TY - JOUR
T1 - Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows
AU - Crucinio, Francesca R.
AU - De Bortoli, Valentin
AU - Doucet, Arnaud
AU - Johansen, Adam M.
N1 - Publisher Copyright:
© 2024 The Author(s)
PY - 2024/7
Y1 - 2024/7
N2 - 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.
AB - 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.
KW - Interacting particle systems
KW - Inverse problems
KW - McKean–Vlasov SDEs
UR - http://www.scopus.com/inward/record.url?scp=85192685608&partnerID=8YFLogxK
U2 - 10.1016/j.spa.2024.104374
DO - 10.1016/j.spa.2024.104374
M3 - Article
AN - SCOPUS:85192685608
SN - 0304-4149
VL - 173
JO - Stochastic Processes and Their Applications
JF - Stochastic Processes and Their Applications
M1 - 104374
ER -