This workshop will highlight emerging methods for computationally efficient Bayesian inference in inverse problems and related high-dimensional models with sparsity-promoting and edge-preserving priors. Such priors often induce heavy-tailed, multimodal posteriors that challenge standard sampling and optimization, motivating new scalable strategies including hierarchical and mixture prior constructions, diffusion- and transport-based sampling, and advanced approaches to uncertainty quantification. A central aim is to connect these methodological developments to practical implementation in modern probabilistic programming frameworks—particularly Turing/Julia and, more broadly, Stan—emphasizing algorithmic advances that enable efficient inference at scale. The program will consist of curated invited talks followed by discussion sessions designed to foster exchange between method developers and users, identify key opportunities for probabilistic programming practitioners, and catalyze cross-community collaboration spanning inverse problems, Bayesian computation, numerical analysis, data assimilation, and scientific machine learning.

The following talks will take place on Friday, August 21, 2026, between 9.00 and 12.00, in Geijersalen (room 6-1023) on Uppsala Universities English Park Campus.

• Lassi Roininen (9.00–9.30): Rough Feature Estimation for Bayesian Inversion

• Yiqiu Dong (9.35–10.05): Sparsity via hyperpriors under empirical Bayes framework

• Jan Glaubitz (10.20–10.50): Computationally efficient inference for sparsity-promoting hierarchical Bayesian models

• Daniel Sharp (10.55–11.25): A hierarchical Bayesian approach to ensemble filtering for hyperbolic conservation laws

• Mirjeta Pasha (canceled): From Prior Conditioning to Low-Rank Riemannian Geometry: Scalable MCMC for Bayesian Inverse Problems

• Lassi Roininen (9.00–9.30), Rough Feature Estimation for Bayesian Inversion:
  We consider the construction of non-Gaussian prior distributions for the stabilisation of inverse problems. The target is to build priors capable of modelling inhomogeneous structures, discontinuities, and high-frequency components. We consider two possible avenues: 1) building up hierarchical priors based on non-stationary Matérn-type priors with stochastic partial differential equation representations, and, 2) directly constructing non-Gaussian priors, with heavy-tailed noise driving the random fields. We first consider the mathematical and statistical properties of these random fields, and then show how to utilise them for practical inverse problems. We explicitly consider the inherent computational cost related to constructing these models, as we end up working with high-dimensional, heavy-tailed, and multimodal posteriors. Hence, obtaining the estimators and quantifying uncertainty are non-trivial tasks, and need to be considered when choosing optimisation and MCMC strategies.

• Yiqiu Dong (9.35–10.05), Sparsity via hyperpriors under empirical Bayes framework:
  The empirical Bayes framework (EBF) provides a powerful approach for sparse learning in inverse problems, yet the role of hyperprior selection is not fully understood. In this talk, I will discuss how hyperpriors influence hyperparameter estimation and, consequently, the sparsity and stability of EBF solutions. We establish theoretical links between the choice of hyperprior and the properties of the resulting estimators, including sparsity promotion and local optimality.

• Jan Glaubitz (10.20–10.50), Computationally efficient inference for sparsity-promoting hierarchical Bayesian models:
  Hierarchical sparsity-promoting priors play a central role in Bayesian inverse problems, enabling adaptive regularization and uncertainty quantification for problems with sparse or piecewise-smooth unknowns. In particular, hierarchical sparsity-promoting scale-mixtures-of-normals models combine conditionally Gaussian priors with heavy-tailed hyperpriors, yielding flexible hierarchical formulations—but also posterior distributions that are high-dimensional, strongly correlated, and often multimodal, making efficient MCMC sampling challenging.

  In this talk, I will present an approach to accelerate MCMC inference in hierarchical sparsity-promoting models via hierarchical prior normalization. The idea is to construct analytic transport maps that transform the full joint sparsity-promoting prior into a standard normal reference prior. I will further demonstrate how sampling the resulting prior-normalized posterior enables the use of efficient, structure-exploiting MCMC methods—such as elliptical slice sampling—and leads to improved mixing and robustness compared to conventional sampling from the original posterior, across a range of linear and nonlinear inverse problems

  This talk is based on joint work with Youssef Marzouk (MIT).

• Daniel Sharp (10.55–11.25), A hierarchical Bayesian approach to ensemble filtering for hyperbolic conservation laws:
  Data assimilation methods are currently used to tackle many practical, applied problems throughout computational science. At their heart, they gauge the state of a targeted dynamical system, and many workhorse methods (EnKF, ETKF, etc.) use ensembles to represent the uncertainty over this target state. We approach these methods with a particular focus toward hyperbolic conservation laws, giving piecewise-smooth states with possibly large discontinuities. While many works focus on representing the uncertainty over the dynamical system state more rigorously by using nonlinear transformations, they are often agnostic of the knowledge over the dynamical system they quantify uncertainty for. Contrasting with the (localized) ensemble Kalman filter, we demonstrate the ability to ensure that each sample drawn from our data assimilation method is a realistic realization of the underlying partial differential equations (PDEs), and further extend our method to take advantage of underlying structure in discontinuous Galerkin formulations of the PDEs.

The workshop will take place on Friday, August 21, 2026, between 9.00 and 12.00, in Geijersalen (room 6-1023) on Uppsala University’s English Park Campus.

• Jan Glaubitz (Assistant Professor, Department of Mathematics, Linköping University, Sweden)

• Yiqiu Dong (Associate Professor, Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark)

• Lassi Roininen (Professor, School of Engineering Sciences, LUT University, Finland)