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As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations.
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This content will become publicly available on January 1, 2026
PARAMETER ESTIMATION FOR THE REDUCED FRACTURE MODEL VIA A DIRECT FILTER METHOD
In this work, we present a numerical method that provides accurate real-time detection for the widthsof the fractures in a fractured porous medium based on observational data on porous medium fluidmass and velocity. To achieve this task, an inverse problem is formulated by first constructing aforward formulation based on the reduced fracture model of the diffusion equation. A parameter estimationproblem is then performed online by utilizing a direct filter method. Numerical experimentsare carried out to demonstrate the accuracy of our method in approximating the target parameters.
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- Award ID(s):
- 2142672
- PAR ID:
- 10596527
- Publisher / Repository:
- Begell House
- Date Published:
- Journal Name:
- Journal of Machine Learning for Modeling and Computing
- Volume:
- 6
- Issue:
- 1
- ISSN:
- 2689-3967
- Page Range / eLocation ID:
- 23 to 40
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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