Monte Carlo MP2 on Many Graphical Processing Units

Abstract

In the Monte Carlo second-order many-body perturbation (MC-MP2) method, the long sum-of-product matrix expression of the MP2 energy, whose literal evaluation may be poorly scalable, is recast into a single high-dimensional integral of functions of electron pair coordinates, which is evaluated by the scalable method of Monte Carlo integration. The sampling efficiency is further accelerated by the redundant-walker algorithm, which allows a maximal reuse of electron pairs. Here, a multitude of graphical processing units (GPUs) offers a uniquely ideal platform to expose multilevel parallelism: fine-grain data-parallelism for the redundant-walker algorithm in which millions of threads compute and share orbital amplitudes on each GPU; coarse-grain instruction-parallelism for near-independent Monte Carlo integrations on many GPUs with few and infrequent interprocessor communications. While the efficiency boost by the redundant-walker algorithm on central processing units (CPUs) grows linearly with the number of electron pairs and tends to saturate when the latter exceeds the number of orbitals, on a GPU it grows quadratically before it increases linearly and then eventually saturates at a much larger number of pairs. This is because the orbital constructions are nearly perfectly parallelized on a GPU and thus completed in a near-constant time regardless of the number of pairs. In consequence, an MC-MP2/cc-pVDZ calculation of a benzene dimer is 2700 times faster on 256 GPUs (using 2048 electron pairs) than on two CPUs, each with 8 cores (which can use only up to 256 pairs effectively). We also numerically determine that the cost to achieve a given relative statistical uncertainty in an MC-MP2 energy increases as O(n) or better with system size n, which may be compared with the O(n) scaling of the conventional implementation of deterministic MP2. We thus establish the scalability of MC-MP2 with both system and computer sizes.

Perspective: Explicitly correlated electronic structure theory for complex systems

“…The operation costs of MC-MP2 and MC-MP2-F12 for a given accuracy (relative statistical error) were observed to be O(n 3 ) and O(n 4 ), respectively, where n is the number of orbitals. 116,118 The memory cost is negligible. With the redundant-walker convergence acceleration scheme, 119 the method can easily achieve >90% of the perfect parallel scalability up to thousands of CPUs, and an unprecedented speedup by a factor of tens of thousands (relative to one CPU core) on hundreds of GPUs.…”

“…With the redundant-walker convergence acceleration scheme, 119 the method can easily achieve >90% of the perfect parallel scalability up to thousands of CPUs, and an unprecedented speedup by a factor of tens of thousands (relative to one CPU core) on hundreds of GPUs. 118 These are by virtue of completely eliminating two-electron integral evaluations and transformations, and by foregoing the conventional dense-matrix algorithms in favor of more scalable stochastic ones. Figure 2 is a result of the MC-MP2-F12 calculation for tetrahydrocannabinol using the cc-pVDZ basis set.…”

The GPU‐enabled divide‐expand‐consolidate RI‐MP2 method (DEC‐RI‐MP2)

“…For instance, basic underline functionality needed for quantum chemistry calculations, such as matrix operations and molecular integral evaluation, has been implemented. Many other programs are now ported to GPUs resulting in methods ranging from density functional and coupled cluster to multireference, full configuration interaciton methods and very recently a Monte Carlo MP2 . The RI‐MP2 method was ported as well using a simple rerouting of a single linear algebra routine .…”

“…Many other programs are now ported to GPUs resulting in methods ranging from density functional [85,86] and coupled cluster [87,88] to multireference, [89,90] full configuration interaciton methods [91] and very recently a Monte Carlo MP2. [92] The RI-MP2 method was ported as well using a simple rerouting of a single linear algebra routine. [67] The authors reported a 4.3 speedup in single point energy calculations of linear alkanes accomplished by using a single precision routine from the compute unified basic linear algebra subprograms (CUBLAS) library.…”

Efficient and portable acceleration of quantum chemical many-body methods in mixed floating point precision using OpenACC compiler directives

“…On that note, we will start by briefly alluding to the existing literature, whichdespite the use of general-purpose GPUs in natural sciences being a rather recent topic-is rich with work devoted to GPU-accelerated quantum chemistry, concerned with diverse topics ranging from the generation of electron repulsion integrals (ERIs), 11-15 over self-consistent Hartree-Fock (HF), complete active space (CAS), and density functional theory (DFT) methods, [16][17][18][19][20][21][22][23][24] to solvent models, 25,26 force fields, 27 and semi-empirical 28 as well as many-body methods. [29][30][31][32][33][34][35][36][37][38] Adding to this quantum, several program packages are today released with full or partial GPU-enabled features, e.g., Terachem, 39 NWChem, 40 among others.…”