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Geometric MCMC for Infinite-Dimensional Inverse Problems
Alexandros Beskos, Mark Girolami, Shiwei Lan, Patrick E. Farrell, Andrew M. Stuart
TL;DR
Infinite-dimensional Bayesian inverse problems require MCMC methods that remain effective under mesh refinement while handling data-informed, non-Gaussian posterior geometry. The paper combines geometric MCMC on an informative finite-dimensional subspace with pCN-like methods elsewhere, reporting up to two orders of magnitude higher sampling efficiency than pCN across challenging inverse problems.
Problem
Sampling is difficult because inverse-problem posteriors are high-dimensional and potentially non-Gaussian, while traditional methods deteriorate under mesh refinement and geometric methods can be computationally intractable in infinite dimensions.
Method
The paper combines infinite-dimensional geometric MALA and HMC with dimension reduction, using geometric adaptation in an informative subspace and robust methods in the complementary space.
Results
The algorithms achieve up to two orders of magnitude improvement in sampling efficiency compared with pCN across challenging inverse problems.
Takeaways & Limitations
Geometric information can provide substantial computational improvements for MCMC in complex inverse problems while retaining methods designed for infinite-dimensional settings.
Takeaways & Limitations
The position-dependent geometric dynamics do not generally preserve the target distribution, so arbitrary integration times lack a theoretical guarantee of good acceptance probabilities.
Abstract
from arXiv · showhide
Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank-Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method.
1. Introduction
The paper studies Bayesian inverse problems with unknown functions in separable Hilbert spaces and finite-dimensional observations. It targets MCMC methods that preserve mesh-independent behavior while exploiting posterior geometry and computationally tractable structure.
- Problem setting: Bayesian inverse problems seek an unknown function u in a separable Hilbert space from finite-dimensional observations linked through a forward PDE operator.The parameter-to-data map may be nonlinear, and the prior is assigned directly to u.
- Problem setting: Sampling is difficult because posteriors can be high-dimensional and non-Gaussian, while likelihood evaluation requires repeated PDE solutions.The posterior may have complex non-elliptic finite-dimensional projections, increasing the challenge of exploration.
- Motivation: Traditional Metropolis-Hastings methods develop deteriorating mixing times as the discretization mesh is refined.This refinement issue motivates methods defined directly on infinite-dimensional Hilbert spaces.
- Motivation: Infinite-dimensional advanced MCMC methods provide mesh-independent mixing times, but the paper seeks to add data-informed geometric adaptation without losing this benefit.The proposed direction combines geometric exploration of complex posterior structure with robust high-dimensional mixing.
- Contributions: The paper generalizes simplified Riemannian MALA and an HMC extension to infinite dimensions, establishes well-posedness, and applies the methods to complex inverse problems.The methodology uses position-dependent curvature and includes a dimension-reduction strategy for practical computation.
2. (Non-Geometric) MCMC on Hilbert Spaces
This section reviews infinite-dimensional MCMC methods, emphasizing pCN's mesh robustness and the use of gradient, curvature, and Hamiltonian information for more data-informed proposals. It also explains the measure-theoretic and computational conditions underlying these methods.
- pCN: pCN is well-defined on the Hilbert space because its proposal preserves the prior, unlike standard random-walk Metropolis.Standard random-walk Metropolis is restricted to finite-dimensional discretizations and loses acceptance as resolution increases.
- pCN: pCN avoids mesh-dependent deterioration but can explore complex posteriors diffusively because its proposal does not generally use the data.Its free parameter ρ controls the proposal move size.
- Gradient-informed methods: Gradient-informed Langevin dynamics use the Fréchet derivative of the potential to construct posterior-informed proposals.The preconditioner is tuned to the prior covariance in the reviewed infinite-dimensional setting.
- Infinite-dimensional MALA: Infinite-dimensional MALA is defined through a Metropolis-Hastings correction under an absolute-continuity condition involving CDΦ(u) and Im(C^1/2).The proposal construction relies on Gaussian measure relationships and a Radon-Nikodym derivative.
- Infinite-dimensional HMC: Infinite-dimensional HMC uses Hamiltonian dynamics and leapfrog maps to make large state-space moves that can suppress random-walk behavior.The dynamics preserve the target measure formally, while the practical integrator uses a Strang-splitting construction.
3. Geometric Metropolis-Hastings Algorithms
The paper develops geometric MCMC methods on infinite-dimensional Hilbert spaces by adapting local posterior curvature while preserving computationally robust proposals. A split preconditioner concentrates geometric computation on data-informed low frequencies and simplifies the complementary directions.
- Local geometric methods: The local Gaussian reference measure makes the target's negative log-density relatively flat near the current position through a second-order approximation.The approximation uses a covariance related to the inverse negative Hessian at the current location.
- Local geometric methods: The algorithms use a location-specific preconditioner K(u) motivated by the inverse negative Hessian to capture local posterior curvature.This replaces the prior covariance as the proposal's local reference structure.
- Local geometric methods: ∞-mMALA uses local reference measures in a semi-implicit scheme, while ∞-mHMC extends it by synthesizing multiple local steps before accept/reject.The methods are defined using location-specific geometric information rather than only the prior covariance.
- Local geometric methods: ∞-mHMC lacks a general guarantee of good acceptance probabilities because its dynamics do not generally preserve the target distribution.With a properly chosen time horizon, the paper reports that ∞-mHMC can outperform ∞-mMALA in numerical examples.
- Split methods: The location-specific geometric calculations can cost O(n^3) after projection to dimension n, motivating a split preconditioner with a finite non-diagonal block.The construction exploits concentration of data information in low-frequency coordinates, leaving the remaining directions simpler.
- Split methods: Split ∞-mMALA and Split ∞-mHMC simplify the required calculations because K(u) is diagonal except over a finite range of coordinates.The split methods retain geometric treatment where the data are informative while reducing computation elsewhere.
2704. Numerical Experiments
The numerical experiments evaluate the proposed MCMC methods on groundwater flow, thermal fin heat conduction, and laminar jet inverse problems using Gaussian Hilbert-space priors and truncated K-L representations. Across these problems, geometric splitting methods improve sampling efficiency while retaining mesh-independent behavior, with gains ranging from close to fourfold to almost two orders of magnitude over pCN.
- Experimental setup: The experiments study groundwater flow, thermal fin heat conductivity, and laminar jet inverse problems with Gaussian priors on Hilbert spaces.The first two examples use MATLAB, while the laminar jet example uses FEniCS.
- Experimental setup: The algorithms sample selected K-L coordinates, while splitting methods truncate the Fisher operator to the first D0 coordinates.Gradients use one additional adjoint solve, and metric actions use two further incremental adjoint solves.
- Groundwater flow: In groundwater flow, Split ∞-mHMC achieves close to 4-fold higher minimum ESS per CPU second than pCN.The non-geometric ∞-MALA and ∞-HMC methods perform worse than pCN after accounting for their additional computation.
- Groundwater flow: Groundwater experiments show that ∞-mMALA and ∞-mHMC autocorrelations do not deteriorate under mesh refinement, indicating mesh-independent equilibration.Posterior information is concentrated mainly in the first 25 eigen-directions.
- Scope and limitations: The thermal fin prior may introduce nonphysical correlations between fins because its construction is not geometry-adapted.Geometry-adapted priors are identified as a possible but more complicated alternative.
- Thermal fin: More than an order of magnitude improvement over pCN is observed for Split ∞-mMALA and Split ∞-mHMC in the thermal fin problem.The posterior mean estimates are consistent across algorithms and closer to the truth with 262 observations for 100 unknown parameters.
- Laminar jet: In the laminar jet problem, geometric methods provide almost 2 orders of magnitude higher sampling efficiency than pCN.Non-geometric methods fail to converge because they require step-sizes of O(10^-4) for non-negligible acceptance rates, while high-frequency estimates remain limited by the data.
5. Conclusion and Discussion
The paper develops infinite-dimensional geometric MALA and HMC methods, connects them, and uses dimension reduction to make them practical for complex inverse problems. It also identifies computational and geometric extensions for future work.
- The paper generalizes simplified Riemannian manifold MALA to infinite dimensions and develops an HMC version for Hilbert-space inverse problems.
- The methods are shown to be well-defined in infinite dimensions and effective across three numerical studies.
- Dimension reduction makes geometric MCMC practical by focusing computation on a finite-dimensional intrinsic subspace.
- Full Riemannian-manifold dynamics would require third-order geometric information and further work to balance improved mixing against computational overhead.
- Future methods could combine fully geometric MCMC in an intrinsic subspace with pCN or ∞-MALA in its complement.
Appendix A. Proof of Theorem 3.10
The appendix proves invariance of the proposed Markov update by relating the leapfrog transformation, velocity symmetry, and Metropolis acceptance step. The proof concludes that the accepted update preserves expectations under the target construction.
- The proof decomposes the proposal transformation into maps and tracks the associated measure-density changes.
- The argument includes the intermediate algebraic identities and invokes similarity to the proof of Theorem 3.1 in.
- The next chain position is selected between the proposal and current state using the Metropolis acceptance probability.
- Velocity reversal and quadratic kinetic energy provide the symmetry used to equate forward and reversed proposal expectations.
Appendix B. Proof of Corollary 3.13
The appendix shows that, under the specified step sizes, one leapfrog step of ∞-mHMC produces the same proposal as ∞-mMALA. Consequently, their Metropolis acceptance probabilities also coincide.
- Under the prescribed step sizes, the proof first establishes coincidence between the ∞-mHMC and ∞-mMALA proposals.
- The one-step ∞-mHMC proposal is written as a cosine-weighted current state plus a sine-weighted velocity term.
- Because the proposals coincide and both use the Metropolis-Hastings ratio, their acceptance probabilities are identical.