In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. This study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for the difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e.scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Finally, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion-reaction equation and on the Navier-Stokes equations. These results demonstrate memory compression ratios up to 450× while maintaining reasonable accuracy in the error-estimates.
Predictive simulations of magnetic confinement fusion and burning plasmas will require the quantification of all uncertainties and errors relating to the simulation capabilities. These include: discretization error (temporal and spatial); incomplete convergence error (nonlinear, linear, etc.); uncertainties in input data (initial conditions, boundary conditions, coefficients, thermo-physical properties, source terms, etc.); and uncertainties in the component models (the specific form and parameter values).13 Typically, there is a focus on model parameter uncertainties but uncertainties about which model to use and about the bias or discrepancy of a particular model (sometimes called model form error) often dominate parameter uncertainty. This whitepaper addresses the challenges of performing uncertainty quantification (UQ) for expensive computational models.
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.
