## Daan Huybrechs (Catholic University of Leuven)

We show how a sparse discretisation matrix can be obtained for an integral operator *A*_{k}
with an oscillatory kernel function *G*_{k}(x, y) = G_{1}(x, y) exp(ikG_{2}(x,y)),
where *k* is a parameter that explicitly determines the frequency of this function. The first result is that,
for a fixed function *u(y)*, the function *(A*_{k}u) (x) can be computed efficiently and with arbitrary
accuracy, even as *k* grows very large. The number of operations is in fact bounded in *k*.
The method can be extended to evaluate *(A*_{k}u_{k}) (x),
where *u*_{k}(y) = u_{1}(y) exp(iku_{2}(y)) may be
increasingly oscillatory.

We proceed to the solution of oscillatory integral equations of the form *(λI+A*_{k}) u = f_{k}.
Such problems are considerablymore complicated than the problem of applying an integral
operator. We show that a sparse discretisation remains possible in the case of scattering
by a convex obstacle, leading immediately to an efficient solution method.