A modern retrospective on probabilistic numerics

Oates, Chris J.; Sullivan, Timothy J.

doi:10.1007/s11222-019-09902-z

Cited by 46 publications

(37 citation statements)

References 92 publications

(106 reference statements)

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“…Such algorithms can output probability measures, instead of point estimates, over the final quantity of inter-est. This approach, now called probabilistic numerics (PN) (Hennig et al 2015;Oates and Sullivan 2019), has in recent years been spelled out for a wide range of numerical tasks, including linear algebra, optimization, integration, and differential equations, thereby working towards the longterm goal of a coherent framework to propagate uncertainty through chained computations, as desirable, e.g., in statistical machine learning.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates of Gaussian ODE filters

2020

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A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss–Markov process $${\varvec{X}}$$ X , which is then iteratively conditioned on information about $${\dot{x}}$$ x ˙ . This article establishes worst-case local convergence rates of order $$q+1$$ q + 1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of $$q=1$$ q = 1 and an integrated Brownian motion prior, and analyses how inaccurate information on $${\dot{x}}$$ x ˙ coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to $$q \in \{2,3,\ldots \}$$ q ∈ { 2 , 3 , … } .

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Section: Introductionmentioning

confidence: 99%

Convergence rates of Gaussian ODE filters

2020

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“…Addressing this issue, probabilistic numerics provides algorithms to quantify this kind of numerical uncertainty [8,30,31]. For sufficiently regular ODE solutions, randomized solvers have the same mean-square convergence rates as their deterministic counterparts, provided the perturbation variance is at most of the same order as the local error [14,32].…”

Section: Probabilistic Solvers For Quantifying Uncertainty In Neural Simulationsmentioning

confidence: 99%

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models

Oesterle

Krämer

Hennig

et al. 2021

Preprint

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Understanding neural computation on the mechanistic level requires biophysically realistic neuron models. To analyze such models one typically has to solve systems of coupled ordinary differential equations (ODEs), which describe the dynamics of the underlying neural system. These ODEs are solved numerically with deterministic ODE solvers that yield single solutions with either no or only aglobal scalar bound on precision. To overcome this problem, we propose to use recently developed probabilistic solvers instead, which are able to reveal and quantify numerical uncertainties, for example as posterior sample paths. Importantly, these solvers neither require detailed insights into the kinetics of themodels nor are they difficult to implement. Using these probabilistic solvers, we show that numerical uncertainty strongly affects the outcome of typical neuroscience simulations, in particular due to the non-linearity associated with the generation of action potentials. We quantify this uncertainty in individual single Izhikevich neurons with different dynamics, a large population of coupled Izhikevich neurons, single Hodgkin-Huxley neuron and a small network of Hodgkin-Huxley-like neurons. For commonly used ODE solvers, we find that the numerical uncertainty in these models can be substantial, possibly jittering spikes by milliseconds or even adding or removing individual spikes from the simulation altogether.

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“…[1] modelled the local error as a stochastic term added to the differential equation and calibrated this as a zero mean, iid Gaussian sequence, with a covariance scaled by h 2p+1 . In this section, this idea is developed and applied to the more complicated local error model in (4).…”

Section: Continuous Time Extension Of Stochastic Local Error Modelmentioning

confidence: 99%

“…The ideas in [1] are developed in [3] to establish stronger convergence results for probabilistic integrators, and as their results apply to general additive noise models, they apply to the the state dependent noise with bias model used to model the local error in this paper. A general survey of probabilistic numerics is provided by [4].…”

Section: Introductionmentioning

confidence: 99%

Representation of simulation errors in single step methods using state dependent noise

Boje

2021

MATEC Web Conf.

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The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.

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A modern retrospective on probabilistic numerics

Cited by 46 publications

References 92 publications

Convergence rates of Gaussian ODE filters

Convergence rates of Gaussian ODE filters

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models

Representation of simulation errors in single step methods using state dependent noise

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