ABSTRACT. I plan to discuss recent work (joint with Roland Griesmaier) on the fixed frequency inverse source problem, emphasizing how we adapted uncertainty principles from the work of Donoho-Stark [1] to the far field splitting and data completion problems.

ABSTRACT. Multiple seismic data sets are often recorded to monitor changes in Earth properties. We test a method for imaging those changes, Alternating Full-Waveform Inversion (AFWI), to determine how errors in the model translate into errors in the final image. The results appear to follow a normal distribution, which opens up the possibility of quantifying these errors.

ABSTRACT. In this paper we present a two-step framework for uncertainty quantification of estimated seismic images and velocity models. First, we combine the adaptive Metropolis-Hastings algorithm with a fast Helmholtz solver to provide uncertainty estimates of a velocity model based on raw waveform data. Second, this error estimate is propagated through an imaging operator to ask meaningful questions about the error. We demonstrate several methods for presenting this uncertainty in a manageable and useful manner.

ABSTRACT. In this work, we present a new posterior dis- tribution to quantify uncertainties in solutions of wave-equation based inverse problems. By introducing an auxiliary variable for the wave- fields, we weaken the strict wave-equation con- straint used by conventional Bayesian approaches. With this weak constraint, the new posterior distribution is a bi-Gaussian distribution with respect to both model parameters and wave- fields, which can be directly sampled by the Gibbs sampling method.

ABSTRACT. We consider inverse medium scattering in two or three dimensions modeled by the Helmholtz equation. To this end, we set up an efficient minimization-based inversion scheme that follows on the one hand the paradigm to, roughly speaking, minimize the discrepancy but on the other hand takes into account various structural a-priori information via suitable penalty terms. This allows for instance to combine sparsity-promoting with total variation-based regularization, while at the same time respecting physical bounds for the inhomogeneous medium. The flexibility of our approach is due to a primal-dual algorithm that we employ to minimize the corresponding Tikhonov functional. We show feasibility and performance of the resulting inversion scheme via reconstructions from synthetic and measured data.

ABSTRACT. We consider the problem of detecting delamination of interfaces in composite materials using acoustic waves or separation between integrated circuit components using electromagnetic waves. We model the separation as a thin opening between two materials of different material properties, and using asymptotic techniques we derive a reduced model where the delaminated region is replaced by jump conditions on the total field and flux. The reduced model has potential singularities due to the edges of the delaminated region, and we show that the forward problem is well posed for a large class of possible delaminations. We then design a special Linear Sampling Method for detecting the shape of the delamination assuming that the background, undamaged, state is known. Finally we show, by numerical experiments, that our algorithm can indeed determine the shape of delaminated regions.

ABSTRACT. This paper investigates the use of Stekloff eigenvalues for Maxwell's equations to detect changes in a scatterer using remote measurements of the scattered wave. Because the Stekloff eigenvalue problem for Maxwell's equations is not a standard eigenvalue problem for a compact operator, we propose a modified Stekloff problem that restores compactness. In order to measure the modified Stekloff eigenvalues of a domain from far field measurements we perturb the usual far field equation of the Linear Sampling Method by using the far field pattern of an auxiliary impedance problem related to the modified Stekloff problem. We show existence of modified Stekloff eigenvalues, well-posedness of the corresponding auxiliary exterior impedance problem and provide theorems that support our claim to be able to detect modified Stekloff eigenvalues from far field measurements. Preliminary numerical results are reported.

ABSTRACT. We consider multiple scattering of electromagnetic waves by randomly distributed, densely packed, spherical particles. For space discretization, we use the discrete exterior calculus (DEC). The time-dependent problem is solved by a wave frequency -corrected time-stepping scheme, and the time-harmonic solution is obtained by the exact controllability method.

ABSTRACT. For many applications in numerical physics, fast convolutions with a Green kernel on unstructured grids are needed to compute in a reasonable time the matrix-vector products. To this aim, many methods have been developed since last decades. There are divided in two major classes, those which use analytical approximation of the Green kernel (FMM, SCSD, etc.) and those based on algebraic compression (ACA, SVD, etc.). Associated to this paper, we provide a new open-source Matlab toolbox (OpenHmX) for the second class of compression.

ABSTRACT. High-order finite difference (FD) methods for evolving simulated waves in time typically permit larger time steps than discontinuous Galerkin (DG) methods of equal order and degrees of freedom. This gap in efficiency widens as the order increases. If, however, one uses finite-difference-style basis functions within a Galerkin formulation, one can enjoy the stability benefits of a built-in discrete energy and upwinding without sacrificing the efficiency of large time steps. We call this new approach the discontinuous Galerkin difference (DGD) method.

ABSTRACT. We present the coupling of a nodal discontinuous Galerkin (DG) scheme with high-order absorbing boundary conditions (HABCs) for the simulation of transient wave phenomena. The HABCs are prescribed on the faces of a cuboidal domain in order to simulate infinite space. To preserve accuracy at the corners and the edges of the domain, novel compatibility conditions are derived. The method is validated using 3D computational results.

ABSTRACT. We aim at determining the acoustic field radiated in 2D by a time-harmonic source in a fluid in flow. We use Goldstein's equations, well-adapted to describe the complex coupling between the radiation of acoustic waves and the transport of acoustic vortices. These involve a vectorial harmonic transport equation which is proved to be well-posed outside a spectrum of frequencies corresponding to resonant streamlines. Then the full model is shown to be well-posed under a coercivity condition, implying a subsonic flow with a small enough vorticity.

ABSTRACT. We consider the generation of incompressible waves in a rapidly-rotating, electrically conducting, Boussinesq fluid stirred by buoyant anomalies, a situation thought to arise in many planetary cores. In the absence of a magnetic field, the dispersion of energy from a localised source is known to be dominated by low-frequency inertial waves, which have wavevectors approximately orthogonal to the rotation axis. We study the modification to this process by a large-scale ambient magnetic field consistent with that found in the outer core of the Earth. We find that the response is again dominated by wavevectors normal to the rotation axis, but these now take the form of hybrid "inertial-Alfven waves". These propagate along the rotation axis at half the speed of conventional low-frequency inertial waves, but also dispatch energy along magnetic field lines at the Alfven velocity. We demonstrate their significance via a simple model problem.

ABSTRACT. We study the propagation of small time-harmonic acoustic perturbations of a stationary fluid flow. We use the Goldstein equations, coupling the acoustic phenomena to the vorticies transport. On a simple toy geometry, allowing explicit calculations, we show the existence of resonant frequencies for recirculating flows, corresponding to an ill-posed problem for the transport of vorticies. We prove that far enough from these resonances, the Goldstein model is well-posed under a coercivity condition. We determine numerically the frequency validity domain of this condition.

ABSTRACT. In ideal compressible hydrodynamics there is an isomorphism between spatially one-dimensional unsteady and two-dimensional steady supersonic flow called piston analogy. This notice shows that this is also true for non-equilibrium magnetosonic flow under alignment of undisturbed flow and magnetic field in case of steady flow. An example for two generic problems, i.e. the signal problem of radiation into a half space and steady flow along a kinked wall bounding a half space, is given.

ABSTRACT. We consider the Cauchy problem for the Kuznetsov equation (a model of non-linear acoustics) and we prove local and global well-posedness results both with and without viscosity. Using these results, we also prove that the solutions of the Kuznetsov equation are approximations of the isentropic Navier-Stokes system solutions.

ABSTRACT. Rocks and concrete have a strong nonlinear behavior. Moreover, the speed of sound diminishes slowly under a dynamic loading. To reproduce this behavior, an internal-variable model of continuum is proposed. It is composed of a constitutive law for the stress and an evolution equation for the internal variable. Qualitatively, the model reproduces the experiments.

ABSTRACT. A fractional time derivative is introduced into Burgers equation to model losses of nonlinear waves arising in acoustics. A diffusive represen- tation of the fractional derivative replaces the nonlocal operator by a continuum of memory variables that satisfy local ordinary differential equations. A quadrature formula yields a sys- tem of local partial differential equations. The quadrature coefficients are computed by opti- mization with a positivity constraint. One re- solves the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Exten- sive details can be found in [3].

ABSTRACT. The objective of this work is to design an efficient and accurate time integration strategy based on exponential integrators and explicit time advancing schemes for the system of time- domain Maxwell equations discretized in space with a high order discontinuous Galerkin scheme formulated on locally refined unstructured meshes.

ABSTRACT. Although the theoretical description of lasing has been studied for many decades, only recently has it become practical to accurately model the lasing process in complex microstructured lasers, such as random lasers or photonic-crystal laser cavities. A key enabling factor is the SALT (steady state ab-initio laser theory) description pioneered by Tureci, Stone, and others starting in 2006, which reduces the complex time-dependent Maxwell-Bloch equations to a much simpler frequency-domain nonlinear eigenproblem for steady-state lasing modes. The SALT equations themselves resisted general solution for several years, but recently we have developed efficient numerical approaches to solving SALT for complex 3d structures. We can even exploit existing linear Maxwell solvers, simply performing a sequence of linear solves in an Anderson-accelerated loop to solve the nonlinear problem. Moreover, given this numerical foundation, a whole host of new analytical and semi-analytical results become possible, via perturbation theory around the SALT modes.

ABSTRACT. In time domain geophysics context, Discontinuous Galerkin (DG) methods are widely studied and used for the simulation of waves propagation. They can be applied to harmonic problems too but their main drawback is that the linear system to solve becomes very huge. Indeed, the number of degrees of freedom is really large as compared to classical finite element methods. We address this issue by considering a new class of DG methods, the hybridizable discontinuous Galerkin (HDG) method. We have formulated and studied the HDG method applied to 2D and 3D elastic waves propagation equations. Then, to be able with realistic 3D geophysical problems, we compare different solvers, a direct one (Mumps) and an hybrid one (Maphys) that combines direct and iterative solvers by using an algebric domain decomposition method.

ABSTRACT. The paper develops a reliable numerical method to solve inverse dynamical problem of seismic waves propagation on the base of nonlinear least squares formulation which is widely known as Full Waveform Inversion (FWI). The key issue on this way is correct reconstruction of macrovelocity component of the model with input seismic data without time frequencies less than 5Hz and reasonable source–recievers offsets. To provide correct macrovelocity reconstruction we modify regular nonlinear leastsquares formulation used in standard versions of FWI by decomposing the model space into two subspaces: • slowly varying in space functions (propagators p) which do not change direction of propagation of seismic energy, but governs travel times; • sharply changing in space functions (space reflectivity r) which do not change travel time, but turn propagation direction towards acquisition.

ABSTRACT. In exploration seismology constructing an accurate velocity model is imperative. One of the algorithms which can lead to an accurate velocity model is Full Waveform Inversion (FWI). Standard FWI uses only scalar data such as pressure to construct a velocity model and does not provide any directivity information about the wavefields. Extending FWI to vector data allows us to use both pressure and velocity components at the same time, giving directivity information about the wavefields. By extending FWI to vector data and thus improving the input data to FWI, we obtain both improved resolution and directivity information.

ABSTRACT. We introduce a weakly conforming discontinuous Petrov-Galerkin method in space and time for the acoustic wave equation in heterogeneous media. The fully implicit high-order discretization is a minimal residual method for the first-order system with discontinuous test spaces on a decomposition of the space-time cylinder and with trace degrees of freedom on the skeleton of this decomposition.

This is applied to a problem in seismic inversion, where the permeability is recovered approximately from measurements of the scattered wave at sampling points. The ill-posed problem in seismic imaging is regularized by an inexact Newton method, where every increment is evaluated by a conjugate gradient iteration. In every iteration step, the residual is computed solving the wave equation, and for the gradient the adjoint wave equation with a right-hand side depending on the full space-time solution is approximated. The efficiency of the method is demonstrated by numerical examples in two space dimensions.

ABSTRACT. We investigate a sampling method to recover the support of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency without knowledge of the geometry of the periodic background media. We analyse the method in a simplified case where the infinite domain is truncated using periodic boundary conditions (which would correspond with the semi-discretized version of the continuous model with respect to the Floquet-Bloch (FB) variable [2]). As a data for the inverse problem, (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and the near field operator associated with single FB-mode measurements then exploit them to built an indicator function of the defect. Numerical validating results are provided for synthetic data in dimension 2.

ABSTRACT. We derive the topological derivatives of the homogenized coefficients associated to a periodic material, with respect of the small size of a penetrable inhomogeneity introduced in the unit cell that defines such material. In the context of antiplane elasticity, this work extends existing results to (i) time-harmonic wave equation and (ii) second-order homogenized coefficients, whose contribution reflects the dispersive behavior of the material.

ABSTRACT. We consider time-harmonic scattering problems of acoustic waves from either a bi-periodic inhomogeneous medium which is absorbing on an open set or a bi-periodic sound-soft obstacle in R^3, both with a local perturbation. For this, the Floquet-transform is used to reformulate the problem as an equivalent system of coupled variational problems on a bounded domain. This system possesses a unique solution for both scattering problems. Furthermore, we calculate the Frechet derivative of the operator, which maps the perturbation to the solution.

ABSTRACT. Unified Transform Method (UTM), alternatively called the Fokas Method, has recently advanced the understanding of boundary value problems (BVPs) in the case of linear and nonlinear integrable equations. This method provides significant advantages computationally and allows for the study of many equations with various boundary conditions using a unified conceptual framework.

The surprising phenomenon of ``dispersive quantization'' describes the solutions to a wide range of dispersive wave models for rough initial data on bounded domains exhibiting fractal profiles at irrational times and quantized, meaning discontinuous but otherwise smooth, at rational times. This is an example of an observed, but as yet poorly understood, dynamical behaviors that depend crucially upon the large wave number asymptotics of the dispersion relation. The UTM will enable a better understanding of the effects of the boundary conditions in linear and nonlinear dispersive models with various boundary conditions and new numerical methods.

ABSTRACT. Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In \cite{DG09} a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here convergence (in the PDE sense) of the LTS-LF method is proved when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

ABSTRACT. The numerical simulation of acoustic waves propagating from the near surface of the Sun can be performed by solving a Helmholtz equation whose main feature is having a coefficient that is exponentially decaying into the atmosphere of the Sun. Using high-order finite element methods, there is a need in truncating the computational domain by introducing a boundary surrounding the Sun. We propose a family of Radiation Boundary Conditions that are derived first from the factorization of the Helmholtz equation. When the Sun is approximated by a sphere, the corresponding mixed problem is well-posed and a series of numerical experiments allows to identify a second-order condition that gives accurate simulations at any frequency. The condition can be used for regular boundaries including the sphere but not only which makes it useful for any application involving a Helmholtz equation set into a heterogeneous medium.

ABSTRACT. We investigate the feasability of constructing local solutions to the Helmholtz equation thanks to high-order DG finite element approximations of the Dirichlet-to-Neumann operator. This is then used in a Trefftz Discontinuous Galerkin method in place of a boundary element method that was succesfully applied to solve the Helmholtz problem in very large domains. We perform comparisons between the two approaches by considering large domains of propagation including heterogeneities.

ABSTRACT. The Boundary Element Method (BEM) is a powerful numerical method for the computational simulation of wave scattering problems in acoustics and electromagnetics. Because of the surface integral representation, the number of degrees of freedom scales favourably compared to volumetric methods. However, solving the dense set of linear equations poses severe limitations on the maximum frequency that can be used on present-day computing platforms. This paper presents the combined use of parallelisation, preconditioning, and compression techniques to achieve large-scale BEM simulations.

ABSTRACT. Adopting a post-Minkowski approximation for the Einstein-matter equations, we describe work towards numerical construction of helically symmetric spacetimes representing binary neutron stars. Established methods for solving the constraints of general relativity, thereby producing initial data for the Einstein-matter equations, start with trial data. We seek trial data without conformal flatness which gives rise to "junk radiation". Our work relies on sparse, modal, spectral-element methods and 2-center domain decompositions. We gratefully acknowledge NSF DMS 1216866 for supporting most of this work.

ABSTRACT. Well-posed variational formulations for resonant time-harmonic Maxwell’s equations are an important matter as they are convenient for finite element methods and they help understand the resonant heating in tokamaks. Still, the limit of the viscous system when the friction parameter goes to 0 is ill-posed and the difficulty is that the solution has singularities of type 1/x. We consider the cold plasma model to study the influence of a radio-frequency (RF) electromagnetic wave sent in a tokamak plasma for heating purpose. Combin- ing the vanishing viscosity principle with some well defined manufactured solution leads to a well-posed variational formulation of the equa- tions in the case of a normal incidence heating wave.

ABSTRACT. In this paper we present a numerical scheme for the wave kinetic equation, based on the theory of jump Markov processes. The scheme is implemented in WKBeam, a code which describes electromagnetic wave beams in realistic nuclear fusion devices, accounting in particular for the effect of density fluctuations due to plasma turbulence.

ABSTRACT. We study a time-harmonic waves problem in a 2D waveguide. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length L. We analyse the behaviour of the complex scattering coefficients R, T as L goes to +\infty and we exhibit situations where non reflectivity (R = 0, |T| = 1), perfect reflectivity (|R| = 1, T = 0) or perfect invisibility (R = 0, T = 1) hold.

ABSTRACT. We present a new energy based discontinuous Galerkin method for Hamiltonian systems. Numerical experiments illustrating the properties of the method when applied to Korteveg de Vries equation are also presented.

ABSTRACT. A three-parameter family of Boussinesq systems for internal waves is considered. The systems describe the propagation of internal waves in a two-layer interface problem with rigid lid assumption and under the Boussinesq regime for both fluids. After analyzing the well-posedness and the existence of solitary wave solutions, in one and two dimensions, numerical studies concerning the generation and dynamics of the waves will be presented.

ABSTRACT. When higher order, beyond KdV, shallow water equations are considered, momentum and energy are no longer exact invariants. However, adiabatic invariants (AI) can be found. Their existence results from the general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. The exactness of these adiabatic invariants is shown in numerical tests