The accurate approximation of high-dimensional functions is an essential

task in uncertainty quantification and many other fields. We consider in

particular the case where the approximated functional is represented by

the output of some expensive and complex numerical solver. The

computational complexity of the construction of such surrogates is then

dominated by the number of evaluations of the functional.

This talk will review the adaptive construction of spectrally accurate

surrogates in the tensor-train format, which exploit an

higher-dimensional notion of rank and the smoothness of the approximated

functional. Results regarding the scaling of this approach with respect

to the dimension of the input space and the regularity of the functional

will be presented. Under the fulfillment of low-rank assumptions on the

functional, the method scales linearly with the dimension.

The efficiency of this approach relies on the existence of a particular

form of low-rank structure among the different input dimensions, leading

to what is known as the "ordering problem." We will formalize this

problem and provide heuristics for its solution.

Examples showing the performance of the method will be provided

throughout the presentation.

# The spectral tensor-train format for high-dimensional function approximation

6 May 2016

12:00 pm

The spectral tensor-train format for high-dimensional function approximation

Daniele Bigoni

Postdoctoral Associate

Aerospace Computational Design Laboratory

Dept. of Aeronautics and Astronautics

MIT