Probabilistic Iterative Methods for Linear Systems
Authors: Jon Cockayne, Ilse C.F. Ipsen, Chris J. Oates, Tim W. Reid
JMLR 2021 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We conclude with an empirical illustration that highlights the potential for probabilistic iterative methods to provide insight into solution uncertainty. [...] The aim of this section is to empirically assess our proposed probabilistic iterative methods. |
| Researcher Affiliation | Academia | Jon Cockayne EMAIL Alan Turing Institute 96 Euston Road London, NW1 2DB, UK; Ilse C.F. Ipsen EMAIL Department of Mathematics North Carolina State University Raleigh, NC 27695-8205, USA; Chris J. Oates EMAIL School of Mathematics and Statistics Newcastle University Newcastle-upon-Tyne, NE1 7RU, UK; Tim W. Reid EMAIL Department of Mathematics North Carolina State University Raleigh, NC 27695-8205, USA |
| Pseudocode | Yes | Example 3 (Conjugate gradient method). In CG, for a symmetric positive-definite matrix A the iteration is of the form xm = xm 1 + αmsm αm = s mrm s m Asm sm+1 = rm + βmsm βm = r mrm r m 1rm 1 where the initial direction s0 is taken to be the initial residual r0, and we recall that recall that rm = b Axm. |
| Open Source Code | Yes | The code to reproduce these results is available on Git Hub https://github.com/jcockayne/probabilistic_iterative_methods_code |
| Open Datasets | No | A data set of size d = 520 was generated, with (zi)i=1,...,d consisting of 60 evenly spaced points in [0, 0.1], 400 evenly spaced points in [0.2, 0.8] and 60 evenly spaced points in [0.9, 1], and yi = f(zi) where f(z) = 1z<0.5 sin(2πz)+1z 0.5 sin(4πz). The paper describes generating a dataset but does not provide access information (link, citation, or repository) for public availability. |
| Dataset Splits | No | The paper defines a generated dataset for an interpolation problem. There is no mention of splitting this data into training, validation, or test sets; the entire dataset is used to form the linear system to be solved. |
| Hardware Specification | No | The paper does not provide any specific details about the hardware (e.g., CPU, GPU models, memory) used to run the experiments. |
| Software Dependencies | No | The paper does not mention any specific software dependencies or their version numbers (e.g., Python, PyTorch, specific libraries). |
| Experiment Setup | Yes | The step size ω was set to either the optimal value in Fig. 1b, ω = 2/(λmin(A) + λmax(A)), that minimises the spectral radius of G, or a default value ω = 2/3 in Fig. 1a. [...] The parameter ℓ= 0.0012 was used, which produces a system for which a direct solver can be used. [...] For the initial distribution µ0 several candidates were considered. Firstly a default choice given by µ0 = N(0, Id). [...] Secondly the natural choice µ0 = N(0, A 1). [...] The third initial distribution we consider is applicable only in settings where a small number of ansatz solutions (i.e. guesses) are provided, perhaps obtained by expert knowledge of the system at hand. Let xi, i = 1, . . . , N, be these ansatz solutions; we use these to estimate the scaling parameter ν2 for an initial distribution µ0 = N(0, ν2Σ0) where Σ0 is fixed. [...] We used N = 5 ansatz solutions. [...] For the second order method, we opted to use the rich initial distribution for x1. [...] For each method, m = 10 iterations were performed. |