Übersicht über Systemidentifikation mit dem Fokus der Inversen Modellierung

The intention behind this literature review is to obtain knowledge about the current status in the field of system identification with special focus put on the inverse modelling step. There the parameters for a model are to be determined by taking data obtained from the true system into account. The application in mind is located in geophysics, especially oil reservoir engineering, so special focus is put on methods which are relevant for system identification problems that arise in that context. Nonetheless the review should be interesting for everybody who works on system identification problems.--- Die Intention des Literaturreviews ist eine Übersicht über den Bereich der Systemidentifikation, im speziellen den Bereich der inversen Modellierung, zu erhalten. In diesem Schritt werden Parameter für ein Modell durch Konditionierung auf gemessene Daten eines realen Systems bestimmt. Das Anwendungsgebiet ist im Bereich der Geophysik, im speziellen Erdöl-Reservoirs, angesiedelt. Daher werden besonders die dort genutzten Methoden betrachtet

Stochastische spektrale Methoden zur linearen Bayes'schen Inferenz

Simulation-based control of dynamic systems is of key importance for many areas of science and industry. To ensure the predictive capabilities, simulation models used for predicting control responses have to be calibrated to available observations. Bayesian approaches to make inference from data on unobservable quantities are used because of their consistent, inherent treatment of diverse sources of uncertainties. Spectral approaches to uncertainty quantification have become popular over the last years. However, their combination with Bayesian inference usually employs expensive probabilistic sampling methods. In this work, a family of linear Bayesian approaches is presented which directly results in a representation of the posterior. A specific implementation is discussed which overcomes some of the difficulties that remained unsolved in related approaches. All implementation details are given, and the applicability is demonstrated on some linear and non-linear numerical examples.Die simulationsbasierte Steuerung von dynamischen Systemen stellt eine Schlüsseltechnologie für weite Bereiche von Forschung und Industrie dar. Um die Vorhersagefähigkeiten von Simulationsmodellen sicherzustellen müssen diese auf die verfügbaren Daten kalibriert werden. Bayes'sche Ansätze für die Erzeugung von Rückschlüssen aus Daten auf unbeobachtbare Modellgrößen sind aufgrund ihrer inhärenten Möglichkeiten, Unsicherheiten in den Rückschlussprozess einzubetten, beliebt. Spektrale Methoden für die Quantifizierung von Unsicherheiten sind über die letzten Jahre populär geworden. Allerdings bedingt ihre Kombination mit Bayes'schen Rückschlussmethoden typischerweise den Einsatz von aufwändigen probabilistischen Abtastverfahren. In dieser Arbeit wird eine Familie von linearen Bayes'schen Vorgehensweisen präsentiert, welche direkt die spektrale à posteriori Repräsentation der unsicheren Zielgröße erzeugen. Eine spezifische Implementierung wird vorgestellt, welche einige der Schwierigkeiten der bisher existierenden Ansätze umgeht. Alle Implementierungsdetails hierzu werden beschrieben, und die Anwendbarkeit anhand von verschiedenen linearen und nicht-linearen numerischen Beispielen belegt

Overview of System Identification with Focus on Inverse Modeling: Literature Review

The intention behind this literature review is to obtain knowledge about the current status in the field of system identification with special focus put on the inverse modelling step. There the parameters for a model are to be determined by taking data obtained from the true system into account. The application in mind is located in geophysics, especially oil reservoir engineering, so special focus is put on methods which are relevant for system identification problems that arise in that context. Nonetheless the review should be interesting for everybody who works on system identification problems.--- Die Intention des Literaturreviews ist eine Übersicht über den Bereich der Systemidentifikation, im speziellen den Bereich der inversen Modellierung, zu erhalten. In diesem Schritt werden Parameter für ein Modell durch Konditionierung auf gemessene Daten eines realen Systems bestimmt. Das Anwendungsgebiet ist im Bereich der Geophysik, im speziellen Erdöl-Reservoirs, angesiedelt. Daher werden besonders die dort genutzten Methoden betrachtet

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.Comment: arXiv admin note: substantial text overlap with arXiv:1312.504

A Deterministic Filter for Non-Gaussian Bayesian Estimation

We present a fully deterministic method to compute sequential updates for stochastic state estimates of dynamic models from noisy measurements. It does not need any assumptions about the type of distribution for either data or measurement — in particular it does not have to assume any of them as Gaussian. It is based on a polynomial chaos expansion (PCE) of the stochastic variables of the model. We use a minimum variance estimator that combines an a priori state estimate and noisy measurements in a Bayesian way. For computational purposes, the update equation is projected onto a finite-dimensional PCE-subspace. The resulting Kalman-type update formula for the PCE coefficients can be efficiently computed solely within the PCE. As it does not rely on sampling, the method is deterministic, robust, and fast. In this paper we discuss the theory and practical implementation of the method. The original Kalman filter is shown to be a low-order special case. In a first experiment, we perform a bi-modal identification using noisy measurements. Additionally, we provide numerical experiments by applying it to the well known Lorenz-84 model and compare it to a related method, the ensemble Kalman filter

Direct Bayesian Update of Polynomial Chaos Representations

We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by a non-Gaussian prior distribution. The description of the introduced random fields is given in a ``white noise'' framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such representation, the probabilistic identification problem is cast in a polynomial chaos expansion setting and the linear Bayesian form of updating. This representation leads to a corresponding new formulation of the minimum squared error estimator, obtained by its additional projection onto the polynomial chaos basis. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is inexpensive to evaluate. In addition, we show that the well-known Kalman filter method is the low order part of this update. The proposed method has been tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

Parameter Identification in a Probabilistic Setting

Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.Comment: 29 pages, 16 figure

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 Wiener's 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

Deterministic Linear Bayesian Updating of State and Model Parameters for a Chaotic Model

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 Wiener's 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

Parametric and Uncertainty Computations with Tensor Product Representations

Part 2: UQ TheoryInternational audienceComputational 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-Loève 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