A high frequency boundary element method for scattering by a class of nonconvex obstacles

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-014-0648-7EPSR

High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.Comment: 24 pages, 9 figures and the implementation is available on https://github.com/popsomer/asyBEM/release

Acoustic scattering : high frequency boundary element methods and unified transform methods

We describe some recent advances in the numerical solution of acoustic scattering problems. A major focus of the paper is the efficient solution of high frequency scattering problems via hybrid numerical-asymptotic boundary element methods. We also make connections to the unified transform method due to A. S. Fokas and co-authors, analysing particular instances of this method, proposed by J. A. De-Santo and co-authors, for problems of acoustic scattering by diffraction gratings

Parallel Controllability Methods For the Helmholtz Equation

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures

Solving forward and inverse Helmholtz equations via controllability methods

Waves are useful for probing an unknown medium by illuminating it with a source. To infer the characteristics of the medium from (boundary) measurements, for instance, one typically formulates inverse scattering problems in frequency domain as a PDE-constrained optimization problem. Finding the medium, where the simulated wave field matches the measured (real) wave field, the inverse problem requires the repeated solutions of forward (Helmholtz) problems. Typically, standard numerical methods, e.g. direct solvers or iterative methods, are used to solve the forward problem. However, large-scaled (or high-frequent) scattering problems are known being competitive in computation and storage for standard methods. Moreover, since the optimization problem is severely ill-posed and has a large number of local minima, the inverse problem requires additional regularization akin to minimizing the total variation. Finding a suitable regularization for the inverse problem is critical to tackle the ill-posedness and to reduce the computational cost and storage requirement. In my thesis, we first apply standard methods to forward problems. Then, we consider the controllability method (CM) for solving the forward problem: it instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period with the conjugate gradient (CG) method, the CMCG method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. Based on the original CM, introduced in 1994 by Bristeau et al., for sound-soft scattering problems, we extend the CMCG method to general boundary-value problems governed by the Helmholtz equation. Numerical results not only show the usefulness, robustness, and efficiency of the CMCG method for solving the forward problem, but also demonstrate remarkably accurate solutions. Second, we formulate the PDE-constrained optimization problem governed by the inverse scattering problem to reconstruct the unknown medium. Instead of a grid-based discrete representation combined with standard Tikhonov-type regularization, the unknown medium is projected to a small finite-dimensional subspace, which is iteratively adapted using dynamic thresholding. The adaptive (spectral) space is governed by solving several Poisson-type eigenvalue problems. To tackle the ill-posedness that the Newton-type optimization method converges to a false local minimum, we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy. Numerical examples illustrate the usefulness of the ASI approach, which not only efficiently and remarkably reduces the dimension of the solution space, but also yields an accurate and robust method