# About
These PDE (partial differential equation) datasets were created for and used in conjunction with Multilevel Training of Neural Networks (MTNN)(https://github.com/LLNL/MTNN).
The datasets are in .npz format and can be accessed in your Python program with `numpy.load` module or with a desktop viewer [npzviewer](https://pypi.org/project/npzviewer/)

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## Convection-Diffusion and Poisson Dataset
These two datasets contain 10,000 steady-state convection-diffusion and pure diffusion problems and their solutions, respectively, on the 2-D unit square.
### Parameters
* F: heat source
* (FluxX, FluxY): flux of heat
* Kappa: heat conductivity
* U: numrical solution
* X, Y: coordinates of the mesh
### Source
The problems were discretized by finite element methods (FEM) using [1,2,4]. The input data include the (convection-) diffusion coefficients, mesh point coordinates and heat source (the right-hand sides of the equations). The target data are the numerical solutions obtained by using [3] and the flux of the heat.

* [1] FEniCS open source PDE solver https://fenicsproject.org/
* [2] DOLFIN C++/Python interface of FEniCS https://pypi.org/project/DOLFIN/
* [3] Numpy open-source Python library https://numpy.org/
* [4] Sympy open-source Python library https://www.sympy.org/en/index.html

## Darcy Flow Data
This dataset contains finite element solutions of the Darcy Flow equation.
### Parameters
* Q: total flux along the outflow boundary
* nx, ny: number of pixels in x, y direction of the simulated problem space
* U: 2D grid with the permeability of each point in the 2D domain
* X_coord, y_coord: coordinates in the domain
### Source
Numerical simulations were performed using tools developed in ParELAG[4], a parallel C++ library for performing numerical upscaling of finite element discretizations and AMG techniques, and ParELAGMC [5], a parallel element agglomeration MLMC library. These libraries use MFEM [3] to generate the fine grid finite element discretization and HYPRE [2] to handle massively parallel linear algebra.

* [1]  GLVis: Opengl finite element visualization tool. glvis.org.
* [2]  HYPRE: High performance preconditioners. http://www.llnl.gov/CASC/hypre/.
* [3]  MFEM: Modular finite element methods library. mfem.org.
* [4]  ParELAG: Element-agglomeration algebraic multigrid and upscaling library, version 2.0. http://github.com/LLNL/parelag, 2015.
* [5]  ParELAGMC: Parallel element agglomeration multilevel Monte Carlo library. http://github.com/
* LLNL/parelagmc, 2018.

# Acknowledgement
We would like to thank Hilary Fairbanks (fairbanks5@llnl.gov) for creating and sharing the DarcyFlow dataset for our work in MTNN.

# Contact
* Colin Ponce, ponce11@llnl.gov
* Ruipeng Li, li50@llnl.gov
* Christina Mao, mao6@llnl.gov