Prof. Bozidar Stojadinovic | ETH Zurich | Switzerland
In the last decade, we observed a significant increase in seismicity caused by anthropogenic activities, such as oil and gas extraction, geothermal projects, and CO2 carbon sequestration. Induced seismicity has characteristics that distinguish it from natural seismicity. In particular, the seismic rate is a consequence of an interaction between time-variant anthropogenic activities (e.g. fluid injection) and natural characteristics (e.g. presence of active faults) that are location-dependent. One peculiar difference between natural and induced seismicity is that both hazard and risk are time-dependent. In this talk, we present a non-uniform Poisson process to model induced seismicity associated with deep underground fluid injection. We describe the time-variant rate of the Poissonian process as a function of the fluid-injection rate and a set of physical parameters describing the ground characteristics. We treat this set of parameters as random variables, and their uncertainty reflects the source-to-source variability. The model, which has two layers of uncertainties, can, therefore, be classified as a hierarchical Bayesian model. A major strength of the Bayesian approach is that it allows uncertainties and expert judgments about the ground parameters to be encoded into a joint prior distribution of the model parameters. Moreover, as soon as the project starts and physical information become available, the Bayesian framework allows for the computation of the posterior distribution for the ground parameters, the formulation of predictive models for the Poissonian process, and borrowing statistical strength from data. After presenting the updating rules and strategies for the proposed Bayesian model, we conclude our talk by presenting a forecast model for predicting the number and the magnitude of induced events for a given future time frame.
Sampling-free Bayesian inversion with adaptive hierarchical tensor representation
The statistical Bayesian approach is a natural setting to alleviate the inherent ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters.
A sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed.
The delicate task to chose a suitable prior density is examined in terms of a coordinate transformation according to likelihood informations.
The proposed sampling-free approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in orthogonal chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood.
This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the respective low-rank (solution) manifolds.
All required computations can then be carried out efficiently in the low-rank format.
An interesting aspect is the evaluation of the exponential of the Bayesian potential by means of an adaptive Runge-Kutta method with tensors.
All discretization parameters are adjusted adaptively based on a posteriori error estimators or indicators.
Numerical experiments, involving affine and log-normal diffusion, demonstrate the performance and confirm the theoretical results.
Measure-Theoretic Stochastic Inversion of Groundwater Problems
In this work, we consider a recently developed measure-theoretic approach for solving stochastic inverse problems. We show that a sample based, non-intrusive, computational algorithm produces exact solutions to the stochastic inverse problem using a certain class of surrogate response surfaces. We use adjoint based techniques to estimate and correct for numerical error in the surrogate while simultaneously increasing the local order of the surrogate response surface. The use of the resulting enhanced surrogates are two-fold where we observe an increase in accuracy and decrease in computational complexity in computation of probabilities of specified events. The methodology may also be utilized in adaptive error control.
Impacts of forcing due to turbulent boundary layer uncertainty on modal response functions in structural acoustics
Dr. Sheri Martinelli | The Pennsylvania State University | United States
PhD Andrew Wixom | The Pennsylvania State University | United States
PhD Micah Shepherd | The Pennsylvania State University | United States
PhD Stephen Hambric | The Pennsylvania State University | United States
PhD Robert Campbell | The Pennsylvania State University | United States
An understanding of the influence of the fluctuating wall-pressure field beneath a turbulent boundary layer (TBL) is critical in the design of structures subject to fatigue loading or noise radiation (e.g., aircraft, watercraft or automobiles). Many models have been developed to provide a statistical description of this forcing, based mainly on flat plate assumptions augmented by fits to empirical data. One such model, by Corcos [G.M. Corcos / J. Sound. Vib. 6: 59–70, 1967], poses an exponential correlation model that incorporates empirically-derived parameters treated as universal constants. In this work, we expand the process of Corcos in terms of its parameters, which we treat as uncertain, using the technique of Karhunen-Loève. We then incorporate the truncated result into a generalized polynomial chaos (gPC) expansion of the modal response – the solution to the forward model. Given the gPC coefficients for a structure of interest, the resulting representation of the forward model can be used as an efficient means to generate data [Y. Marzouk and D. Xiu / Commun. Comput. Phys. 6(4): 826-847, 2009] in order to estimate the posterior distribution of the unknown parameters. The approach also produces expressions for the structure’s modal response, both eigenvalues and eigenfunctions, in terms of the random variables, permitting the study of the impacts of uncertain forcing on the mode shapes and natural frequencies. Results will be compared to standard estimates which typically assume Gaussian densities on system output.
Fast Bayesian model calibration by using non-intrusive interpolating surrogate methods
PhD Benjamin Sanderse | CWI Amsterdam | Netherlands
A popular method in UQ used to assess epistemic uncertainties of a model is Bayesian model calibration, in which the model discrepancy is calibrated via the parameters of the model. This yields a full probability density function on the parameters, called the posterior. Typically Monte Carlo methods are used to sample from the posterior, but this is computationally intractable for many numerical models.
Therefore, we employ a surrogate model, which is constructed by interpolation. Normally the nodes are chosen with respect to an input pdf. In our calibration framework this approach cannot be applied since the posterior is not known explicitly. We propose a new technique: adaptive construction of the surrogate model with weighted Leja nodes. These nodes are by construction nested, stable, and refine in the region of high posterior density.
The application that we consider is wind turbine wake simulation, where many uncertainties have an influence on the efficiency and life time.