A probabilistic model for the numerical solution of initial value problems

10/17/2016
by   Michael Schober, et al.
0

Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a class of algorithms that formulate the solution to an IVP as inference on a latent path that is a draw from a Gaussian process probability measure (or equivalently, the solution of a linear stochastic differential equation). We then show that certain members of this class are connected precisely to generalized linear methods for ODEs, a number of Runge--Kutta methods, and Nordsieck methods. This probabilistic formulation of classic methods is valuable in two ways: analytically, it highlights implicit prior assumptions favoring certain approximate solutions to the IVP over others, and gives a precise meaning to the old observation that these methods act like filters. Practically, it endows the classic solvers with `docking points' for notions of uncertainty and prior information about the initial value, the value of the ODE itself, and the solution of the problem.

READ FULL TEXT
research
06/03/2013

Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

We study a probabilistic numerical method for the solution of both bound...
research
10/22/2021

Probabilistic ODE Solutions in Millions of Dimensions

Probabilistic solvers for ordinary differential equations (ODEs) have em...
research
12/15/2020

Calibrated Adaptive Probabilistic ODE Solvers

Probabilistic solvers for ordinary differential equations (ODEs) assign ...
research
06/03/2015

Probabilistic Numerics and Uncertainty in Computations

We deliver a call to arms for probabilistic numerical methods: algorithm...
research
05/11/2016

Active Uncertainty Calibration in Bayesian ODE Solvers

There is resurging interest, in statistics and machine learning, in solv...
research
12/03/2021

ProbNum: Probabilistic Numerics in Python

Probabilistic numerical methods (PNMs) solve numerical problems via prob...
research
10/19/2020

Probabilistic Linear Solvers for Machine Learning

Linear systems are the bedrock of virtually all numerical computation. M...

Please sign up or login with your details

Forgot password? Click here to reset