# Large-scale Optimization-based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

arXiv: https://arxiv.org/abs/1506.08435 · HTML: https://mlsanity.com/paper/1506.08435/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2015.

## Built on by (most cited first, 1 of 1)

- [A deep learning modeling framework to capture mixing patterns in reactive-transport systems](https://mlsanity.com/paper/2101.04227/) (2021)

## Abstract (arXiv)

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.