Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark.

If a solution of a considered partial differential equation (PDE) is highly sensitive to its parameters,
accurate estimation of the parameters and their uncertainties is essential to obtain a correct approximation of the solution.
Partial observations of the solution are then used to infer uncertain parameters
by solving a PDE-constrained inverse problem.
For instance, one can approach such problems via methods induced by Bayes' formula

When aiming at practical applications as in oil reservoir management

Adaptive Sequential Monte Carlo (SMC) methods are different approaches to approximate the posterior with an

Ensemble Kalman inversion (EKI) approximates primarily the first two moments of the posterior, which makes it computationally attractive for estimating high dimensional parameters

We note that the EKI is an iterative ensemble smoother

As an alternative ansatz one can employ optimal transport resampling that lies at the heart of the ensemble transform particle filter (ETPF) proposed by

In this work we address two issues that have arisen in the context of the TETPF: (i) the immense computational costs of solving the associated optimal transport problem and (ii) the lack of robustness of the TETPF with respect to high dimensional problems. More specifically, the performance of ETPF has been found to be highly dependent on the initial guess. Although tempering restrains any sharp fail in the importance sampling step due to a poor initial ensemble selection, the number of required intermediate steps and the efficiency of ETPF still depend on the initialisation.
The lack of robustness in high dimensions can be addressed via a hybrid approach that combines a Gaussian approximation with a particle filter approximation

Along the lines of

The remainder of the paper is organised as follows: in Sect.

We assume

We consider Sequential Monte Carlo (SMC) methods that approximate the posterior measure

When an easy-to-sample ensemble from the prior

The choice of ESS to define a tempering parameter is supported
by results of

The SMC method with importance sampling (Eq.

Due to the stationarity of the parameters, SMC methods require ensemble perturbation. In the framework of particle filtering for dynamical systems, ensemble perturbation is achieved by rejuvenation, when ensemble members of the posterior measure are perturbed with a random noise sampled from a Gaussian distribution with zero mean and a covariance matrix of the prior measure. The covariance matrix of the ensemble is inflated, and no acceptance step is performed due to the associated high computational costs for a dynamical system.

Since we consider a static inverse problem, for ensemble perturbation we employ a Metropolis–Hastings method (thus we mutate samples) but with a proposal that speeds up the MCMC method for estimating a high dimensional parameter.
Namely, we use the ensemble mutation of

For a Gaussian prior we use the preconditioned Crank–Nicolson MCMC (pcn-MCMC) method:

For a uniform prior

As we have already mentioned in Sect.

The origin of the optimal transport theory lies in finding an optimal way of redistributing mass which was first formulated by

Let us consider a scenario whereby the initial distribution of matter is represented by a probability measure

Let

As discussed above, solving the optimal transport problem has a computational complexity of

For Bayesian inverse problems with Gaussian measures, ensemble Kalman inversion (EKI) is one of the widely used algorithms.
The EKI is an adaptive SMC method that approximates primarily the first two statistical moments of a posterior distribution.
For a linear forward model, the EKI is optimal in the sense that it minimises the error in the mean

The intermediate measures

Since we are interested in an ensemble approximation of the posterior distribution, we update the ensemble members by

Despite the underlying Gaussian assumption, the EKI is remarkably robust in non-linear high dimensional settings as opposed to consistent SMC methods such as the TET(S)PF. For many non-linear problems it is desirable to have better uncertainty estimates while maintaining a level of robustness. This can be achieved by factorising the likelihood given by Eq. (

Then it is possible to alternate between methods with complementing properties such as the EKI and the TET(S)PF updates; e.g. likelihood,

Note that the terminology is also used in the context of data assimilation filters combining variational and sequential approaches.

We consider a steady-state single-phase Darcy flow model defined over an aquifer of a two-dimensional physical domain

We note that a single-phase Darcy flow model, though not a steady-state model, is widely used to model the flow in a subsurface aquifer and to infer uncertain permeability using data assimilation. For example,

Geometrical configuration of channel flow: amplitude d

We consider the following two parameterisations of the permeability function

F1: log permeability over the entire domain

F2: permeability over domain

We assume that the log permeability for both F1 and F2 is drawn from a Gaussian distribution

With

For F1, the prior for log permeability is a Gaussian distribution with mean 5. The grid dimension is

For F2, we assume geometrical parameters

Both the true permeability and an initial ensemble are drawn from the same prior distribution as the prior includes knowledge about geological properties. However, an initial guess is computed
on a coarse grid, and the true solution is computed on a fine grid
that has twice the resolution of the coarse grid.
The synthetic observations of pressure are obtained by

To save computational costs, we choose an ESS threshold

We conduct numerical experiments with ensemble sizes

For geometrical parameters

For F1, we perform numerical experiments using 36 uniformly distributed observations,
which are displayed in circles in Fig.

Application to F1 parameterisation:
using Sinkhorn approximation

Mean log permeability for F1 inference for the lowest error at ensemble size

We plot mean log permeability at ensemble size

For F2, we perform numerical experiments using nine uniformly distributed observations.
which are displayed in circles in Fig.

Application to F2 parameterisation using Sinkhorn approximation.
Box plot over 20 independent simulations of KL divergence for geometrical parameters: amplitude

For frequency, angle and initial point, whose KL divergence is displayed in Fig.

When comparing the TESPF(

The same as Fig.

In Fig.

Posteriors of geometrical parameters for F2 inference: amplitude

Now we investigate adaptive SMC performance for permeability estimation.
First, we display results obtained using Sinkhorn approximation. The box plot shows over 20 independent simulations of the RMSE given by Eq. (

Application to F2 parameterisation using Sinkhorn approximation.
Box plot over 20 independent simulations of RMSE of mean log permeability outside the channel

Next, we compare the TESPF(

The same as Fig.

In Fig.

Mean log permeability for F2 inference for the lowest error at ensemble size

A Sinkhorn adaptation, namely the TESPF, of the previously proposed TETPF has been introduced and numerically investigated for a parameter estimation problem.
The TESPF has similar accuracy results to the TETPF (see Figs.

The table provides an overview of the computational complexity of all the algorithms considered in the paper.

Data and MATLAB codes for generating the plots are available at

SR, SD and JdW designed the research. SD ran the numerical experiments. SR, SD and JdW analysed the results and wrote the paper.

The authors declare that they have no conflict of interest.

The research of Jana de Wiljes and Sangeetika Ruchi have been partially funded by the Deutsche Forschungsgemeinschaft (DFG) SFB 1294/1 – 318763901. Further, Jana de Wiljes has been supported by Simons CRM Scholar-in-Residence Program and ERC Advanced Grant ACRCC (grant no. 339390). Sangeetika Ruchi has been supported by the research programme Shell-NWO/FOM Computational Sciences for Energy Research (CSER), project no. 14CSER007, which is partly financed by the Netherlands Organization for Scientific Research (NWO).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. SFB1294/1 – 318763901), the European Research Council (Advanced Grant ACRCC (grant no. 375 339390)), the Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Stichting voor de Technische Wetenschappen (grant no. Shell-NWO/FOM Computational Sciences for Energy Research (CSER), project no. 14CSER007) and the Simons Foundation (CRM Scholar-in-Residence Program grant).

This paper was edited by Olivier Talagrand and reviewed by Femke Vossepoel and Marc Bocquet.