Uncertainty Quantification for the Homogeneous Landau–Fokker–Planck Equation via Deterministic Particle Galerkin Methods
Abstract.
We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a stochastic Galerkin (sG) representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy dissipation. We provide a regularity result for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.
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History
Submitted: 13 December 2023
Accepted: 27 November 2024
Published online: 2 April 2025
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Funding Information
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM/GNAMPA): CUP E53C22001930001
Funding: This paper has been written within the activities of GNFM group of INdAM (National Institute of High Mathematics). The first author, the second author, and the third authors were supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant 883363) and by the EPSRC grant EP/T022132/1 “Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging.” The third author acknowledges partial support of MIUR-PRIN2020 project 2020JLWP23 and INdAM-GNFM project CUP E53C22001930001. The fourth author acknowledges partial support of MIUR-PRIN2020 project 2020JLWP23.
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