# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## A blob method for inhomogeneous diffusion with applications to multi-agent control and samplingHTML articles powered by AMS MathViewer

by Katy Craig, Karthik Elamvazhuthi, Matt Haberland and Olga Turanova
Math. Comp. 92 (2023), 2575-2654
DOI: https://doi.org/10.1090/mcom/3841
Published electronically: June 14, 2023

## Abstract:

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.

The fact that our method succeeds in capturing the long time behavior of the weighted porous medium equation is significant from the perspective of related problems in quantization. Just as the Fokker-Planck equation provides a way to quantize a probability measure $\bar {\rho }$ by evolving an empirical measure $\rho ^N(t) = \frac {1}{N} \sum _{i=1}^N \delta _{X^i(t)}$ according to stochastic Langevin dynamics so that $\rho ^N(t)$ flows toward $\bar {\rho }$, our particle method provides a way to quantize $\bar {\rho }$ according to deterministic particle dynamics approximating the weighted porous medium equation. In this way, our method has natural applications to multi-agent coverage algorithms and sampling probability measures.

A specific case of our method corresponds to confined mean-field dynamics of training a two-layer neural network for a radial basis activation function. From this perspective, our convergence result shows that, in the overparametrized regime and as the variance of the radial basis functions goes to zero, the continuum limit is given by the weighted porous medium equation. This generalizes previous results, which considered the case of a uniform data distribution, to the more general inhomogeneous setting. As a consequence of our convergence result, we identify conditions on the target function and data distribution for which convexity of the energy landscape emerges in the continuum limit.

References
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Bibliographic Information
• Katy Craig
• Affiliation: Department of Mathematics, South Hall, Room 6607, University of California, Santa Barbara, California 93106-3080
• ORCID: 0000-0001-6085-4022
• Email: kcraig@math.ucsb.edu
• Karthik Elamvazhuthi
• Affiliation: Department of Mechanical Engineering, Bourns Hall A342, 900 University Ave., Riverside, California 92521
• ORCID: 0000-0003-0849-5178
• Email: kelamvazhuthi@engr.ucr.edu
• Matt Haberland
• Affiliation: BioResource and Agricultural Engineering Dept., Agricultural Engineering Bldg. (08), Room 101, San Luis Obispo, California 93407
• MR Author ID: 1436816
• ORCID: 0000-0003-4806-3601
• Email: mhaberla@calpoly.edu
• Olga Turanova
• Affiliation: Department of Mathematics, Michigan State University, 619 Red Cedar Road, C212 Wells Hall, East Lansing, MI 48824
• ORCID: 0000-0002-3129-6989
• Email: turanova@msu.edu
• Received by editor(s): March 9, 2022
• Received by editor(s) in revised form: February 7, 2023, and February 14, 2023
• Published electronically: June 14, 2023
• Additional Notes: The work of the first author was supported by NSF DMS grants 1811012 and 2145900, as well as a Hellman Faculty Fellowship. The first and fourth authors were supported by the Simons Center for Theory of Computing, at which part of this work was completed. The work of the second author was supported by AFOSR grants FA9550-18-1-0502 and FA9550-18-1-0502. The work of the fourth author was supported by NSF DMS grant 1907221 and NSF DMS grant 2204722.