  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 ...
