Crank–Nicolson Langevin sampling
Algorithms AIMech:CrankNicolsonLangevin DRAFT
Aliases: CNL
A Metropolis-corrected Gaussian-reference sampler that combines a Crank–Nicolson treatment of the reference precision with an explicit gradient drift of the remaining target potential.
Identity Notes & Concept Scope
This is Cotter et al. CNL, with unpreconditioned dynamics and a precision-dependent linear solve. The potential gradient is of Phi relative to the Gaussian reference, not of the entire Lebesgue log density. It differs from CN by its gradient term and from pCNL by its preconditioning. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 167 prior native IDs/names/aliases. Full-checkout searches in both worktrees included ignored and hidden files; the 15 lines in each were shared Tierney source-title references or an adaptive-rejection discovery lead, not matching records. Existing Gibbs, Metropolis-within-Gibbs, random-walk and independence Metropolis mechanisms were inspected to resolve specialization boundaries.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For finite-dimensional pi(u) proportional to exp(−Phi(u)) Normal(u;0,C), with fixed symmetric positive-definite C, assume differentiable Phi, choose delta>0 and draw w from Normal(0,C). Solve (2C+delta I)v=(2C−delta I)u−2 delta C grad Phi(u)+sqrt(8 delta C)w. Let q(v|u) be this Gaussian proposal. In finite dimensions accept with min(1,pi(v)q(u|v)/(pi(u)q(v|u))) and otherwise retain u, evaluating the reverse proposal at v. This is the ordinary-density specialization of the source’s general Metropolis construction and its rho-based CNL acceptance. The gradient changes proposal reversibility, so the CN likelihood-only ratio cannot simply be reused.
⚠️ Curated Limitations & Negative Results
- The target must be proper relative to the Gaussian reference. The explicit density-ratio implementation here is finite-dimensional with symmetric positive-definite C; function-space extensions require the source’s operator, regularity and absolute-continuity assumptions.
- Target invariance does not imply independent draws, good exploration or finite-run accuracy. Mesh-robust behavior under stated assumptions is not a universal dimension-free performance guarantee.
- C and the step size are fixed in the stated kernel. Adaptive covariance or step-size policies need separate validity arguments.
- The reprint’s later Equation (6.2) reverses signs relative to Equation (6.1) and the explicit Section 4 proposals. This record uses Section 4 and does not claim a published erratum or use the inconsistent expression.
- The gradient and forward/reverse Gaussian densities must be evaluated consistently. In infinite dimensions one must use the source’s measure-based acceptance under its assumptions, not a nonexistent Lebesgue density.
- No identity with every MALA implementation or guaranteed advantage over CN or pCN is asserted.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | This algorithm constructs target-invariant Markov transitions under its stated assumptions. The exact AIO CLASS denotes a stochastic process/model and is exported beneath Network. This is a method-to-process related match, not equivalence, subclass membership, independence of retained draws or a claim that the algorithm is a trained network. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
S. L. Cotter, G. O. Roberts, A. M. Stuart, D. White
|
primary paper
full_text
|
ID: doi:10.1214/13-STS421
sha256:4be2044ef1d8…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:13:10Z
Updated: 2026-10-11T07:13:10Z
Source File:
data/curated/CrankNicolsonLangevin.yaml