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.

翻译：偏微分方程约束的反问题常因数据含噪、不完整或固有的非唯一性而呈现病态性。全波形反演即为一典型实例，该方法通过拟合波动方程约束的地震观测数据来估计地下介质属性，其病态性源于数据本身的噪声干扰、频带限制、有限孔径观测以及阴影区效应。将反问题置于贝叶斯框架中，可通过后验分布更全面地描述解的特征——该分布不仅刻画了非唯一性，还能通过采样实现不确定性量化。然而，如何将波动方程等硬性物理约束转化为适用于现有采样技术的先验分布，目前尚缺乏明确方法。为此，我们采用增广拉格朗日格式在双空间执行后验采样，将硬约束转化为适用于采样算法的惩罚项，同时确保约束被精确满足。该方法通过交替方向乘子法（ADMM）与基于粒子的采样器——斯坦变分梯度下降（SVGD）的无缝集成实现：在每次迭代中约束被松弛处理，而乘子更新则逐步强化约束满足。这种设计使得约束后验采样能够继承双空间求解器良好的条件数特性——即使当前模型远离真实解，部分约束松弛仍能产生有效的更新。我们在理想化的Rosenbrock条件推断问题、高斯异常体模型的频域全波形反演以及Marmousi~II基准模型上验证了该方法，结果表明：随着数据覆盖度的增加，该方法能提供校准良好的不确定性估计，并实现后验分布的合理收缩。