Philipp Hennig

Probabilistic numerics and uncertainty in computations.

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

Probabilistic Interpretation of Linear Solvers

This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems

Using an Infinite Von Mises-Fisher Mixture Model to Cluster Treatment Beam Directions in External Radiation Therapy

Bx = b

Point-spread functions for backscattered imaging in the scanning electron microscope

with positive definite

Analytical probabilistic modeling for radiation therapy treatment planning

B

Automatic LQR tuning based on Gaussian process global optimization

for

Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers

x

Gaussian Process-Based Predictive Control for Periodic Error Correction

. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of

Nonparametric dynamics estimation for time periodic systems

B

Learning tracking control with forward models

, which can be used to estimate errors. Recent probabilistic interpretations of the secant family of quasi-Newton optimization algorithms are extended. Combined with properties of the conjugate gradient algorithm, this leads to uncertainty-calibrated methods with very limited cost overhead over conjugate gradients, a self-contained novel interpretation of the quasi-Newton and conjugate gradient algorithms, and a foundation for new nonlinear optimization methods.