
Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both bound...
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Active Uncertainty Calibration in Bayesian ODE Solvers
There is resurging interest, in statistics and machine learning, in solv...
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Probabilistic Numerics and Uncertainty in Computations
We deliver a call to arms for probabilistic numerical methods: algorithm...
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On Formal Power Series Solutions of Algebraic Ordinary Differential Equations
We propose a computational method to determine when a solution modulo a ...
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Bayesian Filtering for ODEs with Bounded Derivatives
Recently there has been increasing interest in probabilistic solvers for...
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Probabilistic Linear Multistep Methods
We present a derivation and theoretical investigation of the AdamsBashf...
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Probabilistic Linear Solvers for Machine Learning
Linear systems are the bedrock of virtually all numerical computation. M...
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A probabilistic model for the numerical solution of initial value problems
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 RungeKutta 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.
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