Bojana V. Rosić

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.

Inelastic Media under Uncertainty: Stochastic Models and Computational Approaches

We discuss inelastic media under uncertainty modelled by probabilistic methods. As a prototype inelastic material we consider perfect plasticity. We propose a mathematical formulation as a stochastic variational inequality. The new element vis a vis a stochastic elastic medium is the so-called return map at each Gauss-point. We concentrate on a stochastic version of this, showing how it may be solved via (generalised) polynomial expansion.

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.

Identification of Properties of Stochastic Elastoplastic Systems

This paper presents the parameter identification in a Bayesian setting for the elastoplastic problem, mathematically speaking the variational inequality of a second kind. The inverse problem is formulated in a probabilistic manner in which unknown quantities are embedded in a form of the probability distributions reflecting their uncertainty. With the help of the stochastic functional analysis the update procedure is introduced as a direct, purely algebraic way of computing the posterior, which is comparatively inexpensive to evaluate. Such formulation involves the process of solving the convex minimisation problem in a stochastic setting for which the extension of classical optimization algorithm in predictor-corrector form as the solution procedure is proposed. A validation study of identification procedure is done through a series of virtual experiments taking into account the influence of the measurement error and the order of approximation on the posterior estimate.

Parameter estimation via conditional expectation: a Bayesian inversion

When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp. functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes’s theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, and shortly discuss various numerical approximations.

Stochastic Galerkin Method for the Elastoplasticity Problem with Uncertain Parameters

The mathematical formulation and numerical simulation of an elastic-plastic material with uncertain parameters in the small strain case is considered. Traditional computational approaches to this problem usually use some form of perturbation or Monte Carlo technique. This is contrasted here with more recent methods based on stochastic Galerkin approximations. In addition, we introduce the characterisation of the variational structure behind the discrete equations defining the closest-point projection approximation in stochastic elastoplasticity.

Stochastic Upscaling via Linear Bayesian Updating

In this work we present an upscaling technique for multi-scale computations based on a stochastic model calibration technique. We consider a coarse scale continuum material model described in the framework of generalised standard materials. The model parameters are considered uncertain in this approach, and are approximated using random variables. The update or calibration of these random variables is performed in a Bayesian framework where the information from a deterministic fine scale model computation is used as observation. The proposed approach is independent w.r.t. the choice of models on coarse and fine scales. Simple numerical examples are shown to demonstrate the ability of the proposed approach to calibrate coarse-scale elastic and inelastic material parameters.

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.