Dr. Isabell Franck | IPT - Insight Perspective Technologies GmbH / TUM | Germany
Prof. Stelios Koutsourelakis | TUM Professur fuer Kontinuumsmechanik | Germany
While calibration can almost always be archived it becomes problematic if the underlying model is incorrect, which will lead to wrong predictions and interpretations. Traditional approaches use an additional regression model (e.g. GPs) to account for an underlying model error. This can either violate physical constraints and/or is infeasible in high dimensions. In this work, we open the black box and unfold conservation and constitutive laws to estimate model discrepancies accurately. We use Variational Bayes to decrease computational costs and investigate this problem within a high-dimensional inverse problem.
Optimal reduction of observations for Bayesian inference
We consider the Bayesian inference problem of fitting a Gaussian linear model from overabundant observations corrupted by an independent Gaussian noise. The goal of this work is to determine low dimensional projection subspaces to project the observations, without degrading the posterior distribution of the inferred model. We target in particular applications in big data inversion and assimilation related to geosciences and weather forecasting.
In this work, we address two situations depending on the availability of the actual observations, corresponding to a posteriori reduction when the data are available and a priori reduction when they are not. The projection space is defined as the minimizer of cost functions derived from the information theory. Specifically, we consider and contrast strategies based on the minimization of the (possibly expected) Kullback-Leibler divergence, the log-det divergence, and the Shannon entropy. The optimization problems are formulated in terms of the reduced basis for the projection space. This formulation yields invariance properties (to rotation and scaling) that are subsequently exploited using efficient Riemannian optimization algorithms. We show that these algorithms can be particularly efficient when the noise structure of the linear model is correctly treated.
The strategies are first compared in a Bayesian linear regression setting, monitoring the convergence of the posterior distributions when the dimension of the projection space increases. The robustness of the a priori strategies is also numerically assessed. Finally, we discuss the extension of the approaches to nonlinear models.
In science, engineering and economy a large diversity of inverse problems exists. As is the same for forward problems also solving the inverse problems is generally done under uncertainties. These uncertainties have different origins, e.g. values of parameters which are not included in the inverse problem, the experimental quality or numerical aspects in the repeatedly solved forward problem during the inversion.
Additionally to these sources of uncertainties, regularization parameters required to solve the inverse problem in a robust manner often depend via discrepancy principles on vales (mainly the assumed the noise level) which are also never perfectly known.
The contribution discusses the possible scatter of results of inverse problems and by a sensitivity analysis, i.e. how these uncertainties can be assigned to uncertainties in the input parameters of the inverse problem.
The application examples are inspired by inverse problems occurring in Civil Engineering.
Bayesian techniques for parameter estimation in linear PDEs with noisy boundary conditions
Prof. Marco Iglesias | University of Nottingham | United Kingdom
PhD Zaid Sawlan | King Abdullah University of Science and Technology | Saudi Arabia
Prof. Marco Scavino | Universidad de la República | Uruguay
Prof. Raúl Tempone | King Abdullah University of Science and Technology | Saudi Arabia
Dr. Christopher Wood | University of Nottingham | United Kingdom
In this talk, we present a novel adaptation of a hierarchical Bayesian framework, introduced in [Ruggeri et al., 2016], for parameter estimation in linear time-dependent partial differential equations with noisy boundary conditions. For given model error assumptions, we obtain the joint likelihood of the quantities of interest and the initial and boundary conditions. The nuisance time-dependent boundary conditions are then marginalized out, and the use of a fast analytical approximation technique provided reliable estimates of the parameters of interest.
The application of the proposed method is exemplified to deal with the real problem posed by an experimental case study conducted in an environmental chamber, with measurements recorded every minute from temperature probes and heat flux sensors placed on both sides of a solid brick wall over a five-day period [Iglesias et al., 2017].
The unidimensional heat equation with unknown initial temperature and Dirichlet boundary conditions is used to model the heat transfer through the wall. After marginalizing the boundary conditions that act as nuisance parameters, we obtain the approximate a posteriori distributions for the wall parameters and the initial temperature. The results show that our technique reduces the bias error of the estimates of the wall parameters, compared to other approaches where the boundary conditions are assumed to be non-random. We calculate the information gain from the experiment, to recommend to the user how to efficiently minimize the duration of the measurement campaign and determine the path of the external temperature oscillation.
Finally, we introduce a sequential Bayesian setting for parametric inference in initial-boundary value problems related to linear parabolic partial differential equations. The performance of the new marginalized Ensemble Kalman filter algorithm is compared with the previous method through the analysis of the experimental data collected in the environmental chamber.
Dynamically adaptive data-driven simulation of extreme hydrological flows
Hydrological hazards such as storm surges and tsunamis are physically complex events that are very costly in loss of human life and economic productivity. Such disasters could be mitigated through improved emergency evacuation in real-time, and through development of resilient infrastructure using data-driven computational modeling. We investigate the novel combination of methodologies in forward simulation and data assimilation. The forward geophysical model is based on adaptive mesh refinement (AMR), a process by which a computational mesh adapts in time and space to the current state of a simulation. The forward solution is combined with ensemble based data assimilation methods, whereby observations from an event are assimilated to improve the veracity of the solution. The novelty in our approach is the tight two-way coupling of AMR and ensemble filtering techniques. The technology is tested with twin experiments and actual data from the event of Chile tsunami of February 27 2010.