Advancements in Monte Carlo many body methods

Doran, Alexander

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https://hdl.handle.net/2142/106213 Copy

Description

Title

Advancements in Monte Carlo many body methods

Author(s)

Doran, Alexander

Issue Date

2019-11-25

Director of Research (if dissertation) or Advisor (if thesis)

Hirata, So

Doctoral Committee Chair(s)

Hirata, So

Committee Member(s)

• Gruebele, Martin
• Makri, Nancy
• Schweizer, Kenneth S.

Department of Study

Chemistry

Discipline

Chemistry

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Many-body methods
• Monte Carlo

Abstract

Many-body methods are one of the most powerful tools that may be brought to bear to solve the electronic structure problem. While both of the following theories provide a complete path to obtain any quantity, many-body perturbation theory is principally used to provide molecular energies, while many-body Green's function provides electron binding energies. Importantly, these methods are systematically improvable, which means there is a prescribed route to increase the accuracy of their results. They are also size consistent. Many-body methods are not without weakness, though, the chief of which is their considerable expense. For example, the operational cost to compute the k-order energy correction with many-body perturbation theory (MBPT) is O (n k+3 ), with n being proportional to the molecular size. Additionally, many-body methods are typically cast into complex series of dense matrix-matrix multiplications, which is difficult to parallelize to millions of processors. Because of this, their enormous expense is difficult to mitigate with the highly parallel architecture of modern supercomputing resources. Recently, the Hirata lab has pioneered stochastic versions of MBPT and many-body Green's functions (MBGF), known as MC-MP and MC-GF, collectively MC-MB, respectively, wherein the correlation energy or electron binding energies are obtained through Monte Carlo integration. The motivation to create stochastic implementations of MBPT and MBGF was to prioritize parallelizability, as Monte Carlo integration is trivially parallel, so that modern supercomputer may be used effectively in the challenge of applying many-body methods to chemical relevance. Here, an overview of the current state of the MC-MP and MC-GF methods, including the pioneering work, advancements that I have completed, and several new developments, is archived. Second- and third-order Monte Carlo perturbation theory (MC-MP2 and MC-MP3) are two of the earliest MC-MB methods to have been developed. The algorithmic details of these methods are considered due to their relevance within the MC-MB family of methods. One significant methodological development for the MC-MP methods, the implementation of fourth-order Monte Carlo perturbation theory (MC-MP4), is presented. The development of MC-MP4 represents a major milestone in the MC-MB family of methods timeline, as MP4 is the lowest level of perturbation theory capable of producing benchmark quality energies. Two essential studies on the computational characterization of MC-MP2 and MC-MP3 are presented. In these studies, the cost of the methods is established by numerical experiments. The cost to obtain a result with a specified target relative statistical uncertainty is established to be O(n^3) and O(n 4) for MC-MP2 and MC-MP3, respectively. These cost functions are both two ranks lower than the cost functions of their conventional counterparts and establish the long term viability of MB-MB methods. Second-order Monte Carlo Green's function (MC-GF2) was one of the methods pioneered during the early work on MC-MB methods. However, the early implementation of MC-GF2 was reserved to be only applicable within the frequency-independent diagonal approximation. Recently, the MC-GF family of methods was extended in several directions. The first was to increase the rank of the self-energy to third-order producing the third-order Monte Carlo Green's function (MC-GF3) method. The diagonal approximation for both the MC-GF2 and MC-GF3 methods have been lifted by implementing an efficient algorithm for the calculation of the self-energy matrix. The frequency-independent approximation has been lifted by expanding the self-energy in a truncated Taylor series, which in turn is used to approximate the zeros of the inverse Dyson equation. The derivatives needed to construct the truncated Taylor series are efficiently computed by the MC-GF algorithm. The computational characteristics of MC-GF3 were extensively studied during this work. The most important among these properties was the establishment of the cost function to obtain binding energies within the diagonal frequency-independent and frequency-independent approximations. This experiment is extended here to establish the same for MC-GF2. For the scaling studied, the MC-GF methods are found to have a cost scaling one rank lower than their conventional counterparts. While MC-MB methods have favorable cost scaling with respect to system size, the naive implementation of these methods is computationally demanding. As such, there has been a significant amount of work to accelerate the convergence of MC-MB methods. Five algorithms that alter the original naive MC-MB framework are discussed. (1) The redundant-walker algorithm accelerates the integration of the electronic coordinates, in which more than the minimally required number of electron pairs are propagated, and the relevant energy correction is sampled at all possible permutations of the electron pairs, for all MC-MB methods. Because of disparities in cost between the various steps of MC-MB methods, the redundant-walker algorithm is able to achieve a boost in the sampling rate. Recent analyses of the redundant-walker algorithm suggest that it is in part responsible for the low scaling of MC-MB methods. (2) MC-MB methods are particularly well-suited to utilize many GPUs. This is because GPUs can push the redundant-walker algorithm harder, and thereby attain a higher than expected speedup. (3) Two strategies to integrate the imaginary time-coordinates are presented. In the first strategy, the imaginary-time coordinates are integrated by quadrature, while in the second strategy, they are integrated by direct Monte Carlo. The integration of these degrees of freedom by stochastic means proves to be significantly more efficient, with speedups of 8.4 and 221 for second- and third-order methods being observed. (4) An algorithm to sample the electronic degrees of freedom directly is presented. This is in contrast to the use of the Metropolis algorithm which has thus far been used to sample these coordinates. The new algorithm eliminates autocorrelation in all integrated quantities, thereby reducing their uncertainties. Speedups of up to 3.21 and 1.38 are observed for MC-MP2 and MC-MP3. This algorithm also provides many qualitative improvements to the MC-MB family of methods. (5) Control variates are applied to the MC-MP family of methods. The application of these control variance produces speedups of 13.9, 17.11, and 58.29 for MC-MP2, MC-MP3, and MC-MP4, respectively. Of these algorithms, the first is apart of the seminal work on MC-MB methods. The last two are recently completed.

Graduation Semester

2019-12

Type of Resource

text

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http://hdl.handle.net/2142/106213

Copyright and License Information

2019 by Alexander Doran. All rights reserved.

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