State-Based Adjoint Method for Reduced Order Modeling

Abstract

Introduced here is an adjoint state-based method for model reduction, which provides a single solution to two classes of reduction methods that are currently in the literature. The first class, which represents the main subject of this manuscript, is concerned with linear time invariant problems where one is interested in calculating linear responses variations resulting from initial conditions perturbations. The other class focuses on perturbations introduced in the operator, which result in nonlinear responses variations. Unlike existing adjoint-based methods where an adjoint function is calculated based on a given response, the state-based method employs the state variations to set up a number of adjoint problems, each corresponding to a pseudoresponse. This manuscript extends the applicability of state-based method to generate reduced order models for linear time invariant problems. Previous developments focusing on operator perturbations are reviewed briefly to highlight the common features of the state-based algorithm as applied to these two different classes of problems. Similar to previous developments, the state-based reduction is shown to set an upper-bound on the maximum discrepancy between the reduced and original model predictions. The methodology is applied and compared to other state-of-the-art methods employing several nuclear reactor diffusion and transport models.

An Adjoint Proper Orthogonal Decomposition method for a neutronics reduced order model

“…In this work, an "Adjoint Proper Orthogonal Decomposition" method is proposed which is aimed at combining the feature of the Proper Orthogonal Decomposition for the spatial basis functions and the adjoint function as test functions. This approach can be intended to some extent as the counterpart of what presented in (Bang et al, 2012b) but applied to the projection-based framework (instead of the interpolation-based one). In order to assess the capability of the APOD method, a comparison with other couples of spatial basis functions and test functions is performed.…”

A Novel Adjoint Based Reduced Order Model for Depletion Calculations in Nuclear Reactor Physics

“…They are usually divided into two steps, an offline step carried out once and an online step for each new calculation. In reactor physics, Proper Orthogonal Decomposition [5,6] or Principal Component Analysis in statistics [7] are commonly used for the creation of the basis. The basis can be constructed using a range-finding algorithm which captures the active subspace of our problem and provides an a posteriori estimator for the precision [8].…”

“…For example, Galerkin projection allows solving a reduced order system instead of the full one [10]. Differently, adjoint based reduced order model uses the adjoint equation to calculate the projection coefficients [6]. Hence, one does not solve a reduced order system but simply calculates the projection of the solution on the reduced basis.…”

Hybrid reduced order modeling for assembly calculations

“…In nuclear engineering, efficient basis construction algorithms based on randomized subspace methodologies have been proposed and their feasibility to reactor physics applications has been demonstrated. For example, the state-level reduction technique was incorporated into initial condition perturbation theory to reduce the number of adjoint mode runs [2]. The parameter-level reduction technique was applied to second order sensitivity analysis and uncertainty propagation [3].…”

“…Assembly Model (PB-2 BWR) PWR Assembly Model (WB-2 PWR) SCALE Assembly ModelSuppose that the models can be expressed as: is the variations in system configurations, i.e. fuel/moderator temperatures or nuclide number densities, -0      is the effective macroscopic cross section changes, values of system configurations, effective macroscopic cross sections and k-eigenvalue, respectively, f represents a resonance calculation (BONAMI) g represents a transport calculation (NEWT)2 the nuclides in different mixtures are counted as different nuclides though there are the same isotopes because they are depleted differently.…”