Solvers for ordinary differential equations (ODEs) are one of the basic algorithm classes of numerical mathematics. An ODE is an implicit statement about the relationship of a curve $x:\mathbb{R}\to\mathbb{R}^N$ to its derivative, in the form $x'(t) = f(x(t),t)$, where $x'$ is the derivative of curve, and $f$ is some function. To identify a unique solution of a particular ODE, it is typically also necessary to provide additional statements about the curve, such as its *initial value* $x(t_0)=x_0$. An ODE solver is a rule that maps function and initial value, $(f,x_0)$ to an estimate $x(t)$ for the solution curve. *Good* solvers have certain analytical guarantees about this estimate, such as the fact that its deviation from the true solution is of a high polynomial order in the number of evaluations of $f$ made by the algorithm.

One of the main strains of the group is the development of *probabilistic* versions of these solvers. That is, an algorithm that accepts a probability measure over $p(f,x_0)$ instead of a strict logical definition of the two, and returns a probability measure $p(x(t))$ over the solving curve. A *good* probabilistic solver should have properties analogous to those of a classic one, e.g. the expected value of $x$ under $p$ should be high-polynomially close to the truth. In addition, the width and shape of $p$ around the expected value should also reflect the true distance of the estimate to the exact solution in some sense.

In our work, we have focussed on algorithms of low computational cost, where $p$ is a Gaussian measure. We were able to construct solvers whose mean estimates exactly match the point estimates of classic solvers, and give rise to error measures which are analytically consistent with the true error.

5 results

**A probabilistic model for the numerical solution of initial value problems**
*Statistics and Computing*, Springer US, 2018 (article)

**Active Uncertainty Calibration in Bayesian ODE Solvers**
*Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI 2016)*, pages: 309-318, (Editors: Ihler, A. and Janzing, D.), AUAI Press, 2016 (conference)

**Probabilistic ODE Solvers with Runge-Kutta Means**
In *Advances in Neural Information Processing Systems 27*, pages: 739-747, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

**Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers**
In *Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014, Lecture Notes in Computer Science Vol. 8675*, pages: 265-272, (Editors: P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), Springer, Heidelberg, MICCAI, 2014 (inproceedings)

**Camera-specific Image Denoising**
Eberhard Karls Universität Tübingen, Germany, October 2013 (diplomathesis)