ABSTRACT. Accelerated beams (Airy and Airy-related) correspond to curved caustics of the underlying geometrical rays. The connections will be explained in detail, concentrating on beams associated with the stable caustics classified by catastrophe theory. Some such beams, including the simplest Airy beam in three-dimensional space, are unstable in the mathematical sense: under a symmetry-breaking perturbation, they break up into caustics that are stable. In the Airy case, this is a hyperbolic umbilic catastrophe. Associated with the stable caustics are a variety of exact solutions of the paraxial wave equations.

ABSTRACT. Gravitational waves were predicted in 1916 by Einstein when he discovered wave solutions to the linearized field equations of General Relativity. The first direct gravitational-wave detection was made by the Advanced Laser Interferometer Gravitational Wave Observatory (LIGO) on September 14, 2015. This discovery marked the culmination of a century-long quest to understand the physical nature of these waves and to build instruments sensitive enough to detect them. LIGO's detections mark the beginning of an entirely new form of observational astronomy. Gravitational waves will allow us to explore the nature of gravity and to observe astrophysical processes that are inaccessible to electromagnetic wave observations. I will review the science of gravitational waves and LIGO's discoveries, the key results from LIGO's first observing run, which was conducted September 2015--January 2016, and discuss future directions for the field of gravitational-wave astronomy.

ABSTRACT. Efforts are currently underway to detect gravitational waves with $f\sim10^{-7} - 10^{-9} \, \mathrm{Hz}$ using pulsar timing arrays by collaborations around the world, and detection is expected within the next few years. One such group is the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). I will discuss how pulsar timing arrays are used to detect gravitational waves and what sources we expect to find in the low-frequency regime, with an emphasis on supermassive black hole binaries.

ABSTRACT. Extending the new field of gravitational wave (GW) astronomy into the millihertz band with a space-based GW observatory is a high-priority objective of international astronomy community. This paper summarizes the astrophysical promise and the technological groundwork for such an observatory, concretely focusing on the prospects for the proposed Laser Interferometer Space An- tenna (LISA) mission concept.

ABSTRACT. The goal is to retrieve the shape of an elastic scatterer along with its material parameters from the knowledge of some far field pattern measurements. To this end, we employ a multi-stage strategy based on Tikhonov iterative-like method. Numerical reconstructions for various two-dimensional scatterers will be presented.

ABSTRACT. Qualitative methods are a class of methods that try to retrieve the geometrical support of an obstacle from multistatic data. Recently introduced the Generalized Linear Sampling Method (GLSM) is the extension of the well known Linear Sampling Method (LSM). This extension provides a theoretical justification for the LSM and exhibit better numerical results. The GLSM was introduced for both perfect and noisy data but in the latter case an important ingredient was missing. The proper set up of the weight of regularization with respect to the noise level and ultimately an a priori rules such as Morozov principles. This papers aims at filling this gap under certain constraint on the type of obstacle.

ABSTRACT. We establish the well-posedness of volume integral equations (VIEs) for elastodynamic scattering. Such VIEs are known to be compact perturbations of elastostatic VIEs. We derive a modified version of the latter, which is shown to be unconditionally solvable by Neumann series. The modified VIE is also found to extend the range of applicability of fixed-point (iterated Born) methods.

ABSTRACT. We prove by means of a couple of examples that plasmonic resonances can be used on one hand to classify shapes of nanoparticles with real algebraic boundaries and on the other hand to reconstruct the separation distance between two nanoparticles from measurements of their first collective plasmonic resonances. To this end, we explicitly compute the spectral decompositions of the Neumann-Poincare operators associated with a class of quadrature domains and two nearly touching disks. Numerical results are included in support of our main findings.

ABSTRACT. The talk will contain an exposition of the work over the past decade by the author and his collaborators on modeling wave propagation in unbounded domains. Key to the effectiveness of many developed methods is the special property of midpoint integration completely eliminating the discretization error in half-space impedance that arises from linearfinite element discretization. Impedance preserving discretization complements the method of perfectly matched layers, where impedance preservation is central to its effectiveness. In this talk, we briefly explain the idea of impedance preserving discretization, and show that the midpoint integration is also critical to developing absorbing boundary conditions for more complicated situations with differing signs of phase and group velocities, e.g. anisotropic and periodic media.

ABSTRACT. Discontinuous finite element methods have proven their numerical accuracy and flexibility, but they are still criticized for high number of degrees of freedom used for computation. The Trefftz approach provides a way to overcome these difficulties. The particularity of Trefftz-type methods is in special choice of basis functions: they represent the local solutions of the initial equations. Thus in case of time-dependent problems it requires a space-time mesh.

This approach has been widely used with time-harmonic formulations, while the studies are still limited for reproducing temporal phenomena.

In the present work we develop the theory for coupled elasto-acoustic systems and we present results for the first-order acoustic wave propagation system.

ABSTRACT. It is well-known that generalized Haar wavelets can precondition the single layer operator. This means that these functions are suitable for regularizing the scalar potential part of the electric field integral equation (EFIE). Unfortunately, however, a dual Haar basis is not regular enough to regularize the vector potential part of the EFIE if used in a naive way. In this work, we address this issue by leveraging on an explicit inverse of the dual Haar wavelet transformation matrix and on the scalar Calderón identity. We can prove that this strategy gives rise to a quasi-optimal preconditioner, i.e., the condition number grows polylogarithmically in the number of unknowns. Numerical results demonstrate the effectiveness of our approach.

ABSTRACT. We address the problem of constructing high-order implicit schemes for wave equations. We considered two classes of one-step schemes adapted to linear Ordinary Differential Equations, one based upon Padé approximant of exponential, other one requiring the inversion of an unique linear system such as SDIRK (Singly Diagonaly Implicit Runge Kutta) schemes.

ABSTRACT. A new and efficient two-level, non-overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier (2LM) framework. To accelerate the convergence, the transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are more accurate than the polynomial approximations used in the optimized Schwarz method (OSM). Another important ingredient affecting the convergence of a DD method, namely the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL-based transmission conditions and interface-wave based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains.

ABSTRACT. We describe the theory and practice of Schwarz-type domain decomposition preconditioners with local impedance or PML boundary conditions for solving discretisations of the high-frequency Helmholtz problem. Preconditioners for the pure Helmholtz problem can be constructed by applying the Schwarz algorithm to nearby absorptive problems. We present a new theory which shows that nearly optimal performance for the pure Helmholtz problem can be obtained by solving local impedance problems on the absorptive problem, provided these are combined with suitable restriction and prolongation operators and provided the absorption parameter is properly tuned. We also investigate numerically the benefits of replacing the local impedance solves with PML and discuss the benefits of combining these methods with suitable coarse grid problems .

ABSTRACT. A new weak coupling between the boundary element method and the finite element method for solving harmonic electromagnetic scattering problems is introduced. This method is inspired by domain decomposition techniques. Using suitable approximations of Magnetic-to-Electric operators to build proper transmission conditions allows for good convergence properties.

ABSTRACT. Time-harmonic acoustic scattering problems originally defined on unbounded regions are numerically solved. This is done by coupling the authors' recently developed high order local absorbing boundary condition (ABC) with high order finite difference methods. As a result, high order numerical methods with an overall order of convergence equal to the methods employed in the interior of the computational domain are obtained. These methods are compared in terms of their implementation, complexity, computational cost, and their convergence on several numerical experiments.

ABSTRACT. We study the spectrum of the Neumann-Poincar\'e operator $K^*_\varepsilon$ of a periodic collection of smooth inhomogeneities, as the period $\varepsilon \to 0$. Under the assumption that the pattern of inhomogeneity is strictly included in the periodicity cell, we show that the limit set $\lim_{\varepsilon \to 0} \sigma(K^*_\varepsilon)$ is the union of a Bloch spectrum and of a boundary spectrum, associated with eigenfunctions which are not too small (as functions in $H^1$) near the boundary.

ABSTRACT. Willis proposed effective constitutive relations applicable to the mean wave motion in composites with periodic microstructure. The focus of this work is to represent Willis' (effective) constitutive relations using an eigensystem approach and to explore its asymptotic behavior within the first Brioullin zone.

ABSTRACT. We study approximate models for transmissions problem between homogeneous and periodic half-planes when the period is small regarding to the wavelength. In a previous work, using matched asymptotic expansions techniques, we derived high order transmission conditions. Here, we study an approximate model associated to these high order transmission conditions which consists in replacing the periodic media by an effective one but the transmission conditions are not classical. We establish well-posedness for the approximate problem and error estimates and show some numerical results.

ABSTRACT. We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of cylindrical inclusions of radius $\rho$ and calculate the effective dielectric tensor of the medium. We assume that the dimensionless frequency $\nu \ll 1$ while the concentration of the inclusions is small. The approach is based on the expansion of the magnetic field in a power series in terms of a parameter proportional to the quasimomentum $q$. As a result we obtain an explicit expression of the effective tensor with the accuracy $O\left(\nu^4 + \rho^4 \right)$.

ABSTRACT. The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. Similar phenomena have been ob- served in optics and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory. Ramifications and recent progress on the analysis, numerics,and extensions to nonlinear wave models will be discussed.

ABSTRACT. We report here on the calculation of gravitational radiation and strong-field dynamics from spinning black-hole binaries (BHB). This research is crucial not only for interpreting the data gathered by ground-based interferometric gravitational wave (GW) detectors such as advanced LIGO, VIRGO, and other similar detectors under construction, but it is also important for understanding the astrophysical and cosmological implications of the final kick, mass, and spin of the BHB merger remnant.

ABSTRACT. Gravitational waves are a potential direct probe for the multi-dimensional flow during the first second of core-collapse supernova explosions. Here we outline the structure of the predicted gravitational wave signal from neutrino-driven supernovae of non-rotating progenitors from recent 2D and 3D simulations. We sketch some quantitative dependencies that govern the amplitudes of this signal and its evolution in the time-frequency domain.

ABSTRACT. Neutron stars are complicated objects, touching on all ten volumes of the famous Landau-Lifschitz textbook series on theoretical physics. They can emit several types of gravitational wave signals, from ringdowns of normal modes lasting a fraction of a second to continuous waves lasting longer than human civilization. I summarize the possible signals and describe how they can be used in the future to learn about the physics of matter under the most extreme conditions in the modern universe.

ABSTRACT. Solutions of the wave equation produce Lorentz-holomorphic functions that generate timelike surfaces via generalized Weierstrass-Enneper representations. These solutions can be naturally extended to non-smooth functions, allowing for variants of timelike surfaces with non-smooth features. In this context, we investigate families of isometric timelike surfaces arising from semi-rigid motions, which differ only in the directions of principal curvatures.

ABSTRACT. We consider the efficient numerical approximation of Maxwell's equations in a spatial domain with complex geometry.

For the space discretization discontinuous Galerkin (dG) methods are well-suited since they easily allow to use unstructured, possibly locally refined meshes. For the time integration standard explicit or implicit methods perform suboptimal. The former suffer from a constraint on the time step size (CFL condition). The latter require the solution of a large linear system in each time step.

If the geometry of the problem requires a grid with only few tiny elements, a combination of an explicit and an implicit time integrator provides a promising alternative. These so-called locally implicit methods have been considered in \cite{c1,c2,c3} for central fluxes dG discretizations.

We present an error analysis for the full discretization of Maxwell's equations with the locally implicit scheme \cite{c3} and show how this method can be extended to an upwind fluxes dG discretization.

ABSTRACT. This paper focuses on finding the electromagnetic (EM) field and the time-averaged Poynting vector produced after a time harmonic EM plane wave of an arbitrary fixed (linear) polarization is incident on an infinite perfect electric conducting (PEC) wedge. The aim is to find out how the polarization of this incident EM wave impacts the solution to diffraction by perfectly conducting wedges. We use the z invariance of the scatterer and the PEC boundary conditions to rewrite the EM field governed by Maxwell's equations in terms of two uncoupled scalar potentials or Debye potentials. These potentials will be functions of an arbitrary polarization angle and respectively solve the acoustic (or scalar) wedge problem with Dirichlet or Neumann boundary conditions.

ABSTRACT. Nanoplasmonics forms a major part of the field of nanophotonics, which explores how electromagnetic fields can be confined over dimensions on the order of or smaller than the wavelength. Here, we present an integral-equation formulation of the mathematical model that delivers accurate solutions in small computational times for surface plasmons coupled by periodic corrugations of flat surfaces an extension of single layer configurations to a more challenging case; multilayered configurations. The new configuration is composed of a thin layer of a metal (gold, silver, etc.) with depth larger than skin depth of the material, buried into different epoxies on top (glass/polymer substrate) and the bottom (liquid/water/blood).

ABSTRACT. We identify explicit conditions on geometry and material contrast for creating band gaps in 2-d photonic and 3-d acoustic crystal. The approach applied here makes use of the electrostatic and quasi-periodic source free resonances of the crystal, which deliver a spectral representation for solution operators associated with the propagation of waves inside the periodic high contrast medium. This, together with the Dirichlet spectrum and an auxiliary spectrum associated with the inclusions, delivers conditions for opening band gaps at finite contrast for a given inclusion geometry.

ABSTRACT. Using layer potential techniques we explore the plasmon resonance phenomenon for nanoparticles. We give asymptotic formulas on the size of the particle for: the shift in the resonances as the size of the particles increases, the far and inner field at the plasmonic resonances and the heat generated by them.

ABSTRACT. In this work, we want to solve a scattering problem outside a convex polygonal scatterer for a general class of boundary conditions using the Half-Space Matching Method. This method has been introduced in \cite{to-15} and consists in replacing the problem by a system of coupled integral equations whose unknowns (whose definition depends on the boundary conditions) live on the lines supported by each edge of the polygon. Using the Mellin Transform, we are able to show that this system is coercive + compact in presence of dissipation. Compared to integral methods, this method can be applied for some anisotropic elastic problems where calculating the green function might be expensive or impossible.

ABSTRACT. In this paper we discuss the discontinuous Petrov-Galerkin (DPG) method for high frequency wave propagation problems. The DPG method offers uniform pre-asymptotic stability for any wave number, and this allows for a fully automatic hp-adaptive algorithm. In addition, being a minimum residual method, DPG always delivers a Hermitian positive definite matrix. We introduce a new iterative solution scheme which benefits from these attractive properties. This novel solver is integrated within the DPG adaptive procedure by constructing a two-grid-like preconditioner for the Conjugate Gradient (CG) method. We demonstrate our results using a 2D acoustics problem and show convergence in terms of iterations at a rate independent of the mesh and the wavenumber.

ABSTRACT. When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is notoriously difficult to solve, in fact increasingly so at higher frequencies. Instead of solving the Helmholtz equation in the frequency domain, we thus reformulate it in the time domain and seek a time-periodic solution of the wave equation via Controllability Method (CM). Although straightforward time integration of the wave equation can actually be used to reach the time-periodic solution, its convergence is usually too slow in practice. The CM approach speeds up convergence to the time-periodic solution using the (unknown) initial conditions as control variables. At each iteration the CM requires only the solution of the wave equation and that of a positive definite elliptic problem; hence, it is inherently parallel. To overcome the stringent CFL-condition due to local mesh refinement, we use local time-stepping methods based on explicit Runge-Kutta schemes.

ABSTRACT. We give a brief introduction to stochastic gravitational-wave backgrounds and discuss the standard detection methods used to search for them.

ABSTRACT. I present 3 different ways that gravitational waves can be used to probe fundamental physics and cosmology, focusing on searches for the stochastic gravitational wave background. In particular, I discuss searches for cosmic inflation, cosmic strings, and non-tensor polarizations.

ABSTRACT. General Relativity has passed all Solar System experiments and binary pulsar observations with flying colors. Recent direct detections of gravitational waves from binary black hole coalescences offer us unique testbeds of gravity in the regime where the field is both strong and dynamical. Based on a Fisher analysis, we derive constraints on parameterized deviations in the gravitational waveform from General Relativity. We then map such constraints to those on fundamental pillars of General Relativity, such as the equivalence principle, Lorentz/parity invariance and the dispersion relation of the massless graviton. We find that one can only place relatively weak constraints on generation mechanisms of gravitational waves due to our lack of knowledge of the modified gravitational waveform in the merger regime. On the other hand, one can place relatively strong constraints on modified propagation mechanisms of gravitational waves that are complementary to the existing bounds.

ABSTRACT. In this paper we propose efficient boundary element schemes for the solution of high-frequency convex scattering problems. Our approach is based on frequency dependent changes of variables in forming Galerkin approximation spaces and newly developed asymptotic expansions of the normal derivative of the total field.

ABSTRACT. For problems of time harmonic scattering by polygonal obstacles, embedding formulae provide a useful and frequency-independent means of computing the far field pattern for a large class of incident fields, given the far field pattern of a small set of canonical problems. The number of such problems depends only on the geometry of the scatterer. Whilst the formulae themselves are in principle exact, any implementation will inherit numerical error from the method used to solve the canonical problems, leading to relatively large error at certain points. Here we identify the cause of this problem, and present an alternative approach which overcomes this problem.

ABSTRACT. We propose a new hybrid strategy between the boundary element method (BEM) and ray tracing in order to allow the accurate and quick simulation of high frequency Non Destructive Testing (NDT) configurations involving diffraction phenomena. Results from its implementation to 2D acoustic NDT-like diffraction configurations are presented. The strategy proposed is however generic, and can be extended to three-dimensional configurations and elastodynamic wave propagation.

ABSTRACT. We compute near fields using boundary integral equation methods for 2D acoustic scattering by an obstacle with an analytic boundary. A classical method to approximate the solution everywhere consists of using the same quadrature points as used in the quadrature rule (Nyström method) for the underlying boundary integral equation. It is established that this method creates an O(1) error for a fixed number of quadrature points. Our goal is, for a fixed number of quadrature points and without using high- order Nyström methods, to develop a method to address this O(1) error. Similar to numerical method for approximating singular integrals, we subtract from the associated kernel the asymptotic expansion that captures the nearly singular behavior. Accurate computation of near fields is needed for optical scattering by nanostructures and for other related problems.

ABSTRACT. We are interested in solving sound-hard scattering problems by two obstacles with a large obstacle subject to high-frequency regime relatively to the wavelength and a small one subject to low-frequency regime. The iterative method presented allows to decouple the two obstacles and to use Geometric Optic for the large obstacle and Boundary Element Method for the small obstacle.

ABSTRACT. We present boundary integral equation formulations of scattering problems with mixed boundary conditions that rely on smooth blendings of the different types of boundary conditions.

ABSTRACT. We investigate the spectrum of regular quantum trees through a relation to orthogonal polynomials depending on two variables. For self-adjoint Robin vertex conditions, the behavior of the low eigenvalues is analyzed through the interlacing property of the roots of orthogonal polynomials. The spectrum approaches a band-gap structure as the length of the quantum tree increases. The lowest band becomes negative for large derivative-to-value ratio, and there emerge two isolated eigenvalues below the bands.

ABSTRACT. We focus on the construction of transmission conditions for optimized Schwarz domain decomposition methods applied to time-harmonic elastic wave scattering problems solved numerically with finite element methods. In particular, we investigate different local approximations of the Dirichlet-to-Neumann map, and compare their impact on the convergence rate of the domain decomposition algorithm.

ABSTRACT. When applied to the Helmholtz or time-harmonic Maxwell equations with absorption, classical-additive-Schwarz domain-decomposition preconditioning works well if the absorption is large enough.

ABSTRACT. A deterministic numerical method able to calculate the first two statistical moments of the scattered field by a periodic perfect electric conductor surface with stochastic surface perturbations was developed and implemented. The associated electric field integral equation was solved by using the Galerkin boundary element method with discretization based on piecewise constant hierarchical basis, e.g. Haar’s wavelets. The proposed deterministic approach converges faster than the Monte-Carlo method with lower computational effort.

ABSTRACT. We consider the numerical solution of time-harmonic scattering of acoustic and electromagnetic waves from obstacles with uncertain geometries. Using first-order shape derivatives, we derive deterministic boundary integral equations for the mean field and the two-point correlation function of the random solution for a soft obstacle Dirichlet problem. Sparse tensor Galerkin discretizations of these equations are implemented with the so-called combination technique. We generalize the method to non-nested meshes using a nodal transfer operator. Similar discretization errors for the covariance is achieved with $\mathcal{O}(N \log N)$ degrees of freedom instead of $\mathcal{O}(N^2)$. Performance comparison of our approach to classic Monte-Carlo Galerkin formulation is given for different shapes. Finally, we verify the robustness of the sparse tensor approximation and compare it to low-rank approximations techniques.

ABSTRACT. We prove bounds on the heterogeneous Helmholtz equation $ \nabla\cdot(A \nabla u ) + k^2 n u=-f$ that are explicit in $k$, $A$, and $n$, and then extend these to the case when $A$ and $n$ are random fields.