Large-scale op-miza-on-based non-nega-ve computa-onal framework for diﬀusion equa-ons - Parallel implementa-on and performance studies

Aug 26, 2026

Large-scale optimization-based, non-negative computational framework for anisotropic diffusion equations is presented, motivated by the fact that the Galerkin finite element method fails discrete maximum principles under anisotropic diffusion. Convex optimization techniques are reviewed as a remedy, while the work targets larger and more realistic problems beyond small 2D tests. The approach leverages PETSc TAO and DMPlex features, reports solver iteration and wall-clock time, and evaluates efficiency and speedup under varying processor counts.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Introduc9on and Mo9va9on

Non-nega9ve methodology

Performance studies

Large-‐scale op-miza-on-‐based non-‐nega-ve computa-onal framework for diﬀusion equa-ons: parallel implementa-on and performance studies Jus-n Chang and Kalyana Nakshatrala (University of Houston) Sa9sh Karra (Los Alamos Na9onal Laboratory) PETSc Conference and Workshop June 15th – 18th, 2015 Introduc9on and Mo9va9on • The Galerkin Finite Element Method does not sa9sfy the discrete maximum principles for anisotropic diﬀusion • Recent studies have proposed convex op-miza-on techniques to overcome this setback • So far these have only been tested on small and 2D academic problems Mesh representa9on has a Hasse Diagram Galerkin solu9on (with nega9ve concentra9ons) Aim of this work: Non-‐nega9ve methodology 1. Solve anisotropic diﬀusion leveraging PETSc’s TAO and DMPlex features 2. Ensures non-‐nega9ve solu9ons for larger and more realis9c problems 3. Document the performance metrics Performance studies Solver Iterations 2000 1750 1500 1250 1000 750 500 250 0 A1 B1 C1 A2 B2 C2 A3 B3 C3 Cases Ar ithmet ic Intensity 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 A1 B1 C1 A2 B2 C2 A3 B3 C3 Cases Eff ic iency (%) 100 80 60 40 20 0 Wall-‐clock 9me (s) Solver itera9ons | | | | Mustang-Original
Mustang-Nonnegative | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | ~~ ~~ Wolf-Original
Wolf-Nonnegative

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | A1 B1 C1 A2 B2 C2 A3 B3 C3 Cases Eﬃciency based on the Rooﬂine model Speedup Arithme9c Intensity 64 48 32 16 1 64 48 32 16 1 64 48 32 16 1 64 48 32 16 1 64 48 32 16 1 64 48 32 16 1 \# of processors Chromium contamina9on Wa l l-c lock time (s) 105 104 103 102 16 32 64 128 256 512 1024 2048 \# of processors

FAQs

Why does the Galerkin finite element method have limitations for anisotropic diffusion?

It does not satisfy discrete maximum principles for anisotropic diffusion, which motivates the search for alternative methods.

What is the core aim of the presented work?

Develop a non-negative methodology to compute anisotropic diffusion solutions that remain non-negative for larger, more realistic problems.

How does the framework implement the solver and evaluate performance?

It solves anisotropic diffusion using PETSc TAO and DMPlex features, then documents performance metrics such as solver iterations, wall-clock time, efficiency, and speedup across processor counts.

Aug 26, 2026

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