Julian Hofstadler

Post Doctoral Researcher · University of Bath

Focus - Monte Carlo methods 2025 – Present

Mathematician working on the theory of Monte Carlo methods — convergence analysis of numerical algorithms involving randomness and quantitative error bounds for MCMC and SMC samplers. Within MaThRad I work with Alex Cox at the University of Bath on Markov chain based sampling algorithms for radiation transport, from adaptive MCMC theory to rigorous guarantees for the Monte Carlo codes used in reactor physics. PhD from the University of Passau on adaptive Markov chain Monte Carlo; MSc and BSc in Mathematics from Johannes Kepler University Linz.

Body of work

10 items

Year

Publication 2026

Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo

Stochastic Processes and their Applications · with K. Łatuszyński, G. O. Roberts & D. Rudolf

Establishes almost-sure convergence rates for adaptive increasingly rare MCMC — theoretical guarantees for a class of adaptive sampling algorithms used wherever MCMC is applied. Of relevance to any user of adaptive MCMC: the results certify that the sampler converges, and how fast.

› Preprint at arXiv:2402.12122.
› Follow-up preprint “Error bounds for simultaneous Wasserstein contractive adaptive increasingly rare MCMC” (with D. Rudolf, Jun 2026) extends the theory to error bounds — arXiv:2606.30018.
› Presented at the Austrian Stochastics Days 2025 (Sep 2025).

Linked · DOI Adaptive MCMC Convergence rates

Preprint 2026 · Feb

Solving Poisson's equation for Wasserstein contractive Markov chains

arXiv · sole author

A general framework, built on solutions of Poisson's equation, for proving convergence of Markov chains — with applications to sampling algorithms based on Markov chains such as MCMC and SMC. Improves the theoretical understanding of the samplers used across computational statistics and transport simulation.

Linked · arXiv Markov chains Poisson equation

Algorithm 2026 · in progress

SMC sampler for Bayesian inverse problems with intractable likelihoods

International collaboration · with Alex Cox (Bath) & collaborators in Heidelberg

An algorithm producing samples for Bayesian inverse problems whose likelihoods are intractable — applicable to general inverse problems in particle transport, medical imaging and beyond. Code and a paper draft exist; numerical experiments run on the Heidelberg group's cluster, with submission to a journal as the next step.

› Co-designed through reciprocal research visits — Josef Martinek visiting Bath and Julian visiting Heidelberg.

Asserted SMC Bayesian inverse problems In progress

Theorem 2026 · in progress

Theoretical guarantees for the Monte Carlo algorithms inside SCONE

MaThRad collaboration · University of Bath

Analysis of the explicit and implicit capture algorithms used within the SCONE neutron-transport code — rigorously proving that the sampling schemes nuclear engineers rely on actually work. A paper draft is in development.

Asserted Neutron transport SCONE In progress

Invited Seminar 2025 · Dec

Optimal convergence rates of MCMC integration for functions with unbounded second moment

Scientific Computing Seminar · Heidelberg University

Invited seminar on Markov chain based Monte Carlo algorithms, including an accessible introduction to this class of methods for a broad applied-mathematics audience alongside recent convergence results.

Asserted MCMC Invited talk

Workshop 2025 · May

MaThRad × Sandia National Laboratories workshop

Sandia Labs · neutron transport

Participated in the joint MaThRad–Sandia workshop on open questions in neutron transportation, exploring candidate joint projects between the programme and Sandia's simulation teams.

Asserted International Neutron transport

Working group 2025 → present

Member — MaThRad working groups on k-effective analysis and SMC for inverse problems

MaThRad internal · MaThRad × Heidelberg

Member of two MaThRad working groups, both feeding technical reports into the programme: the k-effective analysis group, analysing the algorithms used within criticality calculations, and the SMC-for-Bayesian-inverse-problems group with Heidelberg, applying the programme's expertise to imaging problems beyond MaThRad.

› Also attended a workshop on Function Approximation and Probability in Będlewo (Sep 2025) to deepen the function-approximation side of the Monte Carlo toolkit.

Asserted Working group Criticality

Co-design pathway 2025 → present

Adaptive-MCMC theory reaching its user community

University of Passau · University of Warwick · MCMC users

Adaptive MCMC is widely used but its theory lagged behind practice. The convergence-rate results co-designed with Passau and Warwick close that gap, and dissemination to the MCMC user community is under way through conference talks and the published paper.

Impact trajectory

Theory built

Published

Disseminated

Adoption

› Presented to MCMC users at the Austrian Stochastics Days 2025.
› Exploratory sibling pathway: the Poisson-equation convergence framework targets the same community — draft available, not yet presented.

Linked · DOI Co-design MCMC

Assurance pathway 2025 → present

Convergence proofs for SCONE's Monte Carlo algorithms

SCONE users · nuclear engineering

SCONE's capture algorithms are widely used but lack published convergence proofs. This analysis sets out to provide them — work that would matter to the nuclear engineers who rely on the code. It is currently a draft, shared so far within the programme at a MaThRad postdoc retreat.

Impact trajectory

Analysis drafted

Proofs complete

Shared with users

Integration

› Presented at the MaThRad postdoc retreat (Nov 2025).

Asserted SCONE Nuclear

Engagement pathway 2025 · Oct

Exploring fusion applications with UKAEA

UKAEA · Culham

Presented current projects at the Culham workshop to explore how the programme's Monte Carlo theory could serve fusion research, opening a channel to UKAEA for future collaboration on fusion-relevant transport problems.

Impact trajectory

Contact made

Engaged

Joint project

Adoption

Asserted Fusion UKAEA

Output over time

2025 — 2026

2025

2026

Research

Joined MaThRad · Monte Carlo theory thread

Adaptive-MCMC paper revised for SPA

Austrian Stochastics Days talk (Sep)

Heidelberg invited seminar (Dec)

Adaptive-MCMC paper published in SPA

Poisson-equation preprint · arXiv (Feb)

Error-bounds follow-up preprint · arXiv (Jun)

SMC for inverse problems · paper target

SCONE guarantees · paper target

Collab

Sandia × MaThRad workshop (May)

Będlewo function-approximation workshop (Sep)

Joined k-effective & SMC working groups

Heidelberg research visits

Impact

Culham workshop · UKAEA engagement (Oct)

SCONE analysis at postdoc retreat (Nov)