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We analyze the recent Multi-index Stochastic Collocation (MISC) method forcomputing statistics of the solution of a partial differential equation (PDEs)with random data, where the random coefficient is parametrized by means of acountable sequence of terms in a suitable expansion. MISC is a combinationtechnique based on mixed differences of spatial approximations and quadraturesover the space of random data and, naturally, the error analysis uses the jointregularity of the solution with respect to both the variables in the physicaldomain and parametric variables. In MISC, the number of problem solutionsperformed at each discretization level is not determined by balancing thespatial and stochastic components of the error, but rather by suitablyextending the knapsack-problem approach employed in the construction of thequasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedyoptimization procedure to select the most effective mixed differences toinclude in the MISC estimator. We apply our theoretical estimates to a linearelliptic PDEs in which the log-diffusion coefficient is modeled as a randomfield, with a covariance similar to a Mat\'ern model, whose realizations havespatial regularity determined by a scalar parameter. We conduct a complexityanalysis based on a summability argument showing algebraic rates of convergencewith respect to the overall computational work. The rate of convergence dependson the smoothness parameter, the physical dimensionality and the efficiency ofthe linear solver. Numerical experiments show the effectiveness of MISC in thisinfinite-dimensional setting compared with the Multi-index Monte Carlo methodand compare the convergence rate against the rates predicted in our theoreticalanalysis.

Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity