Convergence rates of Gaussian ODE filters

Kersting, Hans; Sullivan, Timothy J.; Hennig, Philipp

doi:10.1007/s11222-020-09972-4

Cited by 23 publications

(33 citation statements)

References 32 publications

(100 reference statements)

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“…In a different line of work, probabilistic ODE solvers are constructed using techniques from (nonlinear) Gaussian filtering and smoothing [12,13,[41][42][43][44][45]. These methods have the advantage that instead of repeatedly integrating the initial value problem, they only require a single forward integration and return local uncertainty estimates that are proportional to the local truncation error.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

Oesterle

Krämer

Hennig

et al. 2021

Preprint

Self Cite

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“…For some reviews of research in this area, see [9,17,27]. In probabilistic numerics, ODEs have been considered from many perspectives, including structure-or symmetry-preserving methods [1,40], Bayesian modelling of the unknown solution with Gaussian processes [5,10,33,36,38,40], data-based statistical estimation of discretisation error [24,35], and filtering [19,38]. The papers [10,20] cited earlier also belong to this context.…”

Section: Related Workmentioning

confidence: 99%

Randomised one-step time integration methods for deterministic operator differential equations

Lie

Stahn

Sullivan

2022

Calcolo

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Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065–1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.

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“…Such methods solve sequentially in time, conditioning the GP on acquisition data, i.e. solution and derivative evaluations, at competitive computational cost (compared to classical methods) (Kersting et al, 2020;Krämer et al, 2021). However, integrating IVPs with large time intervals or expensive vector fields using such filters is still a computationally intractable process.…”

Section: Introduction 1motivation and Backgroundmentioning

confidence: 99%

GParareal: A time-parallel ODE solver using Gaussian process emulation

Pentland¹,

Tamborrino²,

Sullivan³

et al. 2022

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Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, "predicting" the solution serially using a cheap (coarse) solver and "correcting" these values using an expensive (fine) solver that runs in parallel on a number of temporal subintervals. In this work, we propose a time-parallel algorithm (GParareal) that solves IVPs by modelling the correction term, i.e. the difference between fine and coarse solutions, using a Gaussian process emulator. This approach compares favourably with the classic parareal algorithm and we demonstrate, on a number of IVPs, that GParareal can converge in fewer iterations than parareal, leading to an increase in parallel speed-up. GParareal also manages to locate solutions to certain IVPs where parareal fails and has the additional advantage of being able to use archives of legacy solutions, e.g. solutions from prior runs of the IVP for different initial conditions, to further accelerate convergence of the method -something that existing time-parallel methods do not do.

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Convergence rates of Gaussian ODE filters

Cited by 23 publications

References 32 publications

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

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

Randomised one-step time integration methods for deterministic operator differential equations

GParareal: A time-parallel ODE solver using Gaussian process emulation

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