  This work develops a numerical solver based on the combination of
isogeometric analysis (IGA) and the tensor train (TT) decomposition for the
approximation of partial differential equations (PDEs) on parameter-dependent
geometries. First, the discrete Galerkin operator as well as the solution for a
fixed geometry configuration are represented as tensors and the TT format is
employed to reduce their computational complexity. Parametric dependencies are
included by considering the parameters that control the geometry configuration
as additional dimensions next to the physical space coordinates. The parameters
are easily incorporated within the TT-IGA solution framework by introducing a
tensor product basis expansion in the parameter space. The discrete Galerkin
operators are accordingly extended to accommodate the parameter dependence,
thus obtaining a single system that includes the parameter dependency. The
system is solved directly in the TT format and a low-rank representation of the
parameter-dependent solution is obtained. The proposed TT-IGA solver is applied
to several test cases which showcase its high computational efficiency and
tremendous compression ratios achieved for representing the parameter-dependent
IGA operators and solutions.
