Vortrag - Institut für Mathematik
(MK) Dr. Carlo Marcati (ETH Zürich)11.02.2020 - W01 0-006 (Wechloy), 16 Uhr c.t. Uhr
Weighted analytic regularity of Schrödinger-type problems and Quantized Tensor Train approximation
Many problems in quantum chemistry (and beyond) have solutions that are not regular in a classical sense. This is the case, for example, of the computation of electronic wave functions in full-electron modelswhere the Coulomb interaction between charged particles gives rise to cusps in the solution. We often can, nonetheless, show that these functions belong to a class of analytic-type weighted Sobolev spaces, with isolated point singularities. This has many relevant consequences on the numerical approximation of such problems, both for linear and nonlinear techniques.
In this talk, we discuss how to obtain weighted analytic estimates on the solutions of linear and nonlinear Schrödinger-type problems, drawing a bridge with the theory of elliptic regularity in polyhedral domains. We also explore the approximability of the wave functions, with a focus on the Quantized Tensor Train (QTT) approximation. This nonlinear technique, based on tensor compression, have recently gained wide popularity both in data analysis and scientific computing. In our case, we will see how QTT representations provide exponentially convergent approximations by exploiting the inherently low-rank structure of the wave functions.