A brief historical perspective of the Wiener-Hopf technique

It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems

Calculating conical diffraction coefficients

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Elastodynamic ray theory and asymptotic methods for direct and inverse scattering problems

In ultrasonic nondestructive evaluation (NDE) testing, sound waves are beamed into a material. Any flaw that may be present in the material will interrupt the incident sound beam and give rise to scattered waves. One important objective of ultrasonic NDE testing is to use this scattered response to determine the nature of the flaw present. In this work, elastodynamic ray theory is employed to obtain both zeroth and first order asymptotic solutions for the far-field leading edge response of a volumetric scatterer caused by incident longitudinal and transverse waves. The explicit results given here for voids have some important implications for the validity of equivalent flaw sizing schemes, such as the inverse Born approximation, and, as shown, can form the basis for new equivalent flaw sizing methods. These results also can be used to develop a more exact flaw sizing algorithm that does not involve any a priori assumption regarding the flaw shape. Other details contained in this work include the asymptotic expansions of integrals defined on arbitrary curved surfaces and a unified treatment of the curvature tensors of the reflected and refracted wavefronts; these results are of fundamental importance for the solution of scattering problems via ray theory

Electromagnetic Wave Scattering by Aerial and Ground Radar Objects

Electromagnetic Wave Scattering by Aerial and Ground Radar Objects presents the theory, original calculation methods, and computational results of the scattering characteristics of different aerial and ground radar objects. This must-have book provides essential background for computing electromagnetic wave scattering in the presence of different kinds of irregularities, as well as Summarizes fundamental electromagnetic statements such as the Lorentz reciprocity theorem and the image principle Contains integral field representations enabling the study of scattering from various layered structures Describes scattering computation techniques for objects with surface fractures and radar-absorbent coatings Covers elimination of "terminator discontinuities" appearing in the method of physical optics in general bistatic cases Includes radar cross-section (RCS) statistics and high-range resolution profiles of assorted aircrafts, cruise missiles, and tanks Complete with radar backscattering diagrams, echo signal amplitude probability distributions, and other valuable reference material, Electromagnetic Wave Scattering by Aerial and Ground Radar Objects is ideal for scientists, engineers, and researchers of electromagnetic wave scattering, computational electrodynamics, and radar detection and recognition algorithms

Digital Color Imaging

This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented us-ing vector-space notation and terminology. Present-day color recording and reproduction systems are reviewed along with the common mathematical models used for representing these devices. Algorithms for processing color images for display and communication are surveyed, and a forecast of research trends is attempted. An extensive bibliography is provided

A contribution to the mathematical theory of diffraction. Part II: Recovering the far-field asymptotics of the quarter-plane problem

We apply the stationary phase method developed in (Assier, Shanin \& Korolkov, QJMAM, 76(1), 2022) to the problem of wave diffraction by a quarter-plane. The wave field is written as a double Fourier transform of an unknown spectral function. We make use of the analytical continuation results of (Assier \& Shanin, QJMAM, 72(1), 2018) to uncover the singularity structure of this spectral function. This allows us to provide a closed-form far-field asymptotic expansion of the field by estimating the double Fourier integral near some special points of the spectral function. All the known results on the far-field asymptotics of the quarter-plane problem are recovered, and new mathematical expressions are derived for the secondary diffracted waves in the plane of the scatterer

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

A study of high performance antenna systems for deep space communication Status report, 15 Dec. 1969 - 15 Jul. 1970

High performance antenna systems for deep space communicatio

Third Conference on Sonic Boom Research

Prediction methods for sonic boom generation and propagation with overpressure minimization in supersonic transport design and operatio

Nonconvex Recovery of Low-complexity Models

Today we are living in the era of big data, there is a pressing need for efficient, scalable and robust optimization methods to analyze the data we create and collect. Although Convex methods offer tractable solutions with global optimality, heuristic nonconvex methods are often more attractive in practice due to their superior efficiency and scalability. Moreover, for better representations of the data, the mathematical model we are building today are much more complicated, which often results in highly nonlinear and nonconvex optimizations problems. Both of these challenges require us to go beyond convex optimization. While nonconvex optimization is extraordinarily successful in practice, unlike convex optimization, guaranteeing the correctness of nonconvex methods is notoriously difficult. In theory, even finding a local minimum of a general nonconvex function is NP-hard – nevermind the global minimum. This thesis aims to bridge the gap between practice and theory of nonconvex optimization, by developing global optimality guarantees for nonconvex problems arising in real-world engineering applications, and provable, efficient nonconvex optimization algorithms. First, this thesis reveals that for certain nonconvex problems we can construct a model specialized initialization that is close to the optimal solution, so that simple and efficient methods provably converge to the global solution with linear rate. These problem include sparse basis learning and convolutional phase retrieval. In addition, the work has led to the discovery of a broader class of nonconvex problems – the so-called ridable saddle functions. Those problems possess characteristic structures, in which (i) all local minima are global, (ii) the energy landscape does not have any ''flat'' saddle points. More interestingly, when data are large and random, this thesis reveals that many problems in the real world are indeed ridable saddle, those problems include complete dictionary learning and generalized phase retrieval. For each of the aforementioned problems, the benign geometric structure allows us to obtain global recovery guarantees by using efficient optimization methods with arbitrary initialization