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*

*On the ensemble Kalman inversion under inequality constraints, 2023, preprint,*

*https://arxiv.org/abs/2312.13804*

*Ensemble Kalman Inversion for Image Guided Guide Wire Navigation in Vascular Systems, 2023, preprint,*

*https://arxiv.org/abs/2312.06460*

**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.

**Ensemble Kalman Inversion for Image Guided Guide Wire Navigation in Vascular Systems**

For our experiments we have images like the left one, where a vertical force is applied to one end of the rod. For the mathematical analysis we apply a distance transformation to it, i.e. after cropping everything out of the image that isn’t the wire and turning the image into a black and white image, we calculate for each pixel a relative distance to the wire (right image). By doing this we define a metric that allows us to measure how ‘close’ images are to another and are able to minimize the difference.

We shorty describe the algorithm,

- Given an initial density and energy dissipation we apply a constant force at one end of the rod while we fix the other. This results in the left image above.
- We crop everything beside the rod out of the picture, turn it into a black and white picture and apply distance transformation to it.
- We calculate the distance of the images to one another.
- We optimize for the density as well as the energy dissipation, w.r.t. to the image distances.

For the optimisation we consider the pixel values of the images, since the amount of pixels is extremly large, standard optimizers as well as the EKI might have computational problems. Due to this we consider the above introduced subsampling approach for optimization.

The left image illustrates the original rod and the images that we obtain through the optimized values of density and energy dissipation. The right hand image shows the computed rods without the original image. We can see that both computed solutions are similar to another as well as similar to the original image. The residuals converge at the same rate, they only differentiate themselves in constant. This can be seen in the image below.

**On the ensemble Kalman inversion under inequality constraints**

Based on a log-barrier approach, we suggest in this project adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. We showcase our results through two examples involving partial differential equations (PDEs).

We illustrate here the solutions obtained through our algorithm for a 2D-Darcy flow. The upper left image illustrates the ground truth, The lower left shows the solution computed by the EKI in its general form, our method, the EKI with boundary constraints, is illustrated in the bottom right and the upper right image shows the reference solution, which is computed through an optimizer. The grey planes illustrate bounds on the diffusion coefficient, we can see that our method satisfies the bounds and is identical to the optimizer solution, whereas the EKI in its general form does not lie in the bounds.

Barrier methods penalize parameter values that move to close to the bounds, they force the particles to stay within the constraints by deactivating directions pointing towards areas that do not fulfill the constraints. This penalty is applied through a parameter, that in theory has to go to infinity for the algorithm to converge to the optimal point (Karush-Kuhn-Tucker point). However, due to the increasing value of the parameter the algorithm becomes computationally slower. To solve this problem we begin the algorithm with a low penalty parameter and increase the parameter over time.

This image illustrates the errors of the parameters (left) as well as the functionals (right). We can see that the algorithm with the largest penalty parameter converges the fastest in both images, however the method with the increasing penalty parameter, reaches the same error level after some time. However, the computational time needed to achieve this result is much lower.