We investigate the use of the local Multiple Trace Formulation (MTF) in solving time-harmonic elastic wave transmission problems. Originally devised for heterogeneous acoustic media, MTF recasts the boundary value problem as a well-posed system of first-kind boundary integral equations, naturally amenable to parallelization and preconditioning. The formulation takes independent displacement and traction unknowns per subdomain, enforces Calderón identities locally, and imposes transmission conditions weakly across interfaces. For the sake of clarity, all derivations in the one-dimensional setting are carried out explicitly, with illustrative examples provided in two dimensions. We analyze the effects of frequency and material contrast on the convergence of the GMRES iterative solver. Finally, we present preliminary results for the square Calderón preconditioner and discuss its potential to further accelerate iterative solvers. We study single homogeneous scatterers as a first step toward MTF techniques for elasticity transmission problems. The heterogeneous case is left for future work.

We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes (2012) for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies proposed.Previous article in issue

Multilayered diffraction gratings are an essential component in many optical devices due to their ability to engineer light. We propose a first-order optimization strategy to maximize diffraction efficiencies of such structures by a fast approximation of the underlying boundary integral equations for polarized electromagnetic fields. A parametric representation of the structure interfaces via trigonometric functions enables the problem to be set as a parametric optimization one while efficiently representing complex structures. Derivatives of the efficiencies with respect to geometrical parameters are computed using shape calculus, allowing a straightforward implementation of gradient descent methods. Examples of the proposed strategy in chirped pulse amplification show its efficacy in designing multilayered gratings to maximize their diffraction efficiency.

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequencies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nyström methods.

Calderón multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderón preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements.

We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance, and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Taylor expansions, we derive tensor deterministic first-kind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique [M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and P. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263--281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128--141]. We supply extensive numerical experiments confirming the predicted error convergence rates with polylogarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.

We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Well-posedness of the discrete problems is established as well as algebraic or even exponential convergence rates depending on the regularities of both arcs and excitations. Our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range. Moreover, we present a suitable compression algorithm that further accelerates computational times.

We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood frequencies, continuity and coercivity results are derived to prove wellposedness of the associated first kind boundary integral equations. 

This work presents the implementation, numerical examples, and experimental convergence study of first- and second-order optimization methods applied to one-dimensional periodic gratings. Through boundary integral equations and shape derivatives, the profile of a grating is optimized such that it maximizes the diffraction efficiency for given diffraction modes for transverse electric polarization. We provide a thorough comparison of three different optimization methods: a first-order method (gradient descent); a second-order approach based on a Newton iteration, where the usual Newton step is replaced by taking the absolute value of the eigenvalues given by the spectral decomposition of the Hessian matrix to deal with non-convexity; and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, a quasi-Newton method. Numerical examples are provided to validate our claims. Moreover, two grating profiles are designed for high efficiency in the Littrow configuration and then compared to a high efficiency commercial grating. Conclusions and recommendations, derived from the numerical experiments, are provided as well as future research avenues.