  We propose an isogeometric solver for Poisson problems that combines
i)low-rank tensor techniques to approximate the unknown solution and the system
matrix, as a sum of a few terms having Kronecker product structure, ii) a
Truncated Preconditioned Conjugate Gradient solver to keep the rank of the
iterates low, and iii) a novel low-rank preconditioner, based on the Fast
Diagonalization method where the eigenvector multiplication is approximated by
the Fast Fourier Transform. Although the proposed strategy is written in
arbitrary dimension, we focus on the three-dimensional case and adopt the
Tucker format for low-rank tensor representation, which is well suited in low
dimension. We show by numerical tests that this choice guarantees significant
memory saving compared to the full tensor representation. We also extend and
test the proposed strategy to linear elasticity problems.
