Advisor(s):
Philipp Hennig
Together with my supervisor Philipp Hennig, I am working on probabilistic numerical methods to solve large linear equation systems.
We believe that numerical algorithms are actually Bayesian inference algorithms:
These algorithms gather data and propose an estimate based on these observations.
Different algorithms usually produce different estimates even if given the same data (e.g. FOM and GMRES) as they usually have different underlying assumptions.
One of our goals is to make these assumptions explicit using the language of probability in form of a prior belief over the solution.
In addition to residual or worst-case error, such a view provides an uncertainty estimate regarding the quality of the approximation.
This uncertainty allows to reason over the average error and it can be propagated to the application that called for the solution.
Furthermore, a Bayesian view allows (at least in theory) to incorporate additional prior knowledge leading to more specifically tailored algorithms for specific applications.
A particular application we are interested in are linear equation systems in which the coefficient matrix stems from a distribution.
For example in Gaussian process regression it is common to assume that the underlying dataset is independently and identically distributed.
This implies that the rows of the kernel matrix are correlated, i.e. a structure that we aim to use to solve such linear equation system faster.
Bayesian Optimization is an increasingly popular approach to industrial and scientific prototyping problems. The basic premise in this setting is that one is looking for a location $x$ in some domain where a fitness function $f(x)$ is (globally) minimized. The additional, sometimes implicit, assumption is t...
Philipp Hennig Simon Bartels Alonso Marco Valle Sebastian Trimpe
Linear algebra methods form the basis for the majority of numerical computations. Because of this foundational, ``inner-loop'' role, they have to satisfy strict requirements on computational efficiency and numerical robustness.
Our work has added to a growing understanding that many widely used&...
Bartels, S., Cockayne, J., Ipsen, I., Hennig, P.
Statistics and Computing, 29(6):1249-1263, 2019 (article)pn
Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F.
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS 2017), 54, pages: 528-536, Proceedings of Machine Learning Research, (Editors: Sign, Aarti and Zhu, Jerry), PMLR, April 2017 (conference)pn
Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F.
Electronic Journal of Statistics, 11, 2017 (article)pn
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