2013/12/18 by Alexander Litvinenko, Litvinenko, Alexander, Hermann G. Matthies +1 · 20 citations
Computer Science · Mathematics · #60H15 #60H25 #62F15 #62P30 #65N21 #74C05 #74G75 #80A23 #Algorithm #Applied mathematics #Artificial intelligence #Bayesian probability #Computation #Computer science #Discretization #FOS: Mathematics #G.1.8 #G.3 #Gaussian Processes and Bayesian Inference #Inverse #J.2 #Machine learning #Markov Chains and Monte Carlo Methods #Mathematical optimization #Mathematics #Monte Carlo method #Numerical Analysis (math.NA) #Quadratic equation #Sampling (signal processing) #Statistical Methods and Inference #Statistics #Uncertainty quantification
paper · pdf · doi:10.48550/arxiv.1312.5048
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2013/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. This is especially the case as 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 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 quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.