Joint Investigation Into the Structure of Ewald Matrices

Research topic and goals

Electrochemical Double Layer Capacitors (EDLCs) have garnered significant interest due to their potential applications in novel energy storage devices. At the Maison de la Simulation, software is currently being developed that can accurately simulate these systems at constant electrostatic potential over long time scales (see, for example, the work of Reed et al. (Reed, Lanning, and Madden 2007) for details of the computational techniques involved). By far the most expensive part of these simulations is the solving of a linear system that represents the interactions between the electrodes and electrolyte ions. Currently, this system is solved using dense linear algebra routines, however these routines scale cubically with the size of the system, which poses severe limitations on the maximum system size and time scales that can be treated.

One alternative would be to use sparse matrix techniques. However, while sparse linear solvers reduce the scaling costs, the need to solve a large linear system at each iteration would impose a large cost and limit the length of simulations that can be practically performed. Recently, at RIKEN a new library called NTPoly (Dawson and Nakajima 2017) has been developed that can compute the full inverse of sparse matrices in linear time. This library would allow the matrix inverse to be precomputed, allowing for fast iteration times and thus long simulations. For this project, we will investigate the properties of the matrices produced in these simulations, and determine if the techniques used in NTPoly are applicable.

Goal Summary:

• Develop a set of benchmark Ewald matrices using the software being developed at the Maison de la Simulation.
• Create a benchmark program based on NTPoly that can be used to assess the accuracy and performance of NTPoly’s sparse matrix techniques.
• Compile a report detailing the feasibility and potential benefits of integrating NTPoly or NTPoly’s techniques into the software being developed at the Maison de la Simulation.

Results for 2017/2018

• Development of a program that can be used to test the accuracy and efficiency of NTPoly when applied to Ewald matrices.
• Initial evaluation of the input parameters for applying NTPoly to Ewald matrices.

Visits and meetings

None yet.

Impact and publications

None yet.

Future plans

Currently we have only investigated medium sized matrices which could be generated in serial using the software developed at the Maison de la Simulation. In the next year, we will take advantage of a parallelized version to generate larger matrices to investigate. We will also explore the use of the other matrix functions implemented in NTPoly for improving the solver.

References

1. DawsNaka2017

   BibTeX Abstract

   Dawson, William, and Takahito Nakajima. 2017. “Massively Parallel Sparse Matrix Function Calculations with NTPoly.” Computer Physics Communications.

   @article{DawsNaka2017,
     author = {Dawson, William and Nakajima, Takahito},
     journal = {Computer Physics Communications},
     publisher = {Elsevier},
     title = {Massively parallel sparse matrix function calculations with NTPoly},
     year = {2017}
   }
   

2. ReedEtAl2007

   BibTeX Abstract

   Reed, Stewart K, Oliver J Lanning, and Paul A Madden. 2007. “Electrochemical Interface between an Ionic Liquid and a Model Metallic Electrode.” The Journal of Chemical Physics 126 (8): 084704.

   @article{ReedEtAl2007,
     author = {Reed, Stewart K and Lanning, Oliver J and Madden, Paul A},
     journal = {The Journal of chemical physics},
     number = {8},
     pages = {084704},
     publisher = {AIP},
     title = {Electrochemical interface between an ionic liquid and a model metallic electrode},
     volume = {126},
     year = {2007}
   }