This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.

High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time. / text

Ο στόχος της παρούσας διδακτορικής διατριβής είναι διττός: η ανάπτυξη και η εφαρμογή αριθμητικών τεχνικών για την επίλυση προβλημάτων που εμπίπτουν στην περιοχή του Μη-Καταστροφικού Ελέγχου. Συγκεκριμένα αναπτύχθηκαν η Μέθοδος των Συνοριακών Στοιχείων (ΜΣΣ) και η Μέθοδος των Τοπικών Ολοκληρωτικών Εξισώσεων χωρίς Διακριτοποίηση για την αριθμητική ανάλυση στατικών και μεταβατικών προβλημάτων στο πεδίο της ελαστικότητας και της αλληλεπίδρασης ελαστικού με ακουστικό μέσο στις δύο διαστάσεις. Σημαντικό μέρος της διδακτορικής διατριβής αποτέλεσε η ανάπτυξη προγράμματος ηλεκτρονικού υπολογιστή, το οποίο επιλύει τα προβλήματα στα οποία πραγματεύεται το παρόν σύγγραμμα. Η διδακτορική διατριβή αποτελείται από τρεις ενότητες. Στην πρώτη ενότητα γίνεται πλήρης περιγραφή της απαραίτητης θεωρίας για την κάλυψη και κατανόηση των αριθμητικών ΜΣΣ αλλά και των Τοπικών Μεθόδων χωρίς Διακριτοποίηση (ΤΜχΔ). Στη δεύτερη ενότητα εφαρμόζονται οι προαναφερθείσες αριθμητικές μέθοδοι για την επίλυση στατικών και δυναμικών (στο πεδίο συχνοτήτων) διδιάστατων προβλημάτων, ώστε να πιστοποιηθεί η ακρίβεια και η αξιοπιστία των εν λόγω μεθοδολογιών. Τέλος, στην τρίτη ενότητα οι αριθμητικές ΜΣΣ και ΤΜχΔ εφαρμόζονται για την επίλυση προβλημάτων κυματικής διάδοσης που εμπίπτουν στο πεδίο του Μη-Καταστροφικού Ελέγχου. Πιο συγκεκριμένα μελετήθηκε η κυματική διάδοση σε ελεύθερες επίπεδες πλάκες και σε κυλινδρικές δεξαμενές αποθήκευσης υγρών καυσίμων. / The aim of this doctoral thesis is twofold: the development and implementation of numerical techniques for solving wave propagation problems in Non-Destructive Testing applications. Particularly, the Boundary Element Method (BEM) and the Local Boyndary Integral Equation Method are developed, so as to numerically solve static and transient problems on the field of elasticity and fluid-structure interaction in two dimensions. A major part of the present research is the construction of a computer program for solving such kind of problems. This textbook consists of three sections. In the first section, a thorough description on the theory of the BEM and the Local Meshless Methods (LMM) is done. The second section is dedicated for the numerical implementation of the BEM and LMM for solving steady state and time-harmonic two dimensional elastic and acoustic problems, in order to verify the accuracy and the ability of the proposed methodologies to solve the above-mentioned problems. Finally in the third section, the wave propagation problems of traction-free plates and cylindrical fuel storage tanks is studied, from the perspective of Non-Destructive Testing. The numerical methods of BEM and LMM are implemented, as well as spectral methods are utilized, for drawing useful conclusions on the wave propagation phenomena.

Cette thèse aborde deux thématiques différentes. On s’intéresse d’abord au développement et à l’analyse de méthodes pour le sens électrique appliqué à la robotique. On considère en particulier la méthode des réflexions permettant, à l’image de la méthode de Schwarz, de résoudre des problèmes linéaires à partir de sous-problèmes plus simples. Ces deniers sont obtenus par décomposition des frontières du problème de départ. Nous en présentons des preuves de convergence et des applications. Dans le but d’implémenter un simulateur du problème direct d’électrolocation dans un robot autonome, on s’intéresse également à une méthode de bases réduites pour obtenir des algorithmes peu coûteux en temps et en place mémoire. La seconde thématique traite d’un problème inverse dans le domaine de la chimie quantique. Nous cherchons ici à déterminer les caractéristiques d’un système quantique. Celui-ci est éclairé par un champ laser connu et fixé. Dans ce cadre, les données du problème inverse sont les états avant et après éclairage. Un résultat d’existence locale est présenté, ainsi que des méthodes de résolution numériques. / This thesis tackle two different topics.We first design and analyze algorithms related to the electrical sense for applications in robotics. We consider in particular the method of reflections, which allows, like the Schwartz method, to solve linear problems using simpler sub-problems. These ones are obtained by decomposing the boundaries of the original problem. We give proofs of convergence and applications. In order to implement an electrolocation simulator of the direct problem in an autonomous robot, we build a reduced basis method devoted to electrolocation problems. In this way, we obtain algorithms which satisfy the constraints of limited memory and time resources. The second topic is an inverse problem in quantum chemistry. Here, we want to determine some features of a quantum system. To this aim, the system is ligthed by a known and fixed Laser field. In this framework, the data of the inverse problem are the states before and after the Laser lighting. A local existence result is given, together with numerical methods for the solving.

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.

No description available.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.