Linear-Time Probabilistic Solution of Boundary Value Problems
Authors: Nicholas Krämer, Philipp Hennig
NeurIPS 2021 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Now that all parts are in place, we evaluate the performance of the solver on a range of scenarios. All experiments use the CPU of a consumer-level laptop. An efficient probabilistic numerical method should provide both a good point estimate (through its posterior mean) and error estimate (through its posterior covariance). First, the approximation error should decrease rapidly with the number of grid points; we report root-mean-square errors the lower, the better. Second, the width of the posterior distribution should be representative of the numerical approximation error (which has, to some extent, been shown in Section 4 already); we use the χ2-statistic [35]. |
| Researcher Affiliation | Academia | Nicholas Krämer University of Tübingen Tübingen, Germany nicholas.kraemer@uni-tuebingen.de Philipp Hennig University of Tübingen and Max Planck Institute for Intelligent Systems Tübingen, Germany philipp.hennig@uni-tuebingen.de |
| Pseudocode | Yes | Algorithm 1: BVP Solver Input: BVP, mesh, order (ν), tolerances. Output: Probabilistic BVP Solution Initialise with bridge and ODE filter; while 1 interval with large error do Run IEKS; Update m0 and C0 (Equation (12)); Update σ ; Compute error between gridpoints; Refine mesh where necessary; end |
| Open Source Code | No | The paper mentions 'Our research code' but does not provide any explicit statement about open-sourcing the code or a link to a repository. |
| Open Datasets | Yes | As a first testbed, we use the seventh in a collection of test problems for BVP solvers by Mazzia [25] (which will feature heavily in the remainder of this work). A simulation of Bratu’s problem [36] for varying tolerances and orders ν suggests that the solver performs well in both metrics (Figure 6). Figure 7 depicts the results of simulating five BVPs (all from Mazzia [25]). |
| Dataset Splits | No | The paper does not provide specific dataset split information (exact percentages, sample counts, citations to predefined splits, or detailed splitting methodology) for training, validation, or testing. It uses predefined 'test problems' rather than a splittable dataset. |
| Hardware Specification | No | The paper states 'All experiments use the CPU of a consumer-level laptop', which is not specific enough to identify hardware details like exact CPU models, memory, or other specifications. |
| Software Dependencies | No | The paper does not provide specific ancillary software details with version numbers, such as programming language environments or libraries used for its implementation. |
| Experiment Setup | Yes | Algorithm 1: BVP Solver Input: BVP, mesh, order (ν), tolerances. Figure 6 caption: 'To show mesh refinement, the initial grid consisted of only three points; the probabilistic solver initialises with EKS and bridge, and uses the standard deviation as an error estimate.' Figure 7 caption: 'The tolerances are 10-1 (GRAY) and 10-6 (BLUE).' |