Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

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...

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Vydáno v:Journal of scientific computing Ročník 70; číslo 1; s. 243 - 271
Hlavní autoři: Chang, J., Karra, S., Nakshatrala, K. B.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2017
Springer Nature B.V
Témata:
Algorithms
Boundary conditions
Computational Mathematics and Numerical Analysis
Contamination
Finite element method
Galerkin method
Libraries
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Optimization
Partial differential equations
Principles
Solvers
Theoretical
Maximum principles
Non-negative constraint
Large-scale optimization
High performance computing
Anisotropic diffusion
ISSN:0885-7474, 1573-7691
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Shrnutí: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. Graphical Abstract This figure shows the fate of chromium after 180 days using the single-field Galerkin formulation. The white regions indicate the violation of the non-negative constraint.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-016-0250-5