IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v173y2024ics0304414924000802.html
   My bibliography  Save this article

Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows

Author

Listed:
  • Crucinio, Francesca R.
  • De Bortoli, Valentin
  • Doucet, Arnaud
  • Johansen, Adam M.

Abstract

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we consider integral equations featuring kernels which may be expressed as scalar multiples of conservative (i.e. Markov) kernels and we adopt a 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.

Suggested Citation

  • Crucinio, Francesca R. & De Bortoli, Valentin & Doucet, Arnaud & Johansen, Adam M., 2024. "Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    • Handle: RePEc:eee:spapps:v:173:y:2024:i:c:s0304414924000802
    DOI: 10.1016/j.spa.2024.104374
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414924000802
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2024.104374?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Delaigle, A. & Gijbels, I., 2004. "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 249-267, March.
    2. Malrieu, F., 2001. "Logarithmic Sobolev inequalities for some nonlinear PDE's," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 109-132, September.
    3. P. G. Bissiri & C. C. Holmes & S. G. Walker, 2016. "A general framework for updating belief distributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1103-1130, November.
    4. Wang Miao & Zhi Geng & Eric J Tchetgen Tchetgen, 2018. "Identifying causal effects with proxy variables of an unmeasured confounder," Biometrika, Biometrika Trust, vol. 105(4), pages 987-993.
    5. Meleard, Sylvie & Roelly-Coppoletta, Sylvie, 1987. "A propagation of chaos result for a system of particles with moderate interaction," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 317-332.
    6. Francesca R. Crucinio & Arnaud Doucet & Adam M. Johansen, 2023. "A Particle Method for Solving Fredholm Equations of the First Kind," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(542), pages 937-947, April.
    7. Ma, Jun, 2011. "Indirect density estimation using the iterative Bayes algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1180-1195, March.
    8. Pui Hing Chau & Wei Ying Li & Paul S. F. Yip, 2020. "Construction of the Infection Curve of Local Cases of COVID-19 in Hong Kong using Back-Projection," IJERPH, MDPI, vol. 17(18), pages 1-8, September.
    9. Chae, Minwoo & Martin, Ryan & Walker, Stephen G., 2018. "Convergence of an iterative algorithm to the nonparametric MLE of a mixing distribution," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 142-146.
    10. Brosse, Nicolas & Durmus, Alain & Moulines, Éric & Sabanis, Sotirios, 2019. "The tamed unadjusted Langevin algorithm," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3638-3663.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kuosmanen, Timo & Johnson, Andrew, 2017. "Modeling joint production of multiple outputs in StoNED: Directional distance function approach," European Journal of Operational Research, Elsevier, vol. 262(2), pages 792-801.
    2. Fabio Canova & Christian Matthes, 2021. "Dealing with misspecification in structural macroeconometric models," Quantitative Economics, Econometric Society, vol. 12(2), pages 313-350, May.
    3. Ali Al-Sharadqah & Majid Mojirsheibani & William Pouliot, 2020. "On the performance of weighted bootstrapped kernel deconvolution density estimators," Statistical Papers, Springer, vol. 61(4), pages 1773-1798, August.
    4. Jourdain, B., 1998. "Convergence of moderately interacting particle systems to a diffusion-convection equation," Stochastic Processes and their Applications, Elsevier, vol. 73(2), pages 247-270, March.
    5. Smith, Simon C. & Timmermann, Allan & Zhu, Yinchu, 2019. "Variable selection in panel models with breaks," Journal of Econometrics, Elsevier, vol. 212(1), pages 323-344.
    6. Malmendier, Ulrike & Pouzo, Demian & Vanasco, Victoria, 2020. "Investor experiences and international capital flows," Journal of International Economics, Elsevier, vol. 124(C).
    7. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    8. Yang Zu, 2015. "A Note on the Asymptotic Normality of the Kernel Deconvolution Density Estimator with Logarithmic Chi-Square Noise," Econometrics, MDPI, vol. 3(3), pages 1-16, July.
    9. Ben-Moshe, Dan, 2018. "Identification Of Joint Distributions In Dependent Factor Models," Econometric Theory, Cambridge University Press, vol. 34(1), pages 134-165, February.
    10. Fei Cao & Sebastien Motsch, 2021. "Derivation of wealth distributions from biased exchange of money," Papers 2105.07341, arXiv.org.
    11. Gael M. Martin & David T. Frazier & Ruben Loaiza-Maya & Florian Huber & Gary Koop & John Maheu & Didier Nibbering & Anastasios Panagiotelis, 2023. "Bayesian Forecasting in the 21st Century: A Modern Review," Monash Econometrics and Business Statistics Working Papers 1/23, Monash University, Department of Econometrics and Business Statistics.
    12. Jun Cai & William C. Horrace & Christopher F. Parmeter, 2021. "Density deconvolution with Laplace errors and unknown variance," Journal of Productivity Analysis, Springer, vol. 56(2), pages 103-113, December.
    13. Parmeter, Christopher F., 2008. "The effect of measurement error on the estimated shape of the world distribution of income," Economics Letters, Elsevier, vol. 100(3), pages 373-376, September.
    14. Romain Fournier & Zoi Tsangalidou & David Reich & Pier Francesco Palamara, 2023. "Haplotype-based inference of recent effective population size in modern and ancient DNA samples," Nature Communications, Nature, vol. 14(1), pages 1-13, December.
    15. DongHyuk Lee & Soumendra N. Lahiri & Samiran Sinha, 2020. "A test of homogeneity of distributions when observations are subject to measurement errors," Biometrics, The International Biometric Society, vol. 76(3), pages 821-833, September.
    16. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    17. Ben Deaner, 2021. "Many Proxy Controls," Papers 2110.03973, arXiv.org.
    18. Zhang, Jeffrey & Li, Wei & Miao, Wang & Tchetgen Tchetgen, Eric, 2023. "Proximal causal inference without uniqueness assumptions," Statistics & Probability Letters, Elsevier, vol. 198(C).
    19. Toru Kitagawa & Hugo Lopez & Jeff Rowley, 2022. "Stochastic Treatment Choice with Empirical Welfare Updating," Papers 2211.01537, arXiv.org, revised Feb 2023.
    20. Zhichao Liu & Catherine Forbes & Heather Anderson, 2017. "Robust Bayesian exponentially tilted empirical likelihood method," Monash Econometrics and Business Statistics Working Papers 21/17, Monash University, Department of Econometrics and Business Statistics.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:173:y:2024:i:c:s0304414924000802. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.