin of the Kramers-Kronig relations, the mathematical foundations of Herglotz functions, the extinction paradox for scattering of waves and particles, and non-forward dispersion relations with application to the prediction of bistatic radar cross sections

On the Uniqueness of Inverse Problems with Fourier-domain Measurements and Generalized TV Regularization

We study the super-resolution problem of recovering a periodic continuous-domain function from its low-frequency information. This means that we only have access to possibly corrupted versions of its Fourier samples up to a maximum cut-off frequency. The reconstruction task is specified as an optimization problem with generalized total-variation regularization involving a pseudo-differential operator. Our special emphasis is on the uniqueness of solutions. We show that, for elliptic regularization operators (e.g., the derivatives of any order), uniqueness is always guaranteed. To achieve this goal, we provide a new analysis of constrained optimization problems over Radon measures. We demonstrate that either the solutions are always made of Radon measures of constant sign, or the solution is unique. Doing so, we identify a general sufficient condition for the uniqueness of the solution of a constrained optimization problem with TV-regularization, expressed in terms of the Fourier samples.Comment: 20 page

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Mathematics and Algorithms in Tomography

This was the ninth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, emission tomography, X-ray CT, and vector tomography along with a wide range of mathematical analysis

Geometric and Numerical analysis of nonholonomic systems

Geometric mechanics is a fairly recent field of mathematics lying in the intersection of at least four different scientific fields: differential geometry, physics, numerical analysis and dynamical systems. Its starting point is to shed light on the underlying geometry behind mechanics and use it to obtain new results which frequently reach a variety of different mathematical fields. One of the practical applications that was made possible by using geometric techniques was the ability to construct \textit{variational integrators}, which are numerical methods reproducing the geometry of the original mechanical system such as symplecticity, conservation of momentum and energy. These methods are often computationally cheaper than standard ones while demonstrating an adequate qualitative behaviour even at low order. However, not all mechanical systems may be approximated using variational integrators. Nonholonomic mechanics is one of such cases, where we lack a variational principle, symplecticity and conservation of momentum, in general. Hence, the investigation of the geometric structure of nonholonomic mechanics must be carried out having into account its non-symplectic and non-variational nature. In this thesis, we will deduce new geometric and analytical properties of nonholonomic systems which hopefully will provide a new insight to the subject. Our main definition, which we will meet across all sections, is the nonholonomic exponential map. This map is a generalization of the well-known Riemannian exponential map and we will see that it plays a role in the description of nonholonomic trajectories as well as on the applications to numerical analysis. After introducing this new object, the thesis may be divided into two parts. In the first part, we take advantage of the nonholonomic exponential map to present new geometric properties of mechanical nonholonomic systems such as the existence of a constrained Riemannian manifold containing radial nonholonomic trajectories with fixed starting point and on which they are geodesics. This is a new and surprising result because it opens the possibility of applying variational techniques to nonholonomic dynamics, which is commonly seen to be non-variational in nature. Also, introduce the notion of a nonholonomic Jacobi field and provide a nonholonomic Jacobi equation. In the second part, which is more applied, we use the nonholonomic exponential map to characterize the exact discrete trajectory of nonholonomic systems. Then we propose a numerical method which is able to generate the exact trajectory. On the last chapter, we discuss contact systems and apply the nonholonomic exponential map to construct an exact discrete Lagrangian function for these systems.N/

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

We consider the Helmholtz equation $-\Delta u + Vu - \lambda u = f$ on $\mathbb{R}^n$ where the potential $V : \mathbb{R}^n\to\mathbb{R}$ is constant on each of the half-spaces $\mathbb{R}^{n−1}\times(−\infty, 0)$ and $\mathbb{R}^{n−1}\times(0,\infty)$. We prove an $L^p − L^q$-Limiting Absorption Principle for frequencies $\lambda > \text{max }V$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations. As a main analytical tool we develop new $L^p − L^q$ estimates for a singular Fourier multiplier supported in an annulus