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Solving Fredholm integral equations of the first kind via Wasserstein gradient flow
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Crucinio, F. R., De Bortoli, V., Doucet, A. and Johansen, Adam M. (2024) Solving Fredholm integral equations of the first kind via Wasserstein gradient flow. Stochastic Processes and their Applications, 173 . 104374. doi:10.1016/j.spa.2024.104374 ISSN 0304-4149.
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Official URL: https://doi.org/10.1016/j.spa.2024.104374
Abstract
Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
Journal or Publication Title: | Stochastic Processes and their Applications | ||||||||
Publisher: | Elsevier Science BV | ||||||||
ISSN: | 0304-4149 | ||||||||
Official Date: | July 2024 | ||||||||
Dates: |
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Volume: | 173 | ||||||||
Article Number: | 104374 | ||||||||
DOI: | 10.1016/j.spa.2024.104374 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 1 May 2024 | ||||||||
Date of first compliant Open Access: | 13 May 2024 | ||||||||
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