  Numerical methods to approximate implicitly defined functions and their values have a long illustrious history in the Mathematical Sciences. These contributions span from the Newton Method of root finding to quadrature methods for approximating integrals, up to more recent developments such as Multi-Grid Methods for the solution of Partial Differential Equations exhibiting multiple scales.

Independently from these advances in numerics, Functional Analysis in the form of Average Case analysis (as opposed to Worst Case analysis) of function approximations required the definition of a probability measure on the Banach space on which the sequence of function approximations are defined. This analysis then is viewed as a dual form represented by classical statistical estimators and logically leads to the proposal of viewing numerical approximation as one of statistical approximation.

In yet another independent stream of enquiry the formal axioms defining the framework of personal probability were laid down in the 1940s by Cox and introduced the inferential framework of Subjectivist Probability leading to the Bayesian school of statistical inference. This Bayesian approach to deal with uncertainty led Diaconis in the mid 80s to suggest that numerical methods could be considered as problems of Bayesian inference (note that randomness and stochasticity plays no role in the Bayesian formalism as it considers all that is unknown). Under the average case and minimax risks then Bayesian estimators of quadrature return well known approximations such as the Trapezoidal Rule.

Underlying all of these independent mathematical viewpoints is the common theme of a function space endowed with a probability measure and thus a natural extension to function approximation is to view such as the definition of a probability measure consistent with all sources of evidence, or constraints imposed by the functional definition. This leads to the emerging area of Probabilistic Numerics and this school will present a study of this overall area that is emerging in the Mathematical Sciences.

Lecture Series

• Philipp Hennig
• 1. Introduction to GP regression
• 2. PN approach to Ordinary Differential Equations (ODEs) I - Skilling, Hennig, Chrekbtii, Hennig
• 3. PN approach to ODEs II  Runge-Kutta Means solvers (including some beautiful algebra)
• 4. PN approach to Linear Algebra
• 5. PN approach to Nonlinear Optimization (including some new applications)
• 6. Bayesian Optimization (including practical use cases), and an outlook
• Chris Oates
• 1. Introduction to Probabilistic Numerics (PN)  Conceptual frameworks: Bayesian Inference, Average Case Analysis, Information Based Complexity
• 2. Bayesian Quadrature I - conjugate case, theorems, methods, algorithms, and examples
• 3. Bayesian Quadrature II - non-conjugate case, control functionals (Monte Carlo, Markov chain Monte Carlo, Quasi Monte Carlo), theorems, methods, algorithms and examples
• 4. Bayesian Quadrature III - Non-conjugate and undefined measure, Gaussian Process (GP) + the Cone of functions, theorems, methods, algorithms and examples
• 5. PN approach to Partial Differential Equations (PDEs) I - Elliptic, strong form, meshless, forward problem, theorems, inverse problem, theorems, applications and examples
• 6. PN approach to PDEs II - Time evolving - Parabolic, Hyperbolic, theorems, methods, forward and inverse problems