In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods

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Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering ∗

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns in classical boundary element methods. In addition, the corresponding discretization matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains accuracy for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretization matrix for the oscillatory problem. Moreover, numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency. The sparse discretization applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.

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A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems

We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.

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Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

Based on the transformation y = g(x), some new efficient Filon-type methods for integration of highly oscillatory function $$\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x$$ with an irregular oscillator are presented. One is a moment-free Filon-type method for the case that g(x) has no stationary points in [a,b]. The others are based on the Filon-type method or the asymptotic method together with Filon-type method for the case that g(x) has stationary points. The effectiveness and accuracy are tested by numerical examples.

Efficient Filon-type methods for $$\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x$$

We argue that two specific wave packet families---curvelets and wave atoms---provide powerful tools for representing linear systems of hyperbolic differential equations with smooth, time-independent coefficients. In both cases, we prove that the matrix representation of the Green's function is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals, whose location is predicted by geometrical optics. This result holds only when the basis elements obey a precise parabolic balance between oscillations and support size, shared by curvelets and wave atoms but not wavelets, Gabor atoms, or any other such transform. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles. We also provide fast digital implementations of tight frames of curvelets and wave atoms in two dimensions. In both cases the complexity is O(N² log N) flops for N-by-N Cartesian arrays, for forward as well as inverse transforms. Finally, we present a geometric strategy based on wave atoms for the numerical solution of wave equations in smoothly varying, 2D time-independent periodic media. Our algorithm is based on sparsity of the matrix representation of Green's function, as above, and also exploits its low-rank block structure after separation of the spatial indices. As a result, it becomes realistic to accurately build the full matrix exponential using repeated squaring, up to some time which is much larger than the CFL timestep. Once available, the wave atom representation of the Green's function can be used to perform 'upscaled' timestepping. We show numerical examples and prove complexity results based on a priori estimates of sparsity and separation ranks. They beat the O(N^3) bottleneck on an N-by-N grid, for a wide range of physically relevant situations. In practice, the current wave atom solver can become competitive over a pseudospectral method in the regime when the wave equation should be solved several times with different initial conditions, as in reflection seismology.

Curvelets, Wave Atoms, and Wave Equations

1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, A.Iserles@damtp.cam.ac.uk 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway, S.P.Norsett@math.ntnu.no 3 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, S.Olver@damtp.cam.ac.uk

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S. Olver

Papers

Highly Oscillatory Quadrature: The Story soFar