Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation

We propose a new method for the e cient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter ω ≥ 0, typically varying in a large interval. Our approach is based, for xed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The di culty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both di cult to evaluate and to approximate. Here we use the quantized-tensor train (QTT) approximation method for functional m-vectors of logarithmic complexity in m in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and e cient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the e ciency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions. AMS subject classi cations: 65F30, 65F50, 65N35, 65D30