We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such par ametric PDEs arise for example in uncertainty quan tification problems in engineering applications. W e propose an algorithm that is a hybrid of the alt ernating least squares and the TT cross methods. I t computes a TT approximation of the whole solutio n\, which is particularly beneficial when multiple quantities of interest are sought. The new algori thm exploits and preserves the block diagonal stru cture of the discretized operator in stochastic co llocation schemes. This disentangles computations of the spatial and parametric degrees of freedom i n the TT representation. In particular\, it only r equires solving independent PDEs at a few paramete r values\, thus allowing the use of existing high performance PDE solvers. We benchmark the new algo rithm on the stochastic diffusion equation against quasi-Monte Carlo and dimension-adaptive sparse g rids methods. For sufficiently smooth random field s the new approach is orders of magnitude faster. LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR