1. 1. Further Investigation of Error Bounds for Reduced Order Modeling Mohammad Abdo2, Congjian Wang1,2 and Hany Abdel-Khalik1,2 1School of Nuclear Engineering, Purdue University, IN, USA 2Nuclear Engineering Department, North Carolina State University, NC, USA Motivations To practically perform intensive reactor physics tasks: Employ Reduced Order Modeling Build high accuracy surrogates Identify active subspaces/manifolds. Equip with error analysis modules. To reduce complexity Use simplified/decoupled physics Use homogenization, MLROM Quarter-core virtual geometry model of a PWR http://www.casl.gov/image_gallery.sht ml Fuel Rod, Control Rod, Burnable Poison Rod Assembl ies Structural Materials Overview of Reduction Algorithms Reduction can be performed on various interfaces (parameter, state space and/or response of interest). Gradient-based (Parameter reduction): o Requires Sensitivity information. o Requires Adjoint runs. Gradient-free Snapshots (Response reduction): o Requires forward runs. Gradient-Based Reduction Algorithm Randomly perturb the input parameters. Sample the sensitivities of the pseudo responses w.r.t the input parameters (G). Find the Range of the sensitivity profile R(G) Using any linear algebra approach. Snapshots (Gradient-Free) Reduction Algorithm State variable interface reduction. Can be represented as a reduction in model A that is passed directly to Model B (loosely-coupled physics). Response of Model A is randomly sampled. Active response subspace is identified. Error Analysis Consider the physics model : y f x Reduction Error : ,: ,: [ ] T T i j y y i x x j ij i j f x i i f x E f x Q Q Q Q Probabilistic Error bounds: 1 ,N iw w where ,w p E E Using Normal distribution can make the bound 1-2 orders of magnitude larger than actual error. Current Contribution Many distributions are inspected and the multiplier is computed such that the actual norm is less than the estimated norm in 90 % of the cases. 0.9w B B A random matrix B is used instead of the error matrix E to reduce the analysis cost. The distribution with the least multiplier is picked as the corresponding estimated norm will be the least conservative and hence the most practical (estimated norm is the closest to the actual norm). Numerical Results Uniform (-1,1) 13.2 Poisson 1.67 Log-normal 1.50 Normal (0,1) 7.98 Exponential 1.49 Beta(0.5,(N-1)/2) 1.65 Binomial (N,0.9) 1.02 Chi-square 1.31 Beta(1,10) 1.44 The analysis shows that the numerically computed multiplier for the normal distribution agreed with the analytic value proposed by [Dixon 1983]. Binomial distribution gives the least multiplier and hence is chosen for future error bound estimation. Numerical Results Both Gaussian and Normal distributions give non practically conservative bounds. Binomial distribution: Least multiplier Linear pattern Slope is close to 1.0 (Even for cases that the prediction fails the estimated norm is very close to the actual norm). Conclusions ROM error estimation requires sampling of the actual error and matrix-vector multiplications by randomly sampled vectors. Using the binomial distribution can remove the unnecessary conservativeness of the bounds coming from the Gaussian or Uniform distributions. These results were employed on realistic neutronic problems. Getting rid of the impractical bounds eased the use of this approach in propagating error bounds across different levels of reduction and hence the use on loosely-coupled multi-physics problems. This is also in MLROM where the subspace was extracted from a pin cell then deployed for a full assembly with no violations for the error bound.