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We present a Multi-Index Quasi-Monte Carlo method for the solution ofelliptic partial differential equations with random coefficients. By combiningthe multi-index sampling idea with randomly shifted rank-1 lattice rules, thealgorithm constructs an estimator for the expected value of some functional ofthe solution. The efficiency of this new method is illustrated on athree-dimensional subsurface flow problem with lognormal diffusion coefficientwith underlying Mat\'ern covariance function. This example is particularlychallenging because of the small correlation length considered, and thus thelarge number of uncertainties that must be included. We show numerical evidencethat it is possible to achieve a cost inversely proportional to the requestedtolerance on the root-mean-square error, for problems with a smoothly varyingrandom field

A Multi-Index Quasi-Monte Carlo Algorithm for Lognormal Diffusion Problems