Probabilistic meshless PDE methods extended to compact manifolds
Original title: Probabilistic Meshless Methods on Manifolds
The paper proves consistency and error bounds, then matches existing accuracy on cortical thickness data.
The paper extends probabilistic meshless methods, which place a Gaussian process prior on a PDE solution, from Euclidean domains to compact manifolds and supplies convergence analysis. For known manifolds it discretizes the PDE operator with generalized finite differences on a point cloud. For implicit manifolds it proposes a free-radius stencil that controls coefficient growth and yields uniform Gram-matrix convergence. Experiments on the circle, sphere, and flat torus support the bounds. On cortical thickness data the method matches established discretizations while using fewer than a tenth of the observations and giving a credible interval at every vertex.
- Posterior error bounds separate collocation error from operator and prior approximation.
- Consistency rate on known manifolds is governed by the point-cloud fill distance.
- The cortical test conditions on fewer than one tenth of observations.
From the original article
1 min read at arxiv.orgProbabilistic meshless methods solve partial differential equations (PDEs) by placing a Gaussian process prior on the solution and conditioning it on the PDE at a finite set of collocation nodes. The posterior provides both an approximation of the solution and a Bayesian quantification of its error.
By Federico Pavesi · arxiv.org
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