  This study introduces a non-intrusive approach in the context of low-rank
separated representation to construct a surrogate of high-dimensional
stochastic functions, e.g., PDEs/ODEs, in order to decrease the computational
cost of Markov Chain Monte Carlo simulations in Bayesian inference. The
surrogate model is constructed via a regularized alternative least-square
regression with Tikhonov regularization using a roughening matrix computing the
gradient of the solution, in conjunction with a perturbation-based error
indicator to detect optimal model complexities. The model approximates a vector
of a continuous solution at discrete values of a physical variable. The
required number of random realizations to achieve a successful approximation
linearly depends on the function dimensionality. The computational cost of the
model construction is quadratic in the number of random inputs, which
potentially tackles the curse of dimensionality in high-dimensional stochastic
functions. Furthermore, this vector valued separated representation-based
model, in comparison to the available scalar-valued case, leads to a
significant reduction in the cost of approximation by an order of magnitude
equal to the vector size. The performance of the method is studied through its
application to three numerical examples including a 41-dimensional elliptic PDE
and a 21-dimensional cavity flow.
