# Applications of Sparse Grids

Sparse grids are not only a numerical tool whose theory can be studied, but they can also be applied to various problems in academia and industry. This page showcases some of the diverse applications of sparse grids.

## Estimation of Cosmological Redshifts

The universe and space itself are expanding. Astronomers see the impact of this expansion in the cosmological redshift: Due to the Doppler effect, light from distant stars and galaxies appears to be more red than it really is. The intensity of the redshift is directly correlated to the distance of the object to Earth. With photometric data (measurements involving optical filters), it is possible to estimate the redshift and thus the distance. In maps of the Sloan Digital Sky Survey (SDSS), each object is associated with seven parameters, mainly involving magnitudes in specific ranges of wavelength. These parameters can be used to estimate the corresponding redshift with sparse grid regression.

Literature: [1]

## Topology Optimization with B-Splines

The task of topology optimization is to determine the optimal shape and topology of some structural component such that some physical quantity is minimized or maximized. One example is a cantilever, which is fixed at one side and pulled down by a load on the other side. Using a homogenization and a two-scale approach, thousands of expensive evaluations of the elasticity tensor, a material property, have to be performed in each optimization iteration. By replacing the elasticity tensor with a sparse grid interpolant, the computational effort is drastically reduced. The choice of hierarchical B-splines as sparse grid basis enables the use of gradient-based optimization algorithms, without having to compute derivatives of the original elasticity tensor.

Literature: [2]

## Uncertainty Quantification for Carbon Capture and Storage

The problem of how to reduce global CO2 emissions is very topical. However, it will not be possible to achieve zero emissions, so we will have to deal with some remaining level. Besides planting trees, one way is carbon capture and storage (CCS), where CO2 is captured from the air and then pumped underground to store it in large subterranean reservoirs. Finding suitable storage sights is key as unsuitable locations might lead to water pollution or artificial earthquakes. Unfortunately, the parameters of the underground soil are not known exactly. Here, sparse grids can be applied to quantify this uncertainty (expectation and variances) to estimate the risk of leaks and to assess suitable locations.

Literature: [3]

[1] D. Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Verlag Dr. Hut, 2010.

[2] J. Valentin, “B-Splines for Sparse Grids: Algorithms and Application to Higher-Dimensional Optimization,” PhD thesis, University of Stuttgart, 2019.

[3] F. Franzelin, “Data-Driven Uncertainty Quantification for Large-Scale Simulations,” PhD thesis, University of Stuttgart, 2018.