Goal-based sensitivity maps using time windows and ensemble perturbations

We present an approach for forming sensitivity maps (or sensitivites) using ensembles. The method is an alternative to using an adjoint, which can be very challenging to formulate and also computationally expensive to solve. The main novelties of the presented approach are: 1) the use of goals, weighting the perturbation to help resolve the most important sensitivities, 2) the use of time windows, which enable the perturbations to be optimised independently for each window and 3) re-orthogonalisation of the solution through time, which helps optimise each perturbation when calculating sensitivity maps. These novel methods greatly reduce the number of ensembles required to form the sensitivity maps as demonstrated in this paper. As the presented method relies solely on ensembles obtained from the forward model, it can therefore be applied directly to forward models of arbitrary complexity arising from, for example, multi-physics coupling, legacy codes or model chains. It can also be applied to compute sensitivities for optimisation of sensor placement, optimisation for design or control, goal-based mesh adaptivity, assessment of goals (e.g. hazard assessment and mitigation in the natural environment), determining the worth of current data and data assimilation. We analyse and demonstrate the efficiency of the approach by applying the method to advection problems and also a non-linear heterogeneous multi-phase porous media problem, showing, in all cases, that the number of ensembles required to obtain accurate sensitivity maps is relatively low, in the order of 10s.

[1]  R. Samelson,et al.  An efficient method for recovering Lyapunov vectors from singular vectors , 2007 .

[2]  M. Piggott,et al.  The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models , 2010, International Journal for Numerical Methods in Fluids.

[3]  Dan G. Cacuci,et al.  Best-Estimate Predictions and Model Calibration for Reactor Thermal Hydraulics , 2012 .

[4]  Patrick Jenny,et al.  Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions , 2009, J. Comput. Phys..

[5]  Yvon Maday,et al.  Adaptive PBDW Approach to State Estimation: Noisy Observations; User-Defined Update Spaces , 2017, SIAM J. Sci. Comput..

[6]  M. Blunt,et al.  Reservoir Modeling for Flow Simulation by Use of Surfaces, Adaptive Unstructured Meshes, and an Overlapping-Control-Volume Finite-Element Method , 2015 .

[7]  Istvan Szunyogh,et al.  A comparison of Lyapunov and optimal vectors in a low-resolution GCM , 1997 .

[8]  James R. Percival,et al.  An ensemble method for sensor optimisation applied to falling liquid films , 2014 .

[9]  E. Kalnay,et al.  Estimating observation impact without adjoint model in an ensemble Kalman filter , 2008 .

[10]  C. Pain,et al.  Adjoint eigenvalue correction for elliptic and hyperbolic neutron transport problems , 2014 .

[11]  M. Meldi,et al.  A reduced order model based on Kalman filtering for sequential data assimilation of turbulent flows , 2017, J. Comput. Phys..

[12]  Todd D. Ringler,et al.  Initiation of ensemble data assimilation , 2006 .

[13]  M. Zupanski.  Maximum Likelihood Ensemble Filter: Theoretical Aspects , 2005 .

[14]  Eugenia Kalnay,et al.  Ensemble Forecasting at NMC: The Generation of Perturbations , 1993 .

[15]  Ionel Michael Navon,et al.  Computational Methods for Data Evaluation and Assimilation , 2019 .

[16]  Dacian N. Daescu,et al.  Effect of random perturbations on adaptive observation techniques , 2012 .

[17]  A. Norwood.  Bred vectors, singular vectors, and Lyapunov vectors in simple and complex models , 2015 .

[18]  R. Leroux,et al.  Time-resolved flow reconstruction with indirect measurements using regression models and Kalman-filtered POD ROM , 2017 .

[19]  Ionel M. Navon,et al.  Data Assimilation for Geophysical Fluids , 2009 .

[20]  Mu Mu,et al.  A new approach to identify the sensitivity and importance of physical parameters combination within numerical models using the Lund–Potsdam–Jena (LPJ) model as an example , 2017, Theoretical and Applied Climatology.

[21]  Zhi Wang,et al.  The second order adjoint analysis: Theory and applications , 1992 .

[22]  E. Kalnay,et al.  Ensemble Forecasting at NCEP and the Breeding Method , 1997 .

[23]  B. Ancell,et al.  Comparing Adjoint- and Ensemble-Sensitivity Analysis with Applications to Observation Targeting , 2007 .

[24]  A. Attia.  Advanced Sampling Methods for Solving Large-Scale Inverse Problems. , 2016 .

[25]  James R. Percival,et al.  A force‐balanced control volume finite element method for multi‐phase porous media flow modelling , 2017 .

[26]  Ralph Shapiro,et al.  Smoothing, filtering, and boundary effects , 1970 .

[27]  Tim N. Palmer,et al.  Ensemble forecasting , 2008, J. Comput. Phys..

[28]  On the Orthogonalization of Bred Vectors , 2010 .

[29]  J. Percival,et al.  A balanced-force control volume finite element method for interfacial flows with surface tension using adaptive anisotropic unstructured meshes , 2016 .

[30]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[31]  Zhihua Xie,et al.  Improving the robustness of the control volume finite element method with application to multiphase porous media flow , 2017 .

[32]  Methods for Data Assimilation for the Purpose of Forecasting in the Gulf of Cambay (Khambhat) , 2017 .

[33]  Dongbin Xiu,et al.  A generalized polynomial chaos based ensemble Kalman filter with high accuracy , 2009, J. Comput. Phys..

[34]  Dan G. Cacuci.  Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology , 2015, J. Comput. Phys..

[35]  D. Cacuci,et al.  A Comparative Review of Sensitivity and Uncertainty Analysis of Large-Scale Systems—II: Statistical Methods , 2004 .

[36]  Dacian N. Daescu,et al.  Adaptive observations in the context of 4D-Var data assimilation , 2004 .

[37]  R. H. Brooks,et al.  Hydraulic properties of porous media , 1963 .

[38]  Clifford F. Mass,et al.  Measuring the Ensemble Spread–Error Relationship with a Probabilistic Approach: Stochastic Ensemble Results , 2007 .

[39]  Ionel M. Navon,et al.  Second-Order Information in Data Assimilation* , 2002 .

[40]  D. Cacuci,et al.  A Comparative Review of Sensitivity and Uncertainty Analysis of Large-Scale Systems—I: Deterministic Methods , 2004 .

[41]  Ionel Michael Navon,et al.  Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-Scale Systems , 2005 .

[42]  Arthur Moncorge,et al.  Reservoir Simulation Prototyping Platform for High Performance Computing , 2014 .

[43]  Omar M. Knio,et al.  Dynamically adaptive data-driven simulation of extreme hydrological flows , 2018 .

[44]  B. P. Leonard,et al.  The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection , 1991 .

[45]  C.R.E. de Oliveira,et al.  Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations , 2001 .

[46]  C. C. Pain,et al.  Adjoint goal-based error norms for adaptive mesh ocean modelling , 2006 .

[47]  G CacuciDan.  Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems , 2015 .

[48]  Lars Nerger,et al.  On the influence of model nonlinearity and localization on ensemble Kalman smoothing , 2014 .