# Dual-space posterior sampling for Bayesian inference in constrained inverse problems

arXiv: https://arxiv.org/abs/2603.00393 · HTML: https://mlsanity.com/paper/2603.00393/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 83 references that are in the map.
- First posted on arXiv 28 Feb 2026.

## Builds on (most cited first, 28 of 28)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems](https://mlsanity.com/paper/1410.1390/) (2014)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Split-and-augmented Gibbs sampler - Application to large-scale inference problems](https://mlsanity.com/paper/1804.05809/) (2018)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [Stein Variational Gradient Descent With Matrix-Valued Kernels](https://mlsanity.com/paper/1910.12794/) (2019)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Efficient MCMC Sampling with Dimension-Free Convergence Rate using ADMM-type Splitting](https://mlsanity.com/paper/1905.11937/) (2019)
- [Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions](https://mlsanity.com/paper/1901.08659/) (2019)
- [Total-variation regularization strategies in full-waveform inversion](https://mlsanity.com/paper/1608.06159/) (2016)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)
- [A penalty method for PDE-constrained optimization in inverse problems](https://mlsanity.com/paper/1504.02249/) (2015)
- [Uncertainty quantification in imaging and automatic horizon tracking: a Bayesian deep-prior based approach](https://mlsanity.com/paper/2004.00227/) (2020)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [Weak deep priors for seismic imaging](https://mlsanity.com/paper/2004.06835/) (2020)
- [WISER: multimodal variational inference for full-waveform inversion without dimensionality reduction](https://mlsanity.com/paper/2405.10327/) (2024)
- [Annealed Stein Variational Gradient Descent for Improved Uncertainty Estimation in Full-Waveform Inversion](https://mlsanity.com/paper/2410.13249/) (2024)
- [Improving full waveform inversion by wavefield reconstruction with the alternating direction method of multipliers](https://mlsanity.com/paper/1809.00891/) (2018)
- [Variational Prior Replacement in Bayesian Inference and Inversion](https://mlsanity.com/paper/2406.04072/) (2024)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)

## Built on by (most cited first, 2 of 2)

- [On the role of memorization in learned priors for geophysical inverse problems](https://mlsanity.com/paper/2603.19629/) (2026)
- [A lift for input-convex neural net training](https://mlsanity.com/paper/2605.24274/) (2026)

## Abstract (arXiv)

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion (FWI), which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning stems from noisy, band-limited, finite-aperture measurements and complex geological structures. A Bayesian framework describes the solution more comprehensively: instead of a single estimate, a posterior distribution of plausible solutions characterizes the non-uniqueness and can be sampled to quantify uncertainty. However, no clear procedure exists for translating hard physical constraints, such as the wave equation, into priors amenable to existing sampling techniques. We address this by sampling the posterior in the dual space via an augmented Lagrangian formulation, which converts hard constraints into penalties suited to sampling algorithms while enforcing them progressively through multiplier updates, so they are satisfied in the limit. We integrate the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD), a particle-based sampler: the constraint is relaxed at each iteration and the multiplier updates progressively enforce its satisfaction. This enables posterior sampling under hard constraints while inheriting the favorable conditioning of dual-space solvers, where partial constraint relaxation permits productive updates even when the current model is far from the true solution. We validate the method on a stylized Rosenbrock conditional inference problem and on frequency-domain FWI for a Gaussian anomaly model and the Marmousi II benchmark, demonstrating physically consistent uncertainty estimates and posterior contraction with increasing data coverage.