A Reduced Basis Ensemble Kalman Method Francesco A. B. Silva1 Cecilia Pagliantini1 Martin Grepl2 and Karen Veroy1

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A Reduced Basis Ensemble Kalman Method

Francesco A. B. Silva1*, Cecilia Pagliantini1, Martin Grepl2

and Karen Veroy1

1*Department of Mathematics and Computer Science, Eindhoven

University of Technology, Eindhoven, 5600 MB, The Netherlands.

2Institute of Geometry and Practical Mathematics, RWTH

Aachen University, Aachen, 52056, Germany.

*Corresponding author(s). E-mail(s): f.a.b.silva@tue.nl;

Contributing authors: c.pagliantini@tue.nl;

grepl@igpm.rwth-aachen.de;k.p.veroy@tue.nl;

Abstract

In the process of reproducing the state dynamics of parameter dependent

distributed systems, data from physical measurements can be incorpo-

rated into the mathematical model to reduce the parameter uncertainty

and, consequently, improve the state prediction. Such a Data Assimila-

tion process must deal with the data and model misﬁt arising from experi-

mental noise as well as model inaccuracies and uncertainties. In this work,

we focus on the ensemble Kalman method (EnKM), a particle-based iter-

ative regularization method designed for a posteriori analysis of time

series. The method is gradient free and, like the ensemble Kalman ﬁlter

(EnKF), relies on a sample of parameters or particle ensemble to identify

the state that better reproduces the physical observations, while preserv-

ing the physics of the system as described by the best knowledge model.

We consider systems described by parameterized parabolic partial diﬀer-

ential equations and employ model order reduction (MOR) techniques to

generate surrogate models of diﬀerent accuracy with uncertain parame-

ters. Their use in combination with the EnKM involves the introduction

of the model bias which constitutes a new source of systematic error.

To mitigate its impact, an algorithm adjustment is proposed accounting

for a prior estimation of the bias in the data. The resulting RB-EnKM

is tested in diﬀerent conditions, including diﬀerent ensemble sizes and

increasing levels of experimental noise. The results are compared to those

obtained with the standard EnKF and with the unadjusted algorithm.

arXiv:2210.02279v1 [math.NA] 5 Oct 2022

2A Reduced Basis Ensemble Kalman Method

Keywords: Inverse Problems, Ensemble Kalman Method, Model Order

Reduction, Representation Error

1 Introduction

The problem of estimating model parameters of static and dynamical systems

is encountered in many applications from earth sciences to engineering. In this

work we focus on the parameter estimation of dynamical systems described

by parameterized parabolic partial diﬀerential equations (pPDEs). Here, we

assume that a limited and polluted knowledge of the solution is available at

multiple time instances through noisy local measurements.

For solving this kind of inverse problem, countless deterministic and

stochastic methods have been proposed. Among them, a widely used tech-

nique is the so-called ensemble Kalman ﬁlter [1], a recursive ﬁlter employing a

series of measurements to obtain improved estimates of the variables involved

in the process. The idea of using the EnKF for reconstructing the parameters

of dynamical systems traces back to [2,3], in which trivial artiﬁcial dynam-

ics for the parameters was assumed to make the estimation possible. This was

naturally accompanied by eﬀorts for improving the performance of the method

in terms of stability, by introducing covariance inﬂation [4,5] and localization

[4,6], and in terms of computational cost. Relevant to the latter have been

the development of multi- level methods [7], the use of model order reduction

techniques [8], and the introduction of further surrogate modeling techniques

[9]. The use of approximated models inevitably led to the study of the impact

of model error on the EnKF [10,11], alongside with other data assimilation

methods [12,13].

Although ensemble Kalman methods were originally meant for sequential

data assimilation, i.e., for real-time applications, they proved to be reliable also

for asynchronous data assimilation [14]. The ﬁrst paper proposing to adapt

the EnKF to a retrospective data analysis was [15]. For analysis, the data

are employed all at once at the end of an assimilation window, which is in

common with a series of methods, e.g., variational methods [16] such as 4D-

VAR [17] and other smoothers [18]. Compared to those approaches, the EnKF

is particularly appealing since it does not require the computation of Fr´echet

derivatives, a major complication for data assimilation algorithms.

In [19], Iglesias et al. introduced what they called the ensemble Kalman

method, an EnKF-based asynchronous data assimilation algorithm. Depending

on the design of the algorithm, this method has connections to Bayesian data

assimilation [20] and to maximum likelihood estimation [21]. In particular, in

the latter case, the method constitutes an ensemble-based implementation of

so-called iterative regularization methods [22]. In the case of perfect models,

the EnKM has already been analyzed in depth in [20,23] and convergence and

identiﬁability enhancements have been proposed in [24,25]. Due to the itera-

tive nature of the EnKM, dealing with high-dimensional parametric problems

A Reduced Basis Ensemble Kalman Method 3

is often computationally challenging. In [26] a multi-level strategy has been

proposed to improve the computational performance of the method.

In this work we propose an algorithm, called Reduced Basis Ensemble

Kalman Method (RB-EnKM), that leverages the computational eﬃciency of

surrogate models obtained with MOR techniques to solve asynchronous data

assimilation problems via ensemble Kalman methods. The use of the EnKM

allows us to avoid adjoint problems that are often diﬃcult to reduce and intrin-

sically depend on the choice of measurement positions. Model order reduction,

already employed in other data assimilation problems [27,28], is used as a

key tool for accelerating the method. However, the use of approximate models

within the EnKM introduces a model error that could hinder the convergence

of the method. In this work, we propose to deal with this error by including a

prior estimation of the bias in the data. Speciﬁcally, we incorporate empirical

estimates of the mean and covariance of the bias in the Kalman gain. In some

instances, those quantities can be computed at a negligible cost by employing

the same training set used for the construction of the reduced model.

The paper is structured as follows: in Section 2we introduce the asyn-

chronous data assimilation problem together with the standard ensemble

Kalman method (Algorithm 1). Subsequently, in Section 3.1, we present an

overview on reduced basis (RB) methods and describe how to use them in

combination with the ensemble Kalman method to derive the RB-EnKM

(Algorithm 2). In Section 4, we test the new method on two numerical

examples. In the ﬁrst example, we estimate the diﬀusivity in a linear advection-

dispersion problem in 2D (Section 4.1), while in the second, we estimate

the hydraulic log-conductivity in a non-linear hydrological problem (Section

4.2). In both cases, we compare the behavior of the full order and reduced

order models in diﬀerent conditions. Section 5provides conclusions and

considerations on the proposed method and on its numerical performances.

2 Problem Formulation

Let Ube a given function space and let P ⊂ RNp, with Np∈N+, be a set

of model parameters. We consider the pPDE: for any parameter µ∈ P, ﬁnd

u(·,·;µ)∈ U such that ∂tu(x, t;µ) = Fµu(x, t;µ) for any x∈Ω⊂Rd

and t∈I:= (0, T ]⊂R+. Here Fµis a generic parameterized diﬀerential

operator and ∂tis the ﬁrst order partial time derivative. This pPDE provides

the constraint to the inverse problem of estimating the unknown parameter

µ?∈ P from data or observations given by

y(µ?,η) = Lu(x, t;µ?) + η

s.t. ∂tu(x, t;µ?) = Fµ?u(x, t;µ?).(1)

Here, L:U → RNm, with Nm∈N+, maps the space of the solutions to the

space of the measurements, simulating the observation process, and ηis an

unknown realization of a Gaussian random variable with zero mean and given

4A Reduced Basis Ensemble Kalman Method

covariance, Σ∈RNm×Nm. Note that both the observed data yand additive

noise ηare Nm-dimensional vector-valued quantities and that Σis a symmetric

positive-deﬁnite matrix deﬁning the inner product k·k2

Σ−1:=kΣ−1/2· k2on

RNm, where k·k2is the Euclidean norm.

To solve this inverse problem, we must explicitly solve the pPDE (1). This

is done using a suitable discretization, in space and time, of the diﬀerential

operator Fµ. To this end, we introduce an approximation space Vh⊂ U so

that the approximate problem reads:

ﬁnd uh(µ) = uh(·,·;µ)∈ Vhs.t. ∂tuh(x, t;µ) = Fh

µuh(x, t;µ).(2)

The discretization of the pPDE can be chosen according to the speciﬁc problem

of interest. In all numerical examples proposed in this work, we employ a space-

time Petrov–Galerkin discretization of (1) with piecewise polynomial trial and

test spaces, as described in Section 4, and we assume (2) to be suﬃciently

accurate such that we can take y(µ?,η) = Luh(x, t;µ?) + η.

To characterize the observation of the solution, we introduce the forward

response map G:P → RNmdeﬁned as G(µ):=Luh(x, t;µ) for any solution

of the pPDE (2). Although the use of the map Gresults in a more compact

notation, omitting its dependence on the solution of the pPDE conceals a

key aspect of the method, i.e., the mapping from the parameter vector to the

corresponding space-time pPDE solution. For this reason, and because it makes

it harder to introduce the problem discretization, it will be used with caution.

2.1 The Ensemble Kalman Method

The data assimilation problem presented above can be recast as a minimization

problem for the cost functional, Φ(µ|y):=ky(µ?,η)− Luh(x, t;µ)k2

Σ−1, rep-

resenting the misﬁt between the experimental data, y(µ?,η), and the forward

response. The optimal parameter estimate µopt(y) is thus given by

µopt(y) = arg min

µ∈P Φ(µ|y)

s.t. ∂tuh(x, t;µ) = Fh

µuh(x, t;µ).

(3)

This is equivalent to a maximum likelihood estimation, given the likelihood

function, l(µ|y) = exp{−1

2Φ(µ|y)}, associated with the probability density

function of the data, y|µ, i.e., the probability of observing yif µis the para-

metric state. The shape of the function follows from the probability density

function of the Gaussian noise realization.

Among various methods proposed to solve this optimization problem, the

EnKM relies on a sequence of parameter ensembles En, with n∈N+, to

estimate the minimum of the cost functional. Each ensemble consists of a col-

lection {µ(j)

n}J

j=1 of J∈N+parameter vectors µ(j)

n, hereby named ensemble

members or particles, whose interaction, guided by the experimental measure-

ments, causes them to cluster around the solution of the problem as iterations

A Reduced Basis Ensemble Kalman Method 5

proceed. At the beginning of each iteration, the solution of the pPDE and its

observations are computed for each j∈ {1, . . . , J}. Subsequently, the ensemble

is updated based on the empirical correlation among parameters and between

parameters and measurements, as well as on the misﬁts between the experi-

mental measurements y(µ?,η) and the particle measurements Luh(x, t;µ(j)

n).

A single iteration, equivalent to the one in [19], is formalized in the following

pseudo algorithm:

Algorithm 1 Iterative ensemble method for inverse problems.

Let E0be the initial ensemble with elements {µ(j)

0}J

j=1 sampled from a distribu-

tion Π0(µ). For n= 0,1, . . .

(i) Prediction step. Compute the measurements of the solution for each particle

in the last updated ensemble:

G(µ(j)

n) = Luh(x, t;µ(j)

s.t. ∂tuh(x, t;µ(j)

n) = Fh

µ(j)

uh(x, t;µ(j)

n) for all j∈ {1,...,J}.(4)

(ii) Intermediate step. From the last updated ensemble measurements and

parameters, deﬁne the sample means and covariances:

Pn=1

j=1

G(µ(j)

n)G(µ(j)

n)>− GnG>

nwith Gn=1

j=1

G(µ(j)

n) (5)

Qn=1

j=1

µ(j)

nG(µ(j)

n)>−µnG>

nwith µn=1

j=1

µ(j)

n.(6)

(iii) Analysis step. Update each particle in the ensemble:

µ(j)

n+1 =µ(j)

n+Qn(Pn+Σ)−1(y(j)

n− G(µ(j)

n)) with y(j)

n∼ N (y,Σ).(7)

In the last step of the algorithm, the cross correlation matrices Pnand Qn

are used to compute the Kalman gain Kn:=Qn(Pn+Σ)−1. This modulates

the extent of the correction: a low-gain corresponds to conservative behavior,

i.e., small changes in the particle positions, while a high-gain involves a larger

correction. Note that the experimental data are perturbed with artiﬁcial noise

sampled from the same distribution assumed for the experimental noise η.

This leads to an improved estimate over the unperturbed case.

A termination criterion for the algorithm is essential for the proper imple-

mentation of the method. The one presented in [19] is based on the discrepancy

principle and consists in stopping the algorithm when the error between the

experimental data and the measurements is comparable to the experimental

noise, that is, when ky− G(µn)k2

Σ−1≤σkηk2

Σ−1for some σ≥1. An alterna-

tive approach is to set a threshold for the norm of the parameter update, i.e.,

to terminate the algorithm when kµn+1 −µnk2≤τkµn+1k2for some τ1.

The latter criterion is more robust to model errors and is therefore used in our

numerical experiments.

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AReducedBasisEnsembleKalmanMethodFrancescoA.B.Silva1*,CeciliaPagliantini1,MartinGrepl2andKarenVeroy11*DepartmentofMathematicsandComputerScience,EindhovenUniversityofTechnology,Eindhoven,5600MB,TheNetherlands.2InstituteofGeometryandPracticalMathematics,RWTHAachenUniversity,Aachen,52056,Germany.*Corre...

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A Reduced Basis Ensemble Kalman Method Francesco A. B. Silva1 Cecilia Pagliantini1 Martin Grepl2 and Karen Veroy1

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