A study of reduced‐order 4DVAR with a finite element shallow water model

Four‐dimensional variational data assimilation (4DVAR) is frequently used to improve model forecasting skills. This method improves a model consistency with available data by minimizing a cost function measuring the model–data misfit with respect to some model inputs and parameters. Associated with this type of method, however, are difficulties related to the coding of the adjoint model, which is needed to compute the gradient of the 4DVAR cost function. Proper orthogonal decomposition (POD) is a model reduction method that can be used to approximate the gradient calculation in 4DVAR. In this work, two ways of using POD in 4DVAR are presented, namely model‐reduced 4DVAR and reduced adjoint 4DVAR (RA‐4DVAR). Both techniques employ POD to obtain a reduced‐order approximation of the forward linear tangent operator. The difference between the two methods lies in the treatment of the forward model. Model‐reduced 4DVAR performs minimization entirely in the POD‐reduced space, thereby achieving very low computational costs, but sacrificing accuracy of the end result. On the other hand, the RA‐4DVAR uses POD to approximate only the adjoint model. The main contribution of this study is a comparative performance analysis of these 4DVAR methodologies on a nonlinear finite element shallow water model. The sensitivity of the methods to perturbations in observations and the number of observation points is examined. The results from twin experiments suggest that the RA‐4DVAR method is easy to implement and computationally efficient and provides a robust approach for achieving reasonable results in the context of variational data assimilation. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficiency of a POD‐based reduced second‐order adjoint model in 4D‐Var data assimilation

Order reduction strategies aim to alleviate the computational burden of the four‐dimensional variational data assimilation by performing the optimization in a low‐order control space. The proper orthogonal decomposition (POD) approach to model reduction is used to identify a reduced‐order control space for a two‐dimensional global shallow water model. A reduced second‐order adjoint (SOA) model is developed and used to facilitate the implementation of a Hessian‐free truncated‐Newton (HFTN) minimization algorithm in the POD‐based space. The efficiency of the SOA/HFTN implementation is analysed by comparison with the quasi‐Newton BFGS and a nonlinear conjugate gradient algorithm. Several data assimilation experiments that differ only in the optimization algorithm employed are performed in the reduced control space. Numerical results indicate that first‐order derivative methods are effective during the initial stages of the assimilation; in the later stages, the use of second‐order derivative information is of benefit and HFTN provided significant CPU time savings when compared to the BFGS and CG algorithms. A comparison with data assimilation experiments in the full model space shows that with an appropriate selection of the basis functions the optimization in the POD space is able to provide accurate results at a reduced computational cost. The HFTN algorithm benefited most from the order reduction since computational savings were achieved both in the outer and inner iterations of the method. Further experiments are required to validate the approach for comprehensive global circulation models. Copyright © 2006 John Wiley & Sons, Ltd.     

A dual‐weighted trust‐region adaptive POD 4‐D Var applied to a finite‐volume shallow water equations model on the sphere

In this paper we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40‐yr ECMWF Re‐analysis (ERA‐40) data sets, in the presence of full or incomplete observations being assimilated in a time interval (window of assimilation) with or without background error covariance terms. As an extension of the work by Chen et al. (Int. J. Numer. Meth. Fluids 2009), we attempt to obtain a reduced order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4D‐Var for a finite volume global shallow water equation model based on the Lin–Rood flux‐form semi‐Lagrangian semi‐implicit time integration scheme. Different approaches of POD implementation for the reduced inverse problem are compared, including a dual‐weighted method for snapshot selection coupled with a trust‐region POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4‐D Var model results combined with the trust‐region adaptivity exhibit similarity in terms of various error metrics to the full 4D Var results, but are obtained using a significantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current work, we conclude that POD 4‐D Var certainly warrants further studies, with promising potential of its extension to operational 3‐D numerical weather prediction models. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient identification of uncertain parameters in a large‐scale tidal model of the European continental shelf by proper orthogonal decomposition

The adjoint method can be used to identify uncertain parameters in large‐scale shallow water flow models. This requires the implementation of the adjoint model, which is a large programming effort. The work presented here is inverse modeling based on model reduction using proper orthogonal decomposition (POD). An ensemble of forward model simulations is used to determine the approximation of the covariance matrix of the model variability and the dominant eigenvectors of this matrix are used to define a model subspace. An approximate linear reduced model is obtained by projecting the original model onto this reduced subspace. Compared with the classical variational method, the adjoint of the tangent linear model is replaced by the adjoint of a linear reduced forward model. The minimization process is carried out in reduced subspace and hence reduces the computational costs. In this study, the POD‐based calibration approach has been implemented for the estimation of the depth values and the bottom friction coefficient in a large‐scale shallow sea model of the entire European continental shelf with approximately 10⁶ operational grid points. A number of calibration experiments is performed. The effectiveness of the algorithm is evaluated in terms of the accuracy of the final results as well as the computational costs required to produce these results. The results demonstrate that the POD calibration method with little computational effort and without the implementation of the adjoint code can be used to solve large‐scale inverse shallow water flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A dual‐weighted trust‐region adaptive POD 4D‐VAR applied to a finite‐element shallow‐water equations model

We consider a limited‐area finite‐element discretization of the shallow‐water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced‐order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4‐D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual‐weighed method for snapshot selection coupled with a trust‐region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual‐weighing choice of snapshots is combined with POD adaptivity of the trust‐region type. Results of ad‐hoc adaptivity of the POD 4‐D VAR turn out to yield less accurate results than trust‐region POD when compared with high‐fidelity model. Directions of future research are finally outlined. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient goal‐based reduced order model approach for targeted adaptive observations

An efficient adjoint sensitivity technique for optimally collecting targeted observations is presented. The targeting technique incorporates dynamical information from the numerical model predictions to identify when, where and what types of observations would provide the greatest improvement to specific model forecasts at a future time. A functional (goal) is defined to measure what is considered important in modelling problems. The adjoint sensitivity technique is used to identify the impact of observations on the predictive accuracy of the functional, then placing the sensors at the locations with high impacts. The adaptive (goal) observation technique developed here has the following features: (i) over existing targeted observation techniques, its novelty lies in that the interpolation error of numerical results is introduced to the functional (goal), which ensures the measurements are a distance apart; (ii) the use of proper orthogonal decomposition (POD) and reduced order modelling for both the forward and backward simulations, thus reducing the computational cost; and (iii) the use of unstructured meshes. The targeted adaptive observation technique is developed here within an unstructured mesh finite element model (Fluidity). In this work, a POD reduced order modelling is used to form the reduced order forward model by projecting the original complex model from a high dimensional space onto a reduced order space. The reduced order adjoint model is then constructed directly from the reduced order forward model. This efficient adaptive observation technique has been validated with two test cases: a model of an ocean gyre and a model of 2D urban street canyon flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A reduced‐order simulated annealing approach for four‐dimensional variational data assimilation in meteorology and oceanography

Four‐dimensional variational data assimilation in meteorology and oceanography suffers from the presence of local minima in the cost function. These local minima arise when the system under study is strongly nonlinear. The number of local minima further dramatically increases with the length of the assimilation period and often renders the solution to the problem intractable. Global optimization methods are therefore needed to resolve this problem. However, the huge computational burden makes the application of these sophisticated techniques unfeasible for large variational data assimilation systems. In this study, a Simulated Annealing (SA) algorithm, complemented with an order‐reduction of the control vector, is used to tackle this problem. SA is a very powerful tool of combinatorial minimization in the presence of several local minima at the cost of increasing the execution time. Order‐reduction is then used to reduce the dimension of the search space in order to speed up the convergence rate of the SA algorithm. This is achieved through a proper orthogonal decomposition. The new approach was implemented with a realistic eddy‐permitting configuration of the Massachusetts Institute of Technology general circulation model (MITgcm) of the tropical Pacific Ocean. Numerical results indicate that the reduced‐order SA approach was able to efficiently reduce the cost function with a reasonable number of function evaluations. Copyright © 2008 John Wiley & Sons, Ltd.     

The reduced‐order hybrid Monte Carlo sampling smoother

Hybrid Monte Carlo sampling smoother is a fully non‐Gaussian four‐dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive owing to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the hybrid Monte Carlo sampling smoother based on reduced‐order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow‐water equations model on Cartesian coordinates. The results reveal that the reduced‐order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full‐order formulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10^?3) versus O(10^?4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.     

Discharge estimation under uncertainty using variational methods with application to the full Saint‐Venant hydraulic network model

Estimating river discharge from in situ and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation‐under‐uncertainty problems, which is based on the ideas of iterative regularization. The method is implemented with SIC², which is a full Saint‐Venant based 1D‐network model. The SIC² software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D‐network model of a commercial status in existence. The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross‐device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission. Copyright © 2016 John Wiley & Sons, Ltd.     

GENERALIZED VARIATIONAL DATA ASSIMILATION METHOD AND NUMERICAL EXPERIMENT FOR NON-DIFFERENTIAL SYSTEM

The generalized variational data assimilation for non-differential dynamical systems is studied. There is no tangent linear model for non-differential systems and thus the general adjoint model can not be derived in the traditional way. The weak form of the original system was introduced, and then the generalized adjoint model was derived. The generalized variational data assimilation methods were developed for non-differential low dimensional system and non-differential high dimensional system with global and local observations. Furthermore, ideas in inverse problems are introduced to 4DVAR ( Fourdimensional variational ) of non-differential partial differential system with local observations.     

Convergence acceleration of continuous adjoint solvers using a reduced‐order model

A novel acceleration technique using a reduced‐order model is presented to speed up convergence of continuous adjoint solvers. The acceleration is achieved by projecting to an improved solution within an iterative process solely using early solution results. This is achieved by forming basis vectors from early iteration adjoint solutions to perform model order reduction of the adjoint equations. The reduced‐order model of the adjoint equations is then substituted into the full‐order discretized governing equations to determine weighting coefficients for each basis vector. With these coefficients, a linear combination of the basis vectors is used to project to an improved solution. The method is applied to 3 inviscid quasi‐1D nozzle flow cases including fully subsonic flow, subsonic inlet to supersonic outlet flow, and transonic flow with a shock. Significant cost reductions are achieved for a single application as well as repeated applications of the convergence acceleration technique.     

Non‐intrusive reduced order modelling with least squares fitting on a sparse grid

This paper presents a non‐intrusive reduced order model for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps, taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model are required by this methodology, and the level of non‐intrusiveness is improved compared with existing reduced order models. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios, and we test it on a couple of problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high‐fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of random perturbations on adaptive observation techniques

An observation sensitivity (OS) method to identify targeted observations is implemented in the context of four‐dimensional variational (4D‐Var) data assimilation. This methodology is compared with the well‐established adjoint sensitivity (AS) method using a nonlinear Burgers equation as a test model. Automatic differentiation software is used to implement the first‐order adjoint model (ADM) to calculate the gradient of the cost function required in the 4D‐Var minimization algorithm and in the AS computations and the second‐order ADM to obtain information on the Hessian matrix of the 4D‐Var cost that is necessary in the OS computations. Numerical results indicate that the observation‐targeting is particularly successful in reducing the forecast error for moderate Reynolds numbers. The potential benefits of the OS targeting approach over the AS are investigated. The effect of random perturbations on the performance of these adaptive observation techniques is also analyzed. Copyright © 2011 John Wiley & Sons, Ltd.     

A high‐fidelity low‐cost aerodynamic model using proper orthogonal decomposition

In this paper a high‐fidelity aerodynamic model is presented for use in parametric studies of weapon aerodynamics. The method employs a reduced‐order model obtained from the proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions with varying parameters. This decomposition produces an optimal linear set of orthogonal basis functions that best describe the ensemble of numerical solutions. These solutions are then projected onto this set of basis functions to provide a finite set of scalar coefficients that represent the solutions. A pseudo‐continuous representation of these projection coefficients is constructed, which allows predictions to be made of parameter combinations not in the original set of observations. The paper explores the performance of a few design‐of‐experiment approaches for the generation of the initial ensemble of computational experiments. Response surface construction methods based on parametric and non‐parametric models for the pseudo‐continuous representation of the projection coefficients are also evaluated. The model has been applied to two‐flow problems related to high‐speed weapon aerodynamics, inviscid flow around a flare‐stabilized hypersonic projectile and supersonic turbulent flow around a fin‐stabilized projectile with drooping nose control. Comparisons of model predictions with high‐fidelity CFD simulations suggest that the POD provides a reliable and robust approach to the construction of reduced‐order models. The practicality of the model is shown to be sensitive to the technique used to generate the ensemble of observations from which the model is constructed, while the accuracy of the approach depends on the pseudo‐continuous representation of the projection coefficients. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of second‐order adjoint technique for conduit flow problem

This paper presents the way to obtain the Newton gradient by using a traction given by the perturbation for the Lagrange multiplier. Conventionally, the second‐order adjoint model using the Hessian/vector products expressed by the product of the Hessian matrix and the perturbation of the design variables has been researched (Comput. Optim. Appl. 1995; 4 :241–262). However, in case that the boundary value would like to be obtained, this model cannot be applied directly. Therefore, the conventional second‐order adjoint technique is extended to the boundary value determination problem and the second‐order adjoint technique is applied to the conduit flow problem in this paper. As the minimization technique, the Newton‐based method is employed. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is applied to calculate the Hessian matrix which is used in the Newton‐based method and a traction given by the perturbation for the Lagrange multiplier is used in the BFGS method. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust design in aerodynamics using third‐order sensitivity analysis based on discrete adjoint. Application to quasi‐1D flows

In this paper, the second‐order second moment approach, coupled with an adjoint‐based steepest descent algorithm, for the solution of the so‐called robust design problem in aerodynamics is proposed. Because the objective function for the robust design problem comprises first‐order and second‐order sensitivity derivatives with respect to the environmental parameters, the application of a gradient‐based method , which requires the sensitivities of this function with respect to the design variables, calls for the computation of third‐order mixed derivatives. To compute these derivatives with the minimum CPU cost, a combination of the direct differentiation and the discrete adjoint variable method is proposed. This is presented for the first time in the relevant literature and is the most efficient among other possible schemes on condition that the design variables are much more than the environmental ones; this is definitely true in most engineering design problems. The proposed approach was used for the robust design of a duct, assuming a quasi‐1D flow model; the coordinates of the Bézier control points parameterizing the duct shape are used as design variables, whereas the outlet Mach number and the Darcy–Weisbach friction coefficient are used as environmental ones. The extension to 2D and 3D flow problems, after developing the corresponding direct differentiation and adjoint variable methods and software, is straightforward. Copyright © 2011 John Wiley & Sons, Ltd.     

Reduced rank static error covariance for high‐dimensional applications

A method for creating static (e.g., stationary) error covariance of reduced rank for potential use in hybrid variational‐ensemble data assimilation is presented. The choice of reduced rank versus full rank static error covariance is made in order to allow the use of an improved Hessian preconditioning in high‐dimensional applications. In particular, this method relies on using block circulant matrices to create a high‐dimensional global covariance matrix from a low‐dimensional local sub‐matrix. Although any covariance used in variational data assimilation would be an acceptable choice for the pre‐defined full‐rank static error covariance, for convenience and simplicity, we use a symmetric Topelitz matrix as a prototype of static error covariance. The methodology creates a square root covariance, which has a practical advantage for Hessian preconditioning in reduced rank, ensemble‐based data assimilation. The experiments conducted examine multivariate covariance that includes the impact of cross‐variable correlations, in order to have a more realistic assessment of the value of the constructed static error covariance approximation. The results show that it may be possible to reduce the rank of matrix to O(10) and still obtain an acceptable approximation of the full‐rank static covariance matrix. Copyright © 2016 John Wiley & Sons, Ltd.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.