Konstantinos Zygalakis

Algorithms for posterior sampling in Bayesian imaging inverse problems

Bayesian (imaging) inverse problems provide a coherent mathematical and algorithmic framework that enables researchers to combine mathematical models with data. The ability to solve such inverse problems depends crucially on the efficient calculation of quantities relating to the posterior distribution, which itself requires the solution of high dimensional optimization and sampling problems. One of the main computational challenges in the case of imaging inverse problems is the non-smoothness of the prior which leads to “stiffness” for the corresponding stochastic differential equations that need to be discretised to perform sampling. We address this issue by using tailored stochastic numerical integrators [1,5] and provide a non-asymptotic analysis of them when the potential is either quadratic, or strongly log-concave based on the recently developed framework in [2]. If there is time we will also discuss an alternative approach for sampling based on augmentation of the state space [3], as well as extensions of our MCMC methods in order to be able to deal with problems where the image is corrupted by non-Gaussian noise [4].

[1] L. Vargas, M. Pereyra, K. C. Zygalakis, Accelerating proximal Markov Chain Monte Carlo by using explicit stabilised methods. SIAM J. Imaging Sci. 13(2), 905-935, (2020).
[2] J. M. Sanz Serna, K. C. Zygalakis, Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic diff erential equations. J. Mach. Learn. Res., 22, 1-37, (2021).
[3] L. Vargas, M. Pereyra, K. C. Zygalakis, The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution. SIAM J. Imaging Sci. 16 (4), 2040-2071, (2023).
[4] S. Melidonis, P. Dobson, Y. Altmann, M. Pereyra, K. C. Zygalakis, Efficient Bayesian computation for low-photon imaging problems. SIAM J. Imaging Sci. 16 (3), 1197-1236,(2023).
[5] T. Klatzer, P. Dobson, Y. Altmann , M. Pereyra, J. M. Sanz-Serna, and K. C. Zygalakis Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling. arXiv:2308.09460, (2023)