  An efficient and robust restart strategy is important for any Krylov-based
method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a
Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This
method can be interpreted as an Arnoldi method applied to a linear and infinite
dimensional eigenvalue problem where the Krylov basis consists of polynomials.
We propose new restart techniques for TIAR and analyze efficiency and
robustness. More precisely, we consider an extension of TIAR which corresponds
to generating the Krylov space using not only polynomials but also structured
functions that are sums of exponentials and polynomials, while maintaining a
memory efficient tensor representation. We propose two restarting strategies,
both derived from the specific structure of the infinite dimensional Arnoldi
factorization. One restarting strategy, which we call semi-explicit TIAR
restart, provides the possibility to carry out locking in a compact way. The
other strategy, which we call implicit TIAR restart, is based on the
Krylov-Schur restart method for linear eigenvalue problem and preserves its
robustness. Both restarting strategies involve approximations of the tensor
structured factorization in order to reduce complexity and required memory
resources. We bound the error in the infinite dimensional Arnoldi factorization
showing that the approximation does not substantially influence the robustness
of the restart approach. We illustrate the approaches by applying them to solve
large scale NEPs that arise from a delay differential equation and a wave
propagation problem. The advantages in comparison to other restart methods are
also illustrated.
