FB 6 Mathematik/Informatik

Institut für Mathematik

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SS 2020

19.05.2020 um 14:15 Uhr Meetingroom

Yuya Suzuki (KU Leuven, Belgien)

Lattice rules for integration over ℝd through orthogonal projections onto periodic spaces

In this talk, we consider numerical integration over ℝd using lattice rules. For integration over the (unit) cube it is known that tent-transformed lattice rules can achieve up to second order convergence in a non-periodic unanchored Sobolev space [1, 2]. We show that it is possible to obtain higher order convergence, including truncation error, in an unanchoresd Sobolev space using lattice rules for integration over ℝd. We make use of an orthogonal projection from an unanchored Sobolev space to a Korobov space. Using this projection we can measure/quantify the non-periodicity of a function, and by imposing some decay condition on the integrand we achieve higher order. 
We also numerically demonstrate our method and show that our method can be superior than methods based on interlaced Sobol sequences and Gauss--Hermite sparse grids.
[1] J. Dick, D. Nuyens and F. Pillichshammer: Lattice rules for nonperiodic smooth integrands, Numer. Math., 126:259-291, 2014.
[2] T. Goda, K. Suzuki and T. Yoshiki: Lattice rules in non-periodic subspaces of Sobolev spaces, Numer. Math., 141:399-427, 2019

16.06.2020 um 14:15 Uhr Meetingroom

Adrian Ebert (RICAM Linz)

Fast construction of good rank-1 lattice rules for multivariate numerical integration in weighted function spaces


05.08.2020 um 14:15 Uhr Meetingroom

Alexander Lindenberger (Johannes Kepler University Linz, Austria)

Banach Spaces with Reproducing Kernels

25.08.2020 um 14:15 Uhr Meetingroom

Paul Catala (ENS, Paris)

A moment approach for off-the-grid Wasserstein group-Lasso

We consider the multi-task super-resolution problem, consisting in the simultaneous recovery of pointwise sources across several similar tasks, given several noisy low-pass measurements. The group-Lasso regularizes this problem by enforcing a common sparse support to the solutions in each task, which is often too restrictive in real applications. In this talk, I will first introduce a new off-the-grid (i.e. without spatial discretization) approach for this multi-task recovery, which integrates a Wasserstein distance between the recovered sources. Our scheme combines an optimization step, based on Lasserre's hierarchy, to recover the denoised moment matrices of the sought measures, and a Prony-based extraction step to reconstruct the measures. I will then discuss an extension of the method for recovering non-sparse measures. Using an approximate joint diagonalization scheme, our algorithm recovers a geometrically faithful discrete approximation of the continuous support.