Probabilistic Linear Multistep Methods
Authors: Onur Teymur, Kostas Zygalakis, Ben Calderhead
NeurIPS 2016 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We provide a rigorous proof of the convergence of these new methods, as well as an empirical investigation (up to fifth order) demonstrating their convergence rates in practice. Furthermore, we have investigated their theoretical properties and provided a rigorous proof of their rates of convergence, Finally we have demonstrated the use and computational efficiency of probabilistic Adams-Bashforth methods by implementing the solvers up to fifth order and providing example solutions of a chaotic system, and well as empirically verifying the convergence rates in a Lotka-Voltera model. |
| Researcher Affiliation | Academia | Onur Teymur Department of Mathematics Imperial College London o@teymur.uk Konstantinos Zygalakis School of Mathematics University of Edinburgh k.zygalakis@ed.ac.uk Ben Calderhead Department of Mathematics Imperial College London b.calderhead@imperial.ac.uk |
| Pseudocode | No | The paper describes methods and processes verbally and mathematically, but does not include a structured pseudocode block or algorithm listing. |
| Open Source Code | No | The paper does not provide any statement or link indicating the availability of open-source code for the described methodology. |
| Open Datasets | No | The paper uses mathematical models (Chua circuit, Lotka-Volterra model) for simulation but does not mention or provide access to publicly available datasets in the traditional sense of empirical data for training/testing. |
| Dataset Splits | No | The paper does not specify any training, validation, or test dataset splits, as it concerns numerical methods for ODEs rather than data-driven machine learning models. |
| Hardware Specification | No | The paper mentions 'Wall-clock time' but does not provide any specific hardware details such as CPU/GPU models, memory, or cluster specifications used for running the experiments. |
| Software Dependencies | No | The paper mentions using 'a (packaged) adaptive Runge-Kutta-Fehlberg solver of 7th order with 8th order error control' but does not provide specific version numbers for this or any other software dependencies. |
| Experiment Setup | Yes | The Chua system is given by x0 = (y (1 + h1)x h3x3), y0 = x y + z, z0 = βy γz. We use parameter values = 1.4157, β = 0.02944201, γ = 0.322673579, h1 = 0.0197557699, h3 = 0.0609273571 and initial conditions x0 = 0, y0 = 0.003, z0 = 0.005. This particular choice is taken from Attractor CE96 in [18]. Using the probabilistic version of the Adams-Bashforth integrator with s > 1, it is possible to delay the point at which numerical path diverges from the truth, with effectively no additional evaluations of f required compared to the one-step method. This is demonstrated in Figure 1. Our approach is therefore able to combine the benefits of classical higher-order methods with the additional insight into solution uncertainty provided by a probabilistic method. |