Benjamin Stamm : Gradient flow finite element discretizations with energy-based adaptivity for the Gross Pitaevskii Equation

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation which consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. The mesh-refinement is solely based on energy minimization. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom.

Stéphane Mallat - Wavelet Scattering for Predictions of Molecule Properties

S. Mallat with M. Eickenberg, G. Exarchakis, M. Hirn,Mallat,and L. Thiry
We present a machine learning algorithm for the prediction of molecule energies inspired by ideas from density functional theory. Using Gaussian-type orbital functions, we create surrogate electronic densities of the molecule, from which we compute invariant solid harmonic scattering coefficients that account for different types of interactions at different scales. Multi-linear regressions of various physical properties of molecules are computed from these invariant coefficients. Numerical experiments show that these regressions have near state of the art performance, even with relatively few training examples. Predictions over small sets of scattering coe cients can reach a DFT precision while being interpretable.

Daniel Kressner : Randomized methods for tensor compression

Probabilistic approaches to low-rank matrix approximation, such as the randomized singular value decomposition, have gained popularity during the last few years, thanks to their simplicity, effectiveness, and flexibility. This talk is concerned with various extensions of these approaches to tensors. In particular, we will focus on the benefits and challenges arising from imposing rank-one structure on the involved random vectors. On the one hand, this can increase the efficiency of randomized methods for tensors, sometimes dramatically. On the other hand, this also significantly complicates the probablistic analysis, e.g., for deriving tail bounds and guaranteeing high accuracy with high probability. This talk is based on joint work with Lana Perisa and Zvonimir Bujanovic.

Martin Vohralik : A posteriori error estimates & adaptivity with balancing error components

We review how to bound the error between the unknown weak solution of a partial differential equation and its computer approximation via a fully computable a posteriori estimate. This allows to certify the numerical simulation result. The particularity of the presented approach is that the derived estimates are valid on each iteration of a linearization procedure, as well as on each algebraic solver iteration; moreover, they allow to distinguish the different error components (discretization/linearization/numerical linear algebra). Fully adaptive algorithms relying on adequate stopping and balancing criteria combined with mesh refinement are designed. Some theoretical results such as efficiency, reliability, robustness with respect to problem or approximation parameters, or convergence and quasi-optimality of the adaptive algorithms are presented and illustrated on numerical experiments. Applications include underground fluid flows and eigenvalue problems.