Real-space electronic-structure calculations with full-potential all-electron precision for transition-metals

We have developed an efficient computational scheme utilizing the real-space finite-difference formalism and the projector augmented-wave (PAW) method to perform precise first-principles electronic-structure simulations based on the density functional theory for systems containing transition metals with a modest computational effort. By combining the advantages of the time-saving double-grid technique and the Fourier filtering procedure for the projectors of pseudopotentials, we can overcome the egg box effect in the computations even for first-row elements and transition metals, which is a problem of the real-space finite-difference formalism. In order to demonstrate the potential power in terms of precision and applicability of the present scheme, we have carried out simulations to examine several bulk properties and structural energy differences between different bulk phases of transition metals, and have obtained excellent agreement with the results of other precise first-principles methods such as a plane wave based PAW method and an all-electron full-potential linearized augmented plane wave (FLAPW) method.Comment: 29 Page

Equivalence Classes and Conditional Hardness in Massively Parallel Computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model

Streaming and Massively Parallel Algorithms for Edge Coloring

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: - Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. - Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors

Massively parallel implicit equal-weights particle filter for ocean drift trajectory forecasting

Forecasting of ocean drift trajectories are important for many applications, including search and rescue operations, oil spill cleanup and iceberg risk mitigation. In an operational setting, forecasts of drift trajectories are produced based on computationally demanding forecasts of three-dimensional ocean currents. Herein, we investigate a complementary approach for shorter time scales by using the recently proposed two-stage implicit equal-weights particle filter applied to a simplified ocean model. To achieve this, we present a new algorithmic design for a data-assimilation system in which all components – including the model, model errors, and particle filter – take advantage of massively parallel compute architectures, such as graphical processing units. Faster computations can enable in-situ and ad-hoc model runs for emergency management, and larger ensembles for better uncertainty quantification. Using a challenging test case with near-realistic chaotic instabilities, we run data-assimilation experiments based on synthetic observations from drifting and moored buoys, and analyze the trajectory forecasts for the drifters. Our results show that even sparse drifter observations are sufficient to significantly improve short-term drift forecasts up to twelve hours. With equidistant moored buoys observing only 0.1% of the state space, the ensemble gives an accurate description of the true state after data assimilation followed by a high-quality probabilistic forecast