Oliver Pajonk

Parameter identification in a probabilistic setting

Abstract The parameters to be identified are described as random variables, the randomness reflecting the uncertainty about the true values, allowing the incorporation of new information through Bayes’s theorem. Such a description has two constituents, the measurable function or random variable, and the probability measure. One group of methods updates the measure, the other group changes the function. We connect both with methods of spectral representation of stochastic problems, and introduce a computational procedure without any sampling which works completely deterministically, and is fast and reliable. Some examples we show have highly nonlinear and non-smooth behaviour and use non-Gaussian measures.

Parametric and Uncertainty Computations with Tensor Product Representations

Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Loeve expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.

Sampling-free linear Bayesian updating of model state and parameters using a square root approach

We present a sampling-free implementation of a linear Bayesian filter based on a square root formulation. It employs spectral series expansions of the involved random variables, one such example being Wieners polynomial chaos. The method is compared to several related methods, as well as a full Bayesian update, on a simple scalar example. Additionally it is applied to a combined state and parameter estimation problem for a chaotic system, the well-known Lorenz-63 model. There, we compare it to the ensemble square root filter (EnSRF), which is essentially a probabilistic implementation of the same underlying estimator. The spectral method is found to be more robust than the probabilistic one, especially for variance estimation. This is to be expected due to the sampling-free implementation.

Inverse Problems in a Bayesian Setting

In a Bayesian setting, inverse problems and uncertainty quantification (UQ) --- the propagation of uncertainty through a computational (forward) model --- are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.

Multiscale Wavelet Analysis Applied to Localization of Covariance Estimates in EnKF

The ensemble Kalman filter (EnKF) has become very popular in the field of assisted history matching for its appealing features. Nonetheless, problems can result from so-called spurious correlations due to the finite ensemble size [e.g. Evensen, 2009], which are considered as unphysical. The result of these correlations is a reduction of ensemble spread at model locations where no related data is available. This may cause an underestimation of the uncertainty and can result in a collapsed ensemble [Hamill, 2001]. Two methods are commonly used to address the unwanted reduction of variance: covariance infla-tion and localization. This contribution presents a new covariance localization approach based on multiscale (or multiresolution) wavelet analysis [Daubechiers, 1992]: the model state vector is transformed to a multiscale wavelet space. Correlations are computed in this space, not in the model space. This procedure allows the application of a new localization scheme, i.e., a different covariance localization function can be applied for each of the scale levels using a standard Schur product approach. Especially it allows us to filter unphysical long range correlations from fine scales while retaining longer correlations on coarser scales. Afterwards EnKF updates are computed and the transformation back to model space is applied. This contribution explains our wavelet-based localization approach and presents numerical results for the application of a synthetic model. The results are compared to standard localization approaches. The application to a real field simulation model is discussed.

Stochastic Optimization Using EA and EnKF – A Comparison

ulation are increasingly included in best-practice workflows in the Oil & Gas industry. Most optimization methods applied to model validation in reservoir simulation, including so-called Evolutionary Algorithms like Genetic Algorithms (GA) and Evolution Strategies (ES), use an objective function definition based on the overall simulation period. The integration of a sequential data assimilation process is conceptually not embedded in those optimization methods. The Ensemble Kalman Filter (EnKF) has entered this field for its appealing features. Sequential data assimilation allows the implementation of real-time model updates where classical optimization techniques require simulating the complete history period. This may have negative effects on efficiency and use of computing time. In contrast, EnKF sequentially assimilates information streams into a set of numerical models. While being a special case of a fully fledged particle filter the EnKF method with application to reservoir simulation has proven to generate results with a reasonable amount of ensemble members. The similarities between a particle filter (Monte Carlo Filter) and an Evolutionary Algorithm (Generic Algorithm) have been previously pointed out from a rather theoretical point of view (1,2). In this work we present a concise overview of Evolutionary Algorithms and the Ensemble Kalman Filter in such a way that the cross-relations become apparent. Similarities are highlighted and the potential for hybrid couplings is discussed. Practical implications for the implementation of these methods are derived. 1. Higuchi, Tomoyuki. Monte carlo filter using the genetic algorithm operators. Journal of Statistical Computation and Simulation. 1997, Vol. 1, 59, pp. 1-23. 2. —. Self-organizing Time Series Model. [book auth.] A. Doucet, J.F.G. de Freitas and N.J. Gordon. Sequential Monte Carlo Methods in Practice. s.l. : Springer, 2001, pp. 429-444.

Comparison of Numerical Approaches to Bayesian Updating

This paper investigates the Bayesian process of identifying unknown model parameters given prior information and a set of noisy measurement data. There are two approaches being adopted in this research: one that uses the classical formula for measures and probability densities and one that leaves the underlying measure unchanged and updates the relevant random variable. The former is numerically tackled by a Markov chain Monte Carlo procedure based on the Metropolis-Hastings algorithm, whereas the latter is implemented via the ensemble/square root ensemble Kalman filters, as well as the functional approximation approaches in the form of the polynomial chaos based linear Bayesian filter and its corresponding square root algorithm. The study attempts to show the principal differences between full and linear Bayesian updates when a direct or a transformed version of measurements are taken into consideration. In this regard the comparison of both strategies is provided on the example of a steady state diffusion equation with nonlinear and transformed linear measurement operators.

Deterministic Linear Bayesian Updating of State and Model Parameters

We present a sampling-free implementation of a linear Bayesian filter. It is based on spectral series expansions of the involved random variables, one such example being Wieners polynomial chaos. The method is applied to a combined state and parameter estimation problem for a chaotic system, the well-known Lorenz-63 model. We compare it to the ensemble Kalman filter (EnKF), which is essentially a stochastic implementation of the same underlying estimator---a fact which is demonstrated in the paper. The spectral method is found to be more reliable for the same computational load, especially for the variance estimation. This is to be expected due to the fully deterministic implementation.