The spectral tensor-train format for high-dimensional function approximation in Uncertainty Quantification


Wednesday, 22 June, 2016 - 11:00

SISSA mathLab seminar, Room A-133, SISSA main campus

Dr Daniele Bigoni,

MIT, Aero/Astro, Cambridge, MA, US


The accurate approximation of high-dimensional functions is an essentialtask in uncertainty quantification and many other fields. We consider inparticular the case where the approximated functional is represented bythe output of some expensive and complex numerical solver. Thecomputational complexity of the construction of such surrogates is thendominated by the number of evaluations of the functional.This talk will review the adaptive construction of spectrally accuratesurrogates in the tensor-train format, which exploits anhigher-dimensional notion of low-rank and the smoothness of theapproximated functional. Results regarding the scaling of this approachwith respect to the dimension of the input space and the regularity ofthe functional will be presented. Under the fulfillment of low-rankassumptions on the functional, the method scales linearly with thedimension.The efficiency of this approach relies on the existence of a particularform of low-rank structure among the different input dimensions, leadingto what is known as the "ordering problem". We will formalize thisproblem and provide heuristics for its solution.Examples showing the performances of the method will be provided throughout the presentation.