Nonintrusive reduced order modelling using locally adaptive sparse grids and local basis interpolation

Abstract

Sensitivity analysis and uncertainty quantification of nuclear reactors requires many expensive high-fidelity simulations. To approximate the dynamics of such a time and parameter dependent system efficiently and effectively, reduced order modelling (ROM) is used. In previous research, a ROM was constructed which used a combination of proper orthogonal decomposition (POD) and a locally adaptive sampling strategy based on sparse grids. To improve the representation of the physics in local parts of the parameter domain, in this thesis, the previous ROM is altered to use multiple local bases instead of one fixed global basis covering the entire parameter domain.

The spatially dependent local bases and local time dependent coefficients are interpolated separately in the parameter domain using the method of interpolation on a tangent space to the Grassmann manifold (ITSGM). The separate interpolators of the local bases and local coefficients are coupled in a space-time coupled approach. The algorithm based on sparse grids is modified so that it can adaptively draw new tangent planes in the parameter domain in a hierarchical manner. This results in the domain being split up into into smaller overlapping subdomains, each having their own tangent plane, number of basis vectors, and interpolators that use radial basis functions (RBF). The algorithm was tested on the Burgers Equation and the Molenkamp test which have an analytical solution. Then, the algorithm was tested on a numerically solved 2D neutron diffusion problem. First the parametric dependence of the modes and coefficients was analyzed, after which the performance of the algorithm was evaluated on these models via different experiments.

The results of the experiments on the 1D Burgers equation and the 2D neutron diffusion problem showed that the algorithm can adaptively and hierarchically draw new tangent planes and can accurately interpolate to new unknown solutions in the subdomains in both a 1D and 2D parameter setting. Higher order non-linear modes and coefficients are harder to interpolate than lower order modes and coefficients. The performance of the interpolator also depends on a combination of the chosen RBF and the size of the subdomains. The results from the Molenkamp test showed that in the smooth setting the new algorithm achieved a higher error with a higher number of evaluations, while in the steep setting the new algorithm performed on par with the previous algorithm, only if the interpolation accuracy threshold is set lower. The results from the neutron
diffusion problem indicate that the first principal angle can act as an indicator of which parts of the parameter domain contain more relatable physics than other parts of the domain.

The dependence of RBF interpolator on the subdomain size is caused by how the RBF values scale for further distanced points from the point of interpolation interest. Also more higher order modes than necessary can be included in the local basis interpolation method without worsening the interpolation accuracy of lower order modes, given that no numerical noise is present in the local bases or coefficients. Interpolating the time-dependent behaviour can be easier in the
space-time coupled approach than in the approach based on the global basis. However, a lower interpolation accuracy threshold should be chosen as full time evolutions are being interpolated instead of single state vectors. The current interpolation method scales worse than the previous algorithm, as the corner points of the parameter domain will always have to be sampled when using the RBF interpolator compared to local linear basis functions. Additionally, far more data is needed to represent this ROM compared to the ROM based on the global basis.

This research presented a method that generalizes reduced order modelling of time and parameter dependent problems on sparse grids, by using multiple local bases instead of a fixed global basis to represent the underlying physics of a model. The novel local basis interpolation scheme was competitive with the global basis approach on test problems, showing that manifold methods
such as ITSGM have great potential to be utilized in reduced order models. However, more research on matrix interpolation methods is needed to improve the overall performance, scalability and efficiency of the algorithm.