Publications

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The emergence of high-concurrency architectures offering unprecedented performance has brought many high-performance partial differential equation (PDE) discretization codes to the precipice of a major refactor. To help address this challenge a workshop titled "Algorithms and Abstractions for Assembly in PDE Codes" was held in the Computer Science Research Institute at Sandia National Laboratories on May 12th-14th, 2014. This document summarizes the goals of the workshop and the results of the presentations and subsequent discussions.

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In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-ofinterest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest. We establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions. Central to the approach is that it offers an element-level estimate of the error in the quantity-ofinterest. As a result, we introduce an adaptive approach that yields efficient, adaptively refined approximations. Several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology. Notably, the results show that the new approach is effective at improving the accuracy per total computational cost.

This report describes work directed towards completion of the Thermal Hydraulics Methods (THM) CFD Level 3 Milestone THM.CFD.P7.05 for the Consortium for Advanced Simulation of Light Water Reactors (CASL) Nuclear Hub effort. The focus of this milestone was to demonstrate the thermal hydraulics and adjoint based error estimation and parameter sensitivity capabilities in the CFD code called Drekar::CFD. This milestone builds upon the capabilities demonstrated in three earlier milestones; THM.CFD.P4.02, completed March, 31, 2012, THM.CFD.P5.01 completed June 30, 2012 and THM.CFD.P5.01 completed on October 31, 2012.

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Existing discretizations for stochastic PDEs, based on a tensor product between the deter ministic basis and the stochastic basis, treat the required resolution of uncertainty as uniform across the physical domain. However, solutions to many PDEs of interest exhibit spatially localized features that may result in uncertainty being severely over or under-resolved by existing discretizations. In this report, we explore the mechanics and accuracy of using a spatially varying stochastic expansion. This is achieved through an adaptive refinement algorithm where simple error estimates are used to independently drive refinement of the stochastic basis at each point in the physical domain. Results are presented comparing the accuracy of the adaptive techinque to the accuracy achieved using uniform refinement.

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This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. We compare the adjoint-based a posteriori error estimation approach with a recent variant of a data-centric verification technique. We provide a brief overview of each technique and then we discuss their relative advantages and disadvantages. We use Drekar::CFD to produce numerical results for steady-state Navier Stokes and SARANS approximations. 3

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In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners. © 2011 Springer Science+Business Media (outside the USA).

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