We present a new linearly scaling three-dimensional fragment (LS3DF) method for large scale ab initio electronic structure calculations. LS3DF is based on a divide-and-conquer approach, which incorporates a novel patching scheme that effectively cancels out the artificial boundary effects due to the subdivision of the system. As a consequence, the LS3DF program yields essentially the same results as direct density functional theory (DFT) calculations. The fragments of the LS3DF algorithm can be calculated separately with different groups of processors. This leads to almost perfect parallelization on tens of thousands of processors. After code optimization, we were able to achieve 35.1 Tflop/s, which is 39percent of the theoretical speed on 17,280 Cray XT4 processor cores. Our 13,824-atom ZnTeO alloy calculation runs 400 times faster than a direct DFT calculation, even presuming that the direct DFT calculation can scale well up to 17,280 processor cores. These results demonstrate the applicability of the LS3DF method to material simulations, the advantage of using linearly scaling algorithms over conventional O(N3) methods, and the potential for petascale computation using the LS3DF method.

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## Scholarly Works (97 results)

An efficient new method is presented to calculate the quantum transports using periodic boundary conditions. This method allows the use of conventional ground state ab initio programs without big changes. The computational effort is only a few times of a normal ground state calculations, thus is makes accurate quantum transport calculations for large systems possible.

Space-filling designs are commonly used in computer experiments and other scenarios for investigating complex systems, but the construction of such designs is challenging. In this thesis, we construct a series of maximin-distance Latin hypercube designs via Williams transformations of good lattice point designs. Some constructed designs are optimal under the maximin L1-distance criterion, while others are asymptotically optimal. Moreover, these designs are also shown to have small pairwise correlations between columns. The procedure is further extended to the construction of multi-level nonregular fractional factorial designs which have better properties than regular designs. Existing research on the construction of nonregular designs focuses on two-level designs. We construct a novel class of multilevel nonregular designs by permuting levels of regular designs via the Williams transformation. The constructed designs can reduce aliasing among effects without increasing the run size. They are more efficient than regular designs for studying quantitative factors. In addition, we explore the application of experimental design strategies to data-driven problems and develop a subsampling framework for big data linear regression. The subsampling procedure inherits optimality from the design matrices and therefore minimizes the mean squared error of coefficient estimations for sufficiently large data. It works especially well for the problem of label-constrained regression where a large covariate dataset is available but only a small set of labels are observable. The subsampling procedure can also be used for big data reduction where computation and storage issues are the primary concern.

### A Linear Scaling Three Dimensional Fragment Method for Large Scale Electronic Structure Calculations

We present a novel linear scaling ab initio total energy electronic structure calculation method, which is simple to implement, easily to parallelize, and produces essentially the same results as the direct ab initio method, while it could be thousands of times faster.