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

Abstract

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy and incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion, which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning is inherent to noisy, band-limited, finite-aperture data and shadow zones. Casting the inverse problem into a Bayesian framework allows for a more comprehensive description of its solution, where instead of a single estimate, the posterior distribution characterizes 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 prior distributions amenable to existing sampling techniques. To address this, we perform posterior sampling in the dual space using an augmented Lagrangian formulation, which translates hard constraints into penalties amenable to sampling algorithms while ensuring their exact satisfaction. We achieve this by seamlessly integrating the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD) -- a particle-based sampler -- where the constraint is relaxed at each iteration and multiplier updates progressively enforce satisfaction. This enables constrained posterior sampling while inheriting the favorable conditioning properties of dual-space solvers, where partial constraint relaxation allows 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 full waveform inversion for a Gaussian anomaly model and the Marmousi~II benchmark, demonstrating well-calibrated uncertainty estimates and posterior contraction with increasing data coverage.

Figures (23)

• Figure 1: Overview of the conditional sampling problem. (a) Prior distribution $p(\bm{x})$ (Rosenbrock) with four test instances (pink triangles). (b) Corresponding noisy observations with observation noise $\sigma = 0.5$.
• Figure 2: Posterior distributions for all four test observations obtained by ADMM-SVGD. Each panel shows prior samples (gray), ADMM-SVGD posterior samples (pink), the noisy observation ($\triangledown$), and the true value ($\triangle$). The posteriors concentrate around the observations while remaining on the Rosenbrock manifold.
• Figure 3: Combined posterior distributions for all four test observations from two methods: ADMM-SVGD (pink) and standard SVGD (green). The noisy observation ($\triangledown$) and true value ($\triangle$) are shown in each panel.
• Figure 4: Quantile--quantile plots comparing the marginal distributions of standard SVGD against ADMM-SVGD for all four test instances. Each column corresponds to one instance; the top and bottom rows show the $x_1$ and $x_2$ marginals, respectively. Points on the diagonal indicate perfect agreement between the two methods.
• Figure 5: Convergence diagnostics for ADMM-SVGD across four test instances. (a) Constraint residual $|z - x_1^2|$ decreasing to near zero. (b) Average log-posterior. (c) Kernel bandwidth. (d)--(e) Posterior mean for $x_1$ and $x_2$. (f) Posterior standard deviation.