  An optimization-based approach for the Tucker tensor approximation of
parameter-dependent data tensors and solutions of tensor differential equations
with low Tucker rank is presented. The problem of updating the tensor
decomposition is reformulated as fitting problem subject to the tangent space
without relying on an orthogonality gauge condition. A discrete Euler scheme is
established in an alternating least squares framework, where the quadratic
subproblems reduce to trace optimization problems, that are shown to be
explicitly solvable and accessible using SVD of small size. In the presence of
small singular values, instability for larger ranks is reduced, since the
method does not need the (pseudo) inverse of matricizations of the core tensor.
Regularization of Tikhonov type can be used to compensate for the lack of
uniqueness in the tangent space. The method is validated numerically and shown
to be stable also for larger ranks in the case of small singular values of the
core unfoldings. Higher order explicit integrators of Runge-Kutta type can be
composed.
