Efficient computation of highly oscillatory integrals by using QTT tensor approximation
Talk by Dr. Alexander Veit
Speaker invited by: Prof. Dr. Stefan A. Sauter
Date: 01.10.14 Time: 16.15 - 17.15 Room: Y27H25
We propose a new method for the efficient 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 fixed but arbitrary oscillator, on the precomputation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to an approximation of the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both difficult 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 efficient storage of these functions for a wide range of grid and frequency parameters.We will present theoretical results and numerical experiments that illustrate the efficiency of the QTT-based numerical integration scheme in one and several spatial dimensions.