Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Abstract

Abstract. We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the blocks of the domain in a bottom-up pass and then extracts the diagonal entries efficiently in a top-down pass by exploiting the hierarchical local dependence of the inverse matrices. The overall cost of our algorithm is O(N 3/2 ) for a two dimensional problem with N degrees of freedom. Numerical results in electronic structure calculation illustrate the efficiency and accuracy of the proposed algorithm.

Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations

“…From the results presented above, it is clear that the CS2CF strategy is well suited for bringing down the computational wall times of large-scale Kohn−Sham calculations. We have already demonstrated 29 the superior computational efficiency of the standard CheFSI strategy within DGDFT compared to existing alternatives based on direct diagonalization (using ScaLA-PACK) and certain sparse-direct solution strategies (namely, PEXSI 14,15,21 ). Since the CS2CF strategy is successful in bringing down the wall times of the standard CheFSI approach, it is the method of choice for large-scale calculations in DGDFT.…”

Stable Heteropolyoxotitanate Nanocluster for Full Solar Spectrum Photocatalytic Hydrogen Evolution

“…Due to the large number of atoms contained in POTi 12 Cr 6 systems, the standard diagonalization (DIAGON) method in SIESTA based on the ScaLAPACK software package becomes prohibitively expensive . Therefore, we use the recently developed PEXSI (Pole Expansion and Selected Inversion) technique to reduce the computational time without sacrificing accuracy even for metallic systems. − The PEXSI technique allows the evaluation of physical quantities such as electron density, total energy, and atomic forces to be performed without calculating any eigenvalue or eigenfunction. Furthermore, the SIESTA-PEXSI method can passively scale to more than 10,000 processors on high-performance computing machines. − …”

Enhanced MMSE-FDE method for TS-OFDM under higher mobile environments

“…In [2], a low-complexity minimum mean square error-frequency domain equalization (MMSE-FDE) method was proposed for CP-OFDM. In this method, the full elements of channel frequency response (CFR) matrix which is converted from the time domain channel impulse response (CIR) is approximated by a banded matrix so as to employ a fast algorithm for inverse matrix calculation [5]. However, the CIR matrix is constructed by using only data period which leads the degradation of bit error rate (BER) performance in deep fading channels.…”

“…The order of complexity required in the fast algorithm for calculation of inverse matrix G becomes O½3ðQ 1 þ 1Þ 2 P which is relatively lower than the direct inverse matrix calculation [5]. In the demodulation of data information, the estimatedX T in (8) is converted to the time domain signalx T by P-points IDFT.…”

“…The calculation of inverse matrix in (7) requires the order of complexity O½P 3 which leads significantly higher complexity. To reduce the complexity, the full elements of CFR matrix H is approximated by a banded matrix B with the lower and upper bandwidth of Q 1 so as to employ the fast algorithm for inverse matrix calculation [5]. By suing the banded matrix B, (7) can be approximated by,…”