Data-Driven Control

**Project Heads**

Claudia Schillings

**Project Members**

Matei Hanu

**Project Duration**

01.03.2022 − 14.09.2025

**Located at**

FU Berlin

underlying process as well as on the minimization of uncertainties through optimal experimental design.

**External Website**

**Related Publications
**

*Matei Hanu, Jonas Latz, and Claudia Schillings. Subsampling in ensemble*

*kalman inversion, 2023, Inverse Problems 39 094002, 10.1088/1361-6420/ace64b*

**Projects
**

**Subsampling in ensemble Kalman inversion**

We consider the Ensemble Kalman Inversion which has been recently introduced as an efficient, gradient-free optimisation method to estimate unknown parameters in an inverse setting. In the case of large data sets, the Ensemble Kalman Inversion becomes computationally infeasible as the data misfit needs to be evaluated for each particle in each iteration. Here, randomised algorithms like stochastic gradient descent have been demonstrated to successfully overcome this issue by using only a random subset of the data in each iteration, so-called subsampling techniques. Based on a recent analysis of a continuous-time representation of stochastic gradient methods, we propose, analyse, and apply subsampling-techniques within Ensemble Kalman Inversion. Indeed, we propose two different subsampling techniques: either every particle observes the same data subset (single subsampling) or every particle observes a different data subset (batch subsampling).

Our first method is single-subsampling, where each particle obtains the same new data set when data is changed.

The second method is batch-subsampling, where each particle can obtain different data sets at each data change.

In case of a linear forward operator both methods converge to the same solution as the EKI. The red line represents the EKI, blue is single-subsampling and green is batch-subsampling. In the left image we illustrate the error to the true solution in the parameter space. The right image illustrates the error in the observation space.

Finally, we also considered single-subsampling for a non-linear forward operator. We can see that our method (right image) computes the same solution as MATLABs fmincon optimizer (left image) as well as the EKI (middle image).

Note that our method computes the same results as the EKI, but at lower computational cost. Hence, this method is a suitable alternative in case of high-dimensional data.