Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems

Tiangang Cui; Youssef M. Marzouk; Karen E. Willcox

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Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems

Tiangang Cui, Youssef M. Marzouk, Karen E. Willcox

arXiv:1403.4290v2Mar 17, 2014stat.COmath.NAstat.ME

TL;DR

Bayesian PDE inverse problems are costly because MCMC repeatedly evaluates the full numerical model. The paper adaptively builds posterior-focused reduced-order models within MCMC and uses them to accelerate inference. Experiments show improved reduced-model accuracy and substantially faster posterior sampling, with a trade-off between unbiased full-posterior sampling and bias-controlled approximate inference.

Problem
Repeated full-model evaluations make MCMC computationally expensive for Bayesian inverse problems governed by PDEs.
Method
The paper adaptively constructs projection-based reduced-order models from posterior samples during MCMC and couples reduced and full models to accelerate sampling.
Results
The data-driven reduced-order model achieves several-orders-of-magnitude lower posterior-averaged output error than prior-based offline reduction, while MCMC acceleration reaches up to two orders of magnitude.
Takeaways & Limitations
Posterior-focused online reduction can improve reduced-model accuracy and make computationally expensive posterior sampling more efficient.
Takeaways & Limitations
The ε-approximate algorithm samples an approximate posterior, so its accuracy depends on reliable error indicators and its estimator can be biased.

Abstract

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One of the major challenges in the Bayesian solution of inverse problems governed by partial differential equations (PDEs) is the computational cost of repeatedly evaluating numerical PDE models, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. This paper proposes a data-driven projection-based model reduction technique to reduce this computational cost. The proposed technique has two distinctive features. First, the model reduction strategy is tailored to inverse problems: the snapshots used to construct the reduced-order model are computed adaptively from the posterior distribution. Posterior exploration and model reduction are thus pursued simultaneously. Second, to avoid repeated evaluations of the full-scale numerical model as in a standard MCMC method, we couple the full-scale model and the reduced-order model together in the MCMC algorithm. This maintains accurate inference while reducing its overall computational cost. In numerical experiments considering steady-state flow in a porous medium, the data-driven reduced-order model achieves better accuracy than a reduced-order model constructed using the classical approach. It also improves posterior sampling efficiency by several orders of magnitude compared to a standard MCMC method.

1. INTRODUCTION AND MOTIVATION

Bayesian inverse problems require repeated, expensive forward-model evaluations during MCMC. The paper addresses this by adapting reduced-order models to posterior exploration and integrating them with MCMC.

Motivation: MCMC repeatedly evaluates computationally intensive forward models to sample posterior distributions in Bayesian inverse problems.Each posterior-density evaluation involves solving the forward model, often a PDE system.
Data-driven model reduction: The data-driven approach adaptively selects posterior samples for reduced-order-model snapshots rather than relying on broad prior-based offline sampling.This targets the concentrated region relevant to the observed data.
Data-driven model reduction: The method integrates reduced-order-model construction into adaptive MCMC, improving numerical accuracy during posterior exploration.The reduced model is enriched online as sampling proceeds.
MCMC integration: The reduced-order model and full model can be coupled in MCMC to accelerate sampling of the full posterior.An alternative directly samples an approximate posterior, but its estimator is biased and requires bias control.
Advantages: For equal accuracy, posterior-focused reduction can require a lower-dimensional basis than offline reduction and may scale better with parameter dimension.The reported advantage follows from retaining accuracy only in a concentrated parameter-space region.

2. SAMPLE-BASED INFERENCE FOR INVERSE PROBLEMS

Bayesian inference combines prior knowledge and observation likelihoods to form posterior distributions for unknown parameters. MCMC estimates posterior expectations, but computational efficiency depends on both chain correlation and the cost of repeated forward-model evaluations.

Bayesian formulation: The forward model maps unknown parameters x to observable outputs d, while prior and likelihood distributions define Bayesian inference.The prior represents knowledge before measurements, and the likelihood describes observations conditional on x.
Bayesian formulation: Measurement noise and model uncertainty are modeled as zero-mean Gaussian noise with covariance Σ_e when additional information is unavailable.This assumption determines the likelihood through the data-misfit function.
Posterior sampling: MCMC methods such as Metropolis-Hastings construct a Markov chain that converges to the target posterior through proposal and acceptance steps.Posterior samples support Monte Carlo estimation of expectations.
Sampling efficiency: MCMC efficiency is measured by effective sample size for a fixed CPU-time budget and is affected by sample correlation and forward-model cost.Reducing correlation improves statistical efficiency, while fast forward approximations permit more steps per unit time.
Sampling efficiency: Sampling directly from an approximate posterior produces a biased Monte Carlo estimator, whereas coupled methods can preserve sampling from the full posterior.Bias may be acceptable when controlled approximation error reduces overall mean squared error.

3. DATA-DRIVEN REDUCED-ORDER MODEL AND FULL TARGET ALGORITHM

The paper builds a reduced-order model online while exploring the posterior, using posterior samples to select snapshots and delayed acceptance to retain full-posterior sampling. The approach reduces state dimension but remains subject to computational and approximation limits.

Reduced-order model construction: The reduced-order model approximates the forward PDE map by projecting the discretized state equations onto a basis formed from posterior-sample snapshots.The basis Vm spans snapshots computed by solving the full model at selected parameter samples.
Computational considerations: A reduced model with low state dimension may still be expensive for general parametric dependence because assembling and projecting full-scale operators can depend on the original dimension n.Hyper-reduction methods such as missing point estimation and empirical interpolation are cited as ways to reduce this cost.
Full target algorithm: Delayed acceptance first explores the computationally fast approximate posterior, then uses full-posterior acceptance or rejection to preserve the target distribution.The first-stage subchain reduces sample correlation, while the second stage corrects the proposal using the full posterior.
Adaptive enrichment: Basis enrichment is triggered when the scaled reduced-model output error exceeds a user-specified threshold, controlling the allowable approximation error.A dual weighted residual error indicator can be used when direct forward-model comparisons are unavailable.
Data-driven model reduction: Posterior samples are selected adaptively for snapshots, so reduced-order model construction and posterior exploration proceed simultaneously.The resulting model is data-driven because it uses posterior information and online because construction occurs during sampling.
Ergodicity and limits: The algorithm converges to the full posterior despite reduced-model inaccuracies, provided the stated proposal irreducibility and finite-adaptation conditions hold.The reduced-basis dimension is capped at M, and adaptation terminates after finite time.

4. PROPERTIES OF THE APPROXIMATION

The paper analyzes how reduced-model error affects the approximate posterior, using an ε-feasible set to separate well-approximated regions from regions where the model remains inaccurate. Under the stated conditions, the posterior error is controlled by ε and the poorly approximated region's posterior measure.

Approximation framework: The ε-feasible set contains parameter values where the reduced-model error is bounded, while its complement measures regions not adequately approximated.The Hellinger-distance analysis uses the posterior measure of this complement together with the user-specified threshold ε.
Approximation framework: On the ε-feasible set, the data-misfit error satisfies |Φ(x) − Φm(x)| ≤ Kε for a constant K.This bound follows from the feasible-set definition and the paper's assumptions.
Posterior error bounds: The Hellinger distance between the full and approximate posteriors is bounded by terms involving ε and the posterior measure of the poorly approximated region.Theorem 7 formalizes this bound for a reduced-order model Fm(x).
Posterior error bounds: As the reduced basis is enriched under the stated technical conditions, the poorly approximated region asymptotically shrinks, reducing its contribution to posterior error.The pointwise reduced-model error decreases asymptotically with reduced-basis dimension.
Finite adaptation: When the posterior is sufficiently well sampled, updating the model until the relevant condition is satisfied yields approximate-posterior error proportional to the user-specified threshold ε.The paper uses heuristic finite-adaptation criteria because the condition cannot be checked directly during MCMC.

5. EXTENSION TO APPROXIMATE BAYESIAN INFERENCE

The section develops an ε-approximate Bayesian inference algorithm that uses reduced-order models to trade controlled posterior approximation error against improved computational efficiency. Its analysis identifies when this trade-off lowers mean squared error and specifies conditions and practical checks for reliable adaptation.

Algorithm and rationale: The ε-approximate algorithm samples an approximate posterior, accepting potential estimator bias when it can reduce mean squared error for a fixed computational effort.This contrasts with the full target algorithm, which samples the full posterior but requires full-model evaluations.
Algorithm and rationale: During adaptation, proposals with large error indicators use the full model, while sufficiently accurate reduced-order evaluations can directly determine acceptance or rejection.The upper threshold ε0 discards unreliable reduced-order evaluations; below the lower threshold ε, the reduced model is used directly.
Adaptive construction: The reduced basis is updated from full-model evaluations at accepted MCMC proposals while adaptation continues under a maximum basis dimension and a finite adaptation criterion.If the basis reaches dimension M before the criterion is satisfied, the algorithm switches to the full target algorithm because the approximate estimator may have unbounded bias.
Error analysis: For a target effective sample size not exceeding τ, sampling the approximate posterior can yield smaller MSE than standard full-posterior MCMC when the stated bound holds.In this regime, the estimator’s variance dominates its bias, making the approximate approach more accurate for a fixed computational cost.
Practical scope: The approach requires reliable error indicators or estimators, while the condition needed for selecting suitable snapshots is checked heuristically through a finite adaptation criterion.The relevant condition is computationally infeasible to verify directly in practice.
Error analysis: The bound τ depends on the error threshold ε and the problem-specific speedup factor S(m(ε)), creating a trade-off between numerical accuracy and reduced-model efficiency.Choosing ε too small can produce an expensive, high-dimensional reduced model whose speedup is insufficient, making approximate-posterior sampling less efficient than full-posterior sampling.

6. NUMERICAL RESULTS AND DISCUSSION

The numerical experiments evaluate the data-driven reduced-order approach on nine-dimensional and high-dimensional porous-media inverse problems. The results show accurate posterior sampling, better reduced-model accuracy than prior-based construction, and substantial computational savings.

Test cases: The experiments use isothermal steady flow in porous media, with nine-dimensional and higher-dimensional permeability inverse problems.The nine-dimensional case represents permeability with radial basis functions; the high-dimensional case uses a log-normal random-field model.
Computational efficiency: Second-stage acceptance probabilities exceed 0.96 for all three ǫ values in the full target algorithm.This indicates that the reduced-order models are reasonably accurate relative to the full model in this test case.
Computational efficiency: The ǫ-approximate algorithm achieves speedup factors 4.7 to 7.4 times higher than the full target algorithm.Larger ǫ values produce larger speedups because they use smaller reduced bases.
Computational efficiency: 4672 effective samples are generated in 115.3 seconds by the ǫ-approximate algorithm, versus about 16 by the reference algorithm.The comparison uses a fixed CPU-time budget and the ǫ = 10−1 setting.
Comparison with a prior-based model: The data-driven reduced-order model has a better convergence rate than the prior-based reduced-order model over the full posterior distribution.The error measure is the posterior expectation of the L∞ norm of the scaled true sensor error; accuracy improves significantly once at least 10 reduced basis vectors are included.
Influence of posterior concentration: As signal-to-noise ratio increases, the posterior concentrates and the required reduced-basis dimension decreases.The experiments attribute this lower dimension to the approach exploiting the increasing concentration of the posterior.

6.4. Remarks on the ǫ-approximate algorithm

The ϵ-approximate algorithm’s speedup depends on the case study and reduced-basis accuracy: larger ϵ can reduce cost, but excessive approximation can erase the efficiency gain. Its advantage is strongest when only limited effective samples are needed.

At most 3.7 speedup was achieved in the second case study with ϵ = 10^-1, while ϵ = 10^-3 performed slightly worse than the full target algorithm.The speedup factor decreased as ϵ decreased in both cases.
For ϵ = 10^-3 in the second test case, the reduced-order model became too expensive relative to the full model, eliminating the algorithm’s efficiency gain.
The ϵ-approximate algorithm can be more advantageous when only a limited number of effective samples is required.A relatively large ϵ keeps reduced-order-model costs low while estimator MSE remains dominated by variance rather than sampling bias.

7. CONCLUSION

The paper introduces posterior-oriented reduced-order modeling integrated with MCMC, alongside algorithms that trade full-posterior fidelity against additional speedups. Case studies show substantial acceleration, comparable sampling accuracy for the ϵ-approximate method, and improved reduced-order accuracy over prior-based construction, especially with informative data.

The approach adaptively selects posterior samples for snapshots while integrating reduced-order-model construction into posterior sampling.This enables simultaneous posterior exploration and model reduction.
The full target algorithm couples full and approximate posterior distributions and preserves ergodicity with respect to the true posterior.The ϵ-approximate algorithm instead samples an approximate posterior and can provide further speedups for some problems.
Up to two orders of magnitude acceleration was demonstrated for both algorithms, while the ϵ-approximate algorithm’s sampling accuracy was comparable to reference full-model MCMC.
For the same number of reduced basis vectors, the data-driven model’s posterior-averaged output error was several orders of magnitude smaller than that of prior-based offline reduction.
The results suggest posterior-oriented reduced-order models are preferable to prior-based models for statistical inverse problems, especially when the data are informative.The posterior-oriented construction concept may also extend to Gaussian process regression and generalized polynomial chaos.