Optimized parametric inference for the inner loop of the Multigrid Ensemble Kalman Filter

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Optimized parametric inference for the inner loop of

the Multigrid Ensemble Kalman Filter

G. Moldovana,∗, G. Lehnascha, L. Cordiera, M. Meldia

aInstitut Pprime, CNRS - ISAE-ENSMA - Universit´e de Poitiers, 11 Bd. Marie et Pierre

Curie, Site du Futuroscope, TSA 41123, 86073 Poitiers Cedex 9, France

Abstract

Essential features of the Multigrid Ensemble Kalman Filter (G. Moldovan, G.

Lehnasch, L. Cordier, M. Meldi, A multigrid/ensemble Kalman ﬁlter strategy for

assimilation of unsteady ﬂows, Journal of Computational Physics 443−110481)

recently proposed for Data Assimilation of ﬂuid ﬂows are investigated and as-

sessed in this article. The analysis is focused on the improvement in performance

due to the inner loop. In this step, data from solutions calculated on the higher

resolution levels of the multigrid approach are used as surrogate observations to

improve the model prediction on the coarsest levels of the grid. The latter rep-

resents the level of resolution used to run the ensemble members for global Data

Assimilation. The method is tested over two classical one-dimensional problems,

namely the linear advection problem and the Burgers’ equation. The analyses

encompass a number of diﬀerent aspects such as diﬀerent grid resolutions. The

results indicate that the contribution of the inner loop is essential in obtaining

accurate ﬂow reconstruction and global parametric optimization. These ﬁnd-

ings open exciting perspectives of application to grid-dependent reduced-order

models extensively used in ﬂuid mechanics applications for complex ﬂows, such

as Large Eddy Simulation (LES).

Keywords: Kalman Filter, Data Assimilation, multigrid algorithms

∗Corresponding author, gabriel-ionut.moldovan@ensma.fr

Preprint submitted to Journal of Computational Physics Thursday 20th October, 2022

arXiv:2210.10157v1 [physics.flu-dyn] 18 Oct 2022

1. Introduction

The analysis and control of complex conﬁgurations for high Reynolds prob-

lems of industrial interest is one of the most distinctive open challenges that

the scientiﬁc community has to face for ﬂuid mechanics applications in the com-

ing decades. The essential non-linearity of this class of ﬂows is responsible for

multiscale interactions and an extreme sensitivity to minimal variations in the

set-up of the problem. Under this perspective, applications using data-driven

tools from Data Assimilation [1, 2], and in particular of sequential tools such as

the Kalman ﬁlter (KF) [3] or the ensemble Kalman ﬁlter (EnKF) [4, 5], have

been recently used to obtain a precise estimation of the physical ﬂow state ac-

counting for bias or uncertainty in the performance of the investigative tool

[6, 7, 8, 9, 10, 11, 12]. Advances in EnKF Data Assimilation for meteoro-

logical application have inspired early studies in computational ﬂuid dynamics

(CFD), dealing in particular with the statistical inference of boundary con-

ditions [13] or the optimization of the behavior of turbulence / subgrid scale

modeling [14, 15, 16]. However, further advances are needed for a systematic

application to complex ﬂows. The ﬁrst problematic aspect is associated with

the computational cost associated with the generation of the ensemble. This

issue has usually been bypassed relying on the use of stationary reduced-order

numerical simulations such as RANS [17], providing some statistical inference of

turbulence modeling [7, 18, 11, 12]. Applications for scale resolving unstation-

ary ﬂows such as direct numerical simulation (DNS) and large eddy simulation

(LES) are much more rare in the literature, because of the computational re-

sources required [10, 19] to generate an acceptably large database to perform

a converged parametric inference. Alternative strategies recently explored to

reduce the computational costs deal with multilevel [20, 21, 22] / multiﬁdelity

[23] ensemble applications.

A second issue to be faced is that the solution at the end of the time step

should, as much as possible, comply with the dynamic equations represented by

the discretized numerical model (conservativity). One may argue that, if the

solution at the beginning of the time step is not accurate, then conservativity

is not an eﬃcient objective to be targeted. However, it is also true that the

correction performed to the state estimation by KF methods may be respon-

sible for non-physical perturbations of the predicted state, which may produce

irreversible numerical instabilities for complex ﬂows.

In a recent work, the Authors have presented a multigrid ensemble Kalman

ﬁlter [24] (MEnKF, renamed from now on MGEnKF). This strategy manipulates

data using multiple meshes with diﬀerent resolutions, exploiting the natural

multilevel structure of multigrid solvers. This algorithm belongs to the class of

multilevel methods whose modern treatment was ﬁrst proposed by Giles [25, 26].

The main advantage of the MGEnKF is that a good level of accuracy of the DA

procedure (comparable to classical application of the EnKF) is obtained with

a signiﬁcant reduction of the computational resources required. In addition,

spurious oscillations of the physical variables due to the state estimation are

naturally damped by the iterative resolution procedure of the multigrid solver.

In the case of the classical Full Approximation Scheme (FAS) two-grid multi-

grid algorithm, the sources of information operating in the MGEnKF are the

following:

•One main simulation whose ﬁnal solution at each time step is provided

on the ﬁne level of the grid.

•An ensemble of low-resolution simulations, which are performed at the

coarse level of the grid.

•Some observations which are provided locally in space and time in the

physical domain.

The data assimilation procedure, which will be described in detail in Sec. 2,

relies on the recursive nature of the multigrid algorithm. The update of the

physical state of the ﬂow as well as its parametric description are obtained via

two sets of operations:

•In the outer loop, a classical EnKF procedure is performed using the results

from an ensemble of simulations and the observation. The EnKF is used

to update the physical state and the parametric description of i) every

member of the ensemble on the coarse grid and ii) the main simulation

on the ﬁne grid level, via a projection operator.

•In the inner loop, the physical state obtained with the main simulation,

which is more precise than the predicted state by the ensemble members, is

used as surrogate observation in a new optimization procedure to improve

the predictive capabilities of the physical model over the coarse grid.

In the ﬁrst article detailing the MGEnKF [24], the focus of the analysis has

been over the performance of the outer loop. This choice was performed due to

the central contribution of this loop to the global data assimilation strategy. For

this reason, the inner loop was suppressed in order to obtain an unambiguous

assessment of this element of the MGEnKF.

In this manuscript, an extensive analysis is performed to assess the eﬀects

of the inner loop over the global accuracy obtained via the MGEnKF. While

the accuracy of the numerical model employed to obtain the predicted states for

the ensemble members is directly aﬀected by the outer loop, further signiﬁcant

improvement is expected with the application of the inner loop for two main

reasons. The ﬁrst one is that the usage of surrogate observation from the main

simulation is consistent with the numerical model used for time advancement

on the diﬀerent reﬁnement levels of the computational grids. Therefore, biases

that can aﬀect data assimilation using very diﬀerent sources of information

(such as experiments and numerical results) are naturally excluded. One can

also expect a faster rate of convergence owing to this property. The second

valuable feature of the inner loop is that, as the whole physical state of the

main simulation is known on the ﬁne grid level, the surrogate observation can

be sampled everywhere in the physical domain. One of the main problematic

aspects when assimilating experimental results in numerical models is that often

the placement of sensors is aﬀected by physical limitations which can preclude

the sampling in highly sensitive locations. This problem is completely bypassed

in the inner loop, where the user can arbitrarily select the number and the

location of sensors. This also opens perspectives of automatic procedures to

determine optimal sensor placement [27, 28], which are not explored in the

present manuscript but will be targeted in future works.

The manuscript is organized as follows. In Sec. 2, the mathematical and

numerical ingredients used in the framework of the MGEnKF model are intro-

duced and discussed. An extensive discussion of the strategy used to perform

the inner loop is provided. In Sec. 3, the numerical models optimized in the in-

ner loop are discussed. In Sec. 4, the results of the numerical simulations for the

analysis of the one-dimensional advection equation are provided and discussed.

In Sec. 5, the analysis is extended to the one-dimensional Burgers’ equation. In

this case, the accurate representation of non-linear eﬀects is investigated. For

both test cases, comparisons between results obtained via the MGEnKF with

or without inner loop are performed. In Sec. 6, concluding remarks and future

developments are discussed.

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OptimizedparametricinferencefortheinnerloopoftheMultigridEnsembleKalmanFilterG.Moldovana,,G.Lehnascha,L.Cordiera,M.MeldiaaInstitutPprime,CNRS-ISAE-ENSMA-UniversitedePoitiers,11Bd.MarieetPierreCurie,SiteduFuturoscope,TSA41123,86073PoitiersCedex9,FranceAbstractEssentialfeaturesoftheMultigridEnsemble...

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Optimized parametric inference for the inner loop of the Multigrid Ensemble Kalman Filter

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