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

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.