Probabilistic Exponential Integrators

Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator -- with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics

Calibrated Adaptive Probabilistic ODE Solvers

Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.Comment: 17 pages, 10 figures

Transforming scholarship in the archives through handwritten text recognition:Transkribus as a case study

Purpose: An overview of the current use of handwritten text recognition (HTR) on archival manuscript material, as provided by the EU H2020 funded Transkribus platform. It explains HTR, demonstrates Transkribus, gives examples of use cases, highlights the affect HTR may have on scholarship, and evidences this turning point of the advanced use of digitised heritage content. The paper aims to discuss these issues. - Design/methodology/approach: This paper adopts a case study approach, using the development and delivery of the one openly available HTR platform for manuscript material. - Findings: Transkribus has demonstrated that HTR is now a useable technology that can be employed in conjunction with mass digitisation to generate accurate transcripts of archival material. Use cases are demonstrated, and a cooperative model is suggested as a way to ensure sustainability and scaling of the platform. However, funding and resourcing issues are identified. - Research limitations/implications: The paper presents results from projects: further user studies could be undertaken involving interviews, surveys, etc. - Practical implications: Only HTR provided via Transkribus is covered: however, this is the only publicly available platform for HTR on individual collections of historical documents at time of writing and it represents the current state-of-the-art in this field. - Social implications: The increased access to information contained within historical texts has the potential to be transformational for both institutions and individuals. - Originality/value: This is the first published overview of how HTR is used by a wide archival studies community, reporting and showcasing current application of handwriting technology in the cultural heritage sector

Source Code for my Master's Thesis

Master's Thesis Mathematics - Evolutionary Game Theor

Pick-and-mix information operators for probabilistic ODE solvers

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.Comment: 13 pages, 7 figure

Fenrir: Physics-Enhanced Regression for Initial Value Problems

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches

Probabilistic ODE Solutions in Millions of Dimensions

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions