Björn Sprungk

Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case

We investigate the stochastic collocation method for parametric, elliptic partial differential equations (PDEs) with lognormally distributed random parameters in mixed formulation. Such problems arise, e.g., in uncertainty quantification studies for flow in porous media with random conductivity. We show the analytic dependence of the solution of the PDE w.r.t. the parameters and use this to show convergence of the sparse grid stochastic collocation method. This work fills some remaining theoretical gaps for the application of stochastic collocation in case of elliptic PDEs where the diffusion coefficient is not strictly bounded away from zero w.r.t. the parameters. We illustrate our results for a simple groundwater flow problem.

On a Generalization of the Preconditioned Crank–Nicolson Metropolis Algorithm

Metropolis algorithms for approximate sampling of probability measures on infinite dimensional Hilbert spaces are considered, and a generalization of the preconditioned Crank–Nicolson (pCN) proposal is introduced. The new proposal is able to incorporate information on the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independently of the state-space dimension and the variance of the observational noise. Moreover, a qualitative convergence result is provided by a comparison argument for spectral gaps. In particular, it is shown that the generalization inherits geometric convergence from the Metropolis algorithm with pCN proposal.

Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems

We analyze the Ensemble and Polynomial Chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.

Bayesian Inverse Problems and Kalman Filters

We provide a brief introduction to Bayesian inverse problems and Bayesian estimators emphasizing their similarities and differences to the classical regularized least-squares approach to inverse problems. We then analyze Kalman filtering techniques for nonlinear systems, specifically the well-known Ensemble Kalman Filter (EnKF) and the recently proposed Polynomial Chaos Expansion Kalman Filter (PCE-KF), in this Bayesian framework and show how they relate to the solution of Bayesian inverse problems.

The information content of credit ratings: evidence from European convertible bond markets

Prior research has investigated the information content of credit ratings for standard financing instruments such as stocks and corporate bonds, while this question has been neglected for convertible bonds (CBs) so far. CBs are simultaneously determined by the bond floor and the conversion value, which makes it more difficult to assess price effects following rating announcements. In this context, we compare price effects of CBs with those of stocks and corporate bonds of the same issuer using robust event study methods. Our findings indicate that rating changes convey new information for investors in European CBs. In terms of the direction of the expected price reaction, we find CBs to react in a more debt-like manner to the announcement of a rating change. Moreover, our results provide evidence that the magnitude of price reactions differs among different types of securities.

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general

Stochastic differential equations with fuzzy drift and diffusion

L^2

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables - with Application to Lognormal Elliptic Diffusion Problems

-convergence theory based on previous work by Bachmayr et al. [ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 341--363] and Chen [ESAIM Math. Model. Numer. Anal., in press, 2018, https://doi.org/10.1051/m2an/2018012] and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We specifically verify for Gauss--Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

Metropolis‐Hastings Importance Sampling Estimator

A new framework for the fuzzification of stochastic differential equations is presented. It allows for a detailed description of the model uncertainty and the non-predictable stochastic law of natural systems, e.g. in ecosystems even the probability law of the random dynamic changes due to unobservable influences like anthropogenic disturbances or climate variation. The fuzziness of the stochastic system is modelled by a fuzzy set of stochastic differential equations which is identified with a fuzzy set of initial conditions, time-dependent drift and diffusion functions. Using appropriate function spaces the extension principle leads to a consistent theory providing fuzzy solutions in terms of fuzzy sets of processes, fuzzy states, fuzzy moments and fuzzy probabilities.