Increasingly refined numerical models that depend on a large number of parameters introduce challenges with respect to the computability and interpretability of the generated data. Methods of uncertainty quantification and sensitivity analysis offer ways to identify relevant parameters for the construction of reduced complexity models. In a well-defined parameter range this ultimately allows to replace the original model by a possibly probabilistic fast surrogate and can provide insight into the main dependencies within their range of uncertainty. The present session focuses on the development and application of such techniques that are useful for a variety of model classes from different fields. In particular this includes Bayesian methods and the interplay of uncertainty quantification with surrogates and low-fidelity models as well as methods from machine learning. Application cases ranging from environmental science and biomechanics to plasma physics will demonstrate features and limitations. They will also show similarities and differences to be taken into account for techniques that aim to be widely applicable.
08:30
profit: A framework for parameter studies via surrogate models with nested uncertainty quantification
Christopher Albert | Max-Planck-Institut für Plasmaphysik | Germany
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Christopher Albert | Max-Planck-Institut für Plasmaphysik | Germany
In numerical modeling and experimental design one often seeks a reconstruction of (possibly probabilistic) input-output relations based on a finite number of samples. Such a surrogate can be used for interpolation and regression within parameter studies as well as for optimization purposes, i.e. to find input parameters that produce an extremum of the output. Due to limited resources it is often necessary to study only parameters of interest (POIs). Remaining parameters influence uncertainties in the result. Here we propose to split them into parameters of local interest for which a detailed uncertainty quantification (UQ) is performed and remaining nuisance parameters with only their combined effect on uncertainties of interest. Based on that separation a probabilistic surrogate of the output including local uncertainty information is constructed.
The profit framework provides a flexible implementation of this procedure, treating the underlying input-output relation as a black box and allowing different backends for construction of UQ and surrogates. The present version relies on polynomial chaos expansion via Chaospy for UQ and Gaussian process regression via GPFlow for surrogates. Practical features include scripts for pre- and postprocessing as well a template engine to produce run files and start computations on batch systems of HPC systems. After outlining general method and usage, new developments of sparse approximations and adaptive sampling will be presented.
Link to code: https://dx.doi.org/10.5281/zenodo.3580489
09:00
Combining MCMC, profit and Bayesian networks to identify processes that contribute to simulated algae growth along the Elbe River
Ulrich Callies | Helmholtz-Zentrum Geesthacht | Germany
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Ulrich Callies | Helmholtz-Zentrum Geesthacht | Germany
Christopher Albert | Max-Planck-Institut für Plasmaphysik | Germany
Udo von Toussaint | Max-Planck-Institut für Plasmaphysik | Germany
Process identification is one of the major objectives when simulating biological systems. In corresponding models, each process or cluster of processes will be represented in terms of equations with a set of parameters to be tuned. Usually these parameters are not fully controlled by existing data. In particular, different parameters often contribute to the simulated counterparts of existing data in a very similar way, implying that different processes could be the reason of observed behaviour.
This study deals with a model for diatom growth along the Elbe river. We study to which degree intermittent lack of silica explains time variability of chlorophyll a concentrations observed at Geesthacht Weir. Using MCMC, we first generate an ensemble of 10^6 successful combinations of 7-8 selected parameters. We then use a Bayesian network (BN) approach to represent their multivariate distribution, yielding a basic outcome of model calibration. The interactive BN responds to user-assigned fixed values for any subset of parameters. Thereby options for all other parameters get narrowed, subject to the constraint of a successful reproduction of existing data. In this way, responsibility for choosing a specific parameter set is delegated to the user in a transparent way. The degree to which parameters interact can be studied by entering or retracting evidence for any parameters. The optimum parameter ensemble is then used as a basis for sensitivity studies within the profit framework.
09:30
- NEW - Uncertainty quantification of Hamiltonian maps using polynomial chaos expansion
Katharina Rath | Ludwig-Maximilians-Universität München | Germany
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Katharina Rath | Ludwig-Maximilians-Universität München | Germany
Dynamics of many systems in classical physics are described in terms of Hamiltonian equations. If the Hamiltonian is time-independent it corresponds to the total energy of the system and is conserved. Often, initial conditions are only known within uncertainty bounds. The volume of the region of uncertainty in phase space is preserved over time due to symplecticity of the Hamiltonian flow. Here, this uncertainty is studied using polynomial chaos expansion (PCE) that
propagates uncertainty through a dynamical system. The approach is applied to examine the quality of non-symplectic and symplectic integrators and maps for such systems. A symplectic integration scheme preserves the structure of the Hamiltonian, which results in long-time stability and good approximate conservation of integrals of motion. As a reduced model for Hamiltonian systems it is convenient to construct an interpolated map without having to integrate the whole trajectory, but only study intersections with a given surface. In addition to conserved invariants that are a quality criterion for integration schemes and maps, the Sobol coefficients resulting from PCE allow to perform a sensitivity analysis with respect to the distribution of initial conditions. In comparison to Lyapunov exponents that are a measure for local stability, Sobol coefficients assess regional stability. The approach is tested to investigate the quality of different integration schemes on a 1D pendulum and the 2D Hénon-Heiles system.
10:00
Fully Bayesian multi-fidelity uncertainty quantification with Gaussian processes applied to computational fluid dynamics of the human aorta
Sascha Ranftl | Technische Universität Graz | Austria
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Sascha Ranftl | Technische Universität Graz | Austria
Thomas Müller | Technische Universität Graz | Austria
Günter Brenn | Technische Universität Graz | Austria
Wolfgang von der Linden | Technische Universität Graz | Austria
In 1998, Kennedy and O’Hagan devised a Bayesian model for uncertainty quantification that combines several levels of fidelity of a simulation, e.g. a coarse and a fine mesh used if finite element simulations. They assumed each level to be a Gaussian process, and used low fidelity simulations to reduce the number of costly high fidelity simulations, effectively speeding up the computation. Departing from there, we move away from the un-Bayesian practice of optimizing hyperparameters, and marginalize them rather instead. In this contribution, we integrate out the linear parameters analytically. Then, typically only a small number of non-linear hyperparameters in the kernel matrices is left, allowing for numeric integration, and a feasible fully Bayesian treatment thus. We further provide a presentation that makes generalization to an arbitrary number of code levels rather easy. We scrutinize the method with mock data, and apply it to simulations of pulsatile, non-Newtonian blood flow in the human aorta. Particularly, we investigate the wall shear stresses and hydrodynamic pressures on the arterial wall, which are hypothesized to lead to inflammation of the tissue and aortic dissection eventually.