Multigrid Methods in Lattice Field Computations

The multigrid methodology is reviewed. By integrating numerical processes at all scales of a problem, it seeks to perform various computational tasks at a cost that rises as slowly as possible as a function of $n$, the number of degrees of freedom in the problem. Current and potential benefits for lattice field computations are outlined. They include: $O(n)$ solution of Dirac equations; just $O(1)$ operations in updating the solution (upon any local change of data, including the gauge field); similar efficiency in gauge fixing and updating; $O(1)$ operations in updating the inverse matrix and in calculating the change in the logarithm of its determinant; $O(n)$ operations per producing each independent configuration in statistical simulations (eliminating CSD), and, more important, effectively just $O(1)$ operations per each independent measurement (eliminating the volume factor as well). These potential capabilities have been demonstrated on simple model problems. Extensions to real life are explored.Comment: 4

Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with significantly extended numerical experiments. Some unused material is remove