2014/08/22 by Boris N. Khoromskij, Khoromskij, Boris, Alexander Veit +1 · 2 citations
Mathematics · #65D30 #65F30 #65F50 #FOS: Mathematics #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1408.5224
openalex publication_date 2014/08/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We propose a new method for the efficient approximation of a class of highly\noscillatory weighted integrals where the oscillatory function depends on the\nfrequency parameter \ω \≥ 0, typically varying in a large interval. Our\napproach is based, for fixed but arbitrary oscillator, on the pre-computation\nand low-parametric approximation of certain \ω-dependent prototype\nfunctions whose evaluation leads in a straightforward way to recover the target\nintegral. The difficulty that arises is that these prototype functions consist\nof oscillatory integrals and are itself oscillatory which makes them both\ndifficult to evaluate and to approximate. Here we use the quantized-tensor\ntrain (QTT) approximation method for functional m-vectors of logarithmic\ncomplexity in m in combination with a cross-approximation scheme for TT\ntensors. This allows the accurate approximation and efficient storage of these\nfunctions in the wide range of grid and frequency parameters. Numerical\nexamples illustrate the efficiency of the QTT-based numerical integration\nscheme on various examples in one and several spatial dimensions.\n