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Multifidelity variational inference for derivative-free PDE-constrained inverse problems

Eric J. Parish

arXiv:2610.37359v1math.NAcs.NA

TL;DR

Large PDE-constrained inverse problems can be expensive when forward solves are costly and PDE gradients are unavailable. The paper combines multifidelity score-function estimators, second-order optimization, and adaptive reduced-order models; across four benchmark problems with up to nine parameters, it demonstrates the proposed approach.

  • Problem

    Large-scale PDE-constrained inverse problems can require many costly forward solves, while black-box models may lack usable PDE gradients.

    Source: §1 · p. 1, §1.1 · p. 2

  • Method

    MF-BBVI combines score-function gradient and Hessian estimators with approximate control variates, regularized second-order updates, and adaptive reduced-order models.

    Source: §1.2 · p. 3

  • Results

    Across four benchmark problems with up to nine parameters, numerical results demonstrated the proposed multifidelity approach.

    Source: §6 · p. 44

  • Takeaways & Limitations

    The framework provides a derivative-free VI approach for PDE-constrained inverse problems, with its control-variate unbiasedness dependent on sampling, gain estimation, and ROM adaptation.

    Source: §1.2 · p. 3

    Estimating the ACV gain and correction from the same samples can introduce bias, and using a retrained ROM immediately can also bias the gradient estimator.

    Source: §4.2.1 · p. 16, §4.2.2 · p. 16

Abstract

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We develop a multifidelity black-box variational inference (MF-BBVI) approach for PDE-constrained Bayesian inverse problems in which forward solves are expensive and PDE gradients are unavailable. The method is built on score-function estimators for the evidence lower bound (ELBO), which avoid differentiation through the PDE model but are known to suffer from high variance. To address this high variance, we employ multifidelity approximate control-variate estimators that combine a small number of high-fidelity samples with a larger number of low-fidelity samples to reduce estimator variance and the number of high-fidelity evaluations required for optimization. We further develop a PDE-derivative-free second-order optimization strategy based on both gradient and Hessian score-function estimators. For low-fidelity model generation, we propose an on-the-fly strategy that builds and adaptively updates a reduced-order model (ROM) as the variational distribution evolves throughout the optimization. We investigate strategies for adapting the ROM and estimating the approximate control-variate gains, examining tradeoffs between estimator bias, variance reduction, and computational cost. The resulting approach facilitates derivative-free variational inference for large-scale inverse problems through a reduction in overall computational cost. We demonstrate the method on four numerical examples: a linear benchmark with an exact posterior and three PDE-constrained inverse problems.

1. Introduction

Large PDE-constrained inverse problems are costly when each likelihood evaluation requires expensive forward solves, and black-box models may lack usable derivatives. The paper develops multifidelity, derivative-free VI and evaluates it across four inverse problems.

1. Introduction

  • Expensive PDE solves and unavailable or unreliable gradients make large-scale black-box inverse problems difficult for gradient-free inference.Gradient-free methods require forward evaluations and their simulation burden can grow with parameter dimension.Source: §1 · p. 1, §1.1 · p. 2

1.1. Related work

  • MCMC and ensemble Kalman methods offer alternatives, but can face computational costs or fail to target the exact posterior except under restrictive assumptions.The paper situates its approach among existing surrogate, multifidelity, and reduced-order variants.Source: §1.1 · p. 2

1.1.1. Variational inference

  • Score-function VI avoids PDE derivatives, but its high-variance estimators can make optimization slow, noisy, and unstable.Prior multifidelity score-function work addressed stochastic optimization rather than ELBO optimization with on-the-fly PDE surrogates.Source: §1.1.1 · pp. 2–3

1.1.2. Optimization

  • The authors identify PDE-derivative-free second-order score-function estimators for VI as an unexplored combination.Related second-order methods mostly require forward-model derivatives, while score-function Hessian estimators had not targeted VI.Source: §1.1.2 · p. 3

1.2. Our contributions

  • MF-BBVI combines score-function gradient and Hessian estimates with approximate control variates, using few high-fidelity and more inexpensive low-fidelity evaluations.The paper notes that unbiasedness depends on sampling, gain-estimation, and ROM-adaptation strategies.Source: §1.2 · p. 3
  • The proposed stochastic Newton-type optimizer uses diagonal curvature approximations, lagged Hessian averaging, and regularization.Source: §1.2 · p. 3
  • An on-the-fly ROM is updated as the variational distribution evolves, and four inverse problems are used to assess MF-BBVI and its design choices.The benchmarks range from a linear problem with an analytic posterior to nonlinear PDE-constrained inverse problems.Source: §1.2 · p. 4

2. Variational inference

The section formulates Bayesian inference with an expensive PDE forward model and unavailable PDE gradients, then describes VI and score-function estimators. It also reviews optimization and control-variate tools for addressing estimator variance.

2. Variational inference

  • The target is a posterior over uncertain parameters constrained by a discretized PDE, in a setting where likelihood evaluations are expensive and PDE gradients are unavailable.Source: §2 · p. 4
  • VI approximates the posterior with a tractable search distribution and minimizes its KL divergence, equivalently maximizing the ELBO.Source: §2 · p. 4

2.1. Optimization approach

  • Fixed-step gradient ascent can be inefficient or unstable as the variational scale changes; natural gradients instead measure steps using KL distance but require a Fisher matrix that may be intractable.Source: §2.1 · p. 5

2.2. Black-box VI

  • Score-function estimators rewrite ELBO gradients using derivatives of the variational density alone, avoiding PDE gradients, although their Monte Carlo estimates can have high variance.Source: §2.2 · pp. 5–6, §2.4 · p. 6

2.3. Mean-field variational family

  • Mean-field VI factorizes the variational distribution into independent components, often Gaussian, with log standard deviations used to enforce positive variance.Source: §2.3 · p. 6

2.4. Variance reduction through control variates

  • Control variates reduce estimator variance by adding a correlated auxiliary quantity with known mean; estimating its covariance from the same samples as score functions can introduce bias.For unbiased Monte Carlo estimators, covariance scales at O(λ^-1), though its magnitude can depend strongly on dimension.Source: §2.4 · pp. 6–7

3. Multifidelity black-box variational inference

The paper proposes multifidelity black-box VI to address high estimator variance and slow convergence, combining approximate control variates, regularized stochastic second-order updates, and adaptive surrogates.

3. Multifidelity black-box variational inference

  • MF-BBVI targets high-variance estimators and slow convergence that can make black-box VI costly for large-scale applications.Source: §3 · p. 7
  • The approach uses approximate control-variate estimators to reduce estimator variance.Source: §3 · p. 7
  • It also combines regularized stochastic second-order updates with on-the-fly adaptive surrogates.Source: §3 · p. 7

3.1. Approximate control variates

The paper estimates high-fidelity quantities with approximate control variates built from correlated low-fidelity evaluations. Its multifidelity VI estimator uses nested samples, and under fixed gain and low-fidelity model assumptions can reduce variance while remaining unbiased.

3.1. Approximate control variates

  • Approximate control variates replace an unavailable low-fidelity expectation with an estimate from additional low-fidelity samples, preserving unbiasedness under an appropriate sampling scheme.Source: §3.1 · p. 7
  • The estimator uses a high-fidelity average corrected by the difference between low-fidelity averages on nested high- and low-fidelity sample sets.The low-fidelity sample count can greatly exceed the high-fidelity count; correlation between models can substantially reduce variance.Source: §3.1 · pp. 7–8
  • The gain minimizes estimator variance, but small high-fidelity batches can make fitted gains unreliable.The authors use scalar and componentwise gains; unrestricted matrix gains require more coefficients and can be poorly conditioned.Source: §3.1 · p. 8

3.1.1. Multifidelity VI estimators

  • For VI, the high- and low-fidelity ELBO integrands share the prior, variational-density, and bounded-parameter Jacobian terms; their difference is only in log-likelihood.Source: §3.1.1 · p. 9
  • The multifidelity gradient estimator is unbiased with fixed gain and low-fidelity model, and can lower variance enough to require fewer high-fidelity samples for a target accuracy.Source: §3.1.1 · p. 9

3.2. Regularized stochastic diagonal-Newton optimization

The section extends PDE-gradient-free score-function optimization with Hessian estimates for Newton-type updates. It addresses noisy curvature and same-sample gradient–Hessian dependence through lagging and Fisher-scaled diagonal regularization.

3.2.1. Hessian estimator

  • Hessian score-function estimates enable Newton-type updates using derivatives of the search distribution, without derivatives of the PDE model.The method applies the second-order score identity to the log-joint term; entropy is handled separately and has zero Hessian for the stated mean-field Gaussian family.Source: §3.2 · p. 9, §3.2.1 · pp. 9–10

3.2.2. Lagged Hessian estimator

  • A same-sample Hessian and gradient can yield a nonzero expected update even when the expected gradient is zero, so the method uses an exponential moving average of preceding Hessian estimates.Independent sample sets would resolve the issue but incur substantial computational cost; after initialization, the current Hessian estimate contributes to the next iteration's curvature.Source: §3.2.2 · p. 10

3.2.3. Fisher-scaled Hessian regularization

  • The method keeps only diagonal curvature and regularizes it with the current Fisher matrix to reduce sensitivity to noisy cross-coordinate estimates and wrong-sign curvature.It retains negative curvature and replaces near-zero or wrong-sign entries with a natural-gradient fallback; the work uses a fixed unit step size.Source: §3.2.3 · pp. 10–11

3.3. Additional ingredients

Additional ingredients pair leave-one-out centering with coordinate transformations for bounded physical parameters. The transformations keep the variational distribution Gaussian in unconstrained coordinates while evaluating the likelihood and prior at mapped physical parameters.

3.3.1. Leave-one-out baseline for variance reduction

  • A leave-one-out baseline subtracts the other samples’ average ELBO integrand and is used to reduce score-function estimator variance, particularly for small sample sizes.For independent parameter samples, centering preserves the expectation of each score-function term and the multifidelity estimator for fixed gain and low-fidelity model.Source: §3.3.1 · p. 11
  • The leave-one-out procedure also centers Hessian terms while retaining the decomposition that differentiates entropy analytically.The centered gradient and Hessian estimates feed the Newton update and lagged Hessian average, respectively.Source: §3.3.1 · p. 12

3.3.2. Variable bounds

  • For bounded parameters, the method uses a Gaussian variational distribution in transformed coordinates and evaluates the likelihood and prior at mapped physical parameters.The change of variables includes a Jacobian contribution in the high- and low-fidelity log-joint terms used by the gradient and Hessian estimators.Source: §3.3.2 · p. 12

4. On-the-fly adaptive low-fidelity modeling

This section presents an on-the-fly, data-driven ROM built from high-fidelity evaluations and updated as the variational distribution evolves, addressing tradeoffs in ROM adaptation and multifidelity gain estimation. It also describes implementation abstractions for high- and low-fidelity models and concurrent ensemble evaluation.

4. On-the-fly adaptive low-fidelity modeling

  • The on-the-fly ROM uses HF evaluations generated during inference and updates as the variational distribution evolves, avoiding a separate offline training stage.Training data are concentrated in the region explored by the current distribution.Source: §4 · p. 13

4.1. Training and adaptation from high-fidelity samples

  • The ROM is fitted using a rolling window of recent HF sample sets and retrained when its error on the current HF batch exceeds a threshold.The threshold test evaluates the previously retained ROM before deciding whether to retrain.Source: §4.1 · p. 13

4.2. Adaptation and estimation of correlations

  • Gradient and Hessian gains depend on HF–LF correlations that change with the search distribution and ROM, so online gain estimation requires paired HF–LF evaluations.Source: §4.2 · p. 14

4.2.1. Same-sample estimation of ACV gain introduces bias

  • Same-sample gain fitting can bias the gradient estimator, whereas leave-one-out (LOO) fitting is unbiased for a fixed or delayed ROM but does not guarantee lower variance.Independent paired-sample gain fitting is also unbiased for a fixed or delayed ROM, but requires additional HF evaluations and likewise does not guarantee lower variance.Source: §4.2.1 · p. 16

4.2.2. Same-sample LF adaptation introduces bias

  • Training the active ROM on the current HF batch can bias the gradient estimator because those samples also enter the multifidelity correction.Delayed adaptation separates training from the current correction batch without additional HF evaluations, but responds more slowly to changes in the search distribution.Source: §4.2.2 · pp. 16–17

4.2.3. Discussion

  • Delayed ROM adaptation with LOO gradient-gain estimation is the default, yielding an unbiased gradient estimator after initialization without extra HF cost; same-sample approaches performed comparably in experiments.Unbiasedness does not ensure greater variance reduction; inaccurate gains or weak HF–LF correlation can still leave substantial variance.Source: §4.2.3 · p. 18

4.3. Automated model reduction via Gaussian processes

  • The automated non-intrusive POD–GP ROM uses parameter–response pairs, enabling construction without access to governing equations or their derivatives.POD reduces the QoI responses before GP regression predicts the retained coefficients.Source: §4.3 · pp. 18–19

4.4. Software implementation

  • The rom-tools-and-workflows package manages outer VI algorithms through a QoiModel interface for run setup, PDE-solver execution, and QoI extraction.Users provide a concrete QoiModel for their application.Source: §4.4 · pp. 19–20
  • QoiModelBuilderWithTrainingData builds low-fidelity models from training directories, parameters, and QoIs; its POD–GP ROM implementation can be replaced by another model.The builder returns a new low-fidelity QoiModel.Source: §4.4 · p. 20
  • Concurrent evaluation allows high- and low-fidelity model batches to run in parallel at each iteration, enabling substantial wall-time gains.The implementation supports concurrent evaluation of both model ensembles.Source: §4.4 · p. 20

5. Numerical experiments

The section evaluates MF-BBVI on four inverse problems, comparing single- and multifidelity variants with Adam and stochastic Newton against high-fidelity references.

5. Numerical experiments

  • Four inverse problems are evaluated: an analytic sine-series benchmark, convection–diffusion–reaction, a premixed H2-air flame, and shallow-water bathymetry.The comparisons include single-fidelity BBVI (Adam and Newton) and their multifidelity counterparts.Source: §5 · p. 20
  • All multifidelity methods use delayed ROM updates and a scalar leave-one-out gain by default, yielding an unbiased estimator.Newton uses lagged curvature with β = 0.1; Adam preconditions the ELBO gradient using a regularized Fisher matrix.Source: §5 · p. 21
  • The Gaussian-process ROM uses the four most recent high-fidelity batches, with a 1 × 10^-3 update tolerance and 0.999999 retained POD energy.Parameters and responses are normalized; the covariance kernel is squared-exponential.Source: §5 · p. 21
  • GP hyperparameters are selected per retained POD mode by maximizing log marginal likelihood over 81 length-scale and signal-variance pairs.The grid uses nine logarithmically spaced values for each parameter.Source: §5 · p. 21
  • Computational work is measured by cumulative high-fidelity evaluations, while accuracy is assessed using KL divergence and variational parameters against high-fidelity references.The sine-series reference is an exact Gaussian posterior; the other references are numerical VI solutions, so their KL values measure agreement with those references, not the exact posterior.Source: §5 · p. 21
  • For convection–diffusion–reaction, air-flame, and shallow-water cases, Gaussian parameters are represented in latent coordinates, with physical means and variances evaluated by one-dimensional quadrature.Source: §5 · p. 21

5.1. Gaussian sine-series inverse problem

The sine-series benchmark has an exact diagonal-Gaussian posterior, enabling direct comparison of BBVI and multifidelity BBVI with Adam or Newton optimization. Multifidelity Newton performs best in both the standard and 30-term configurations, while sample-allocation and ROM/gain studies examine tradeoffs in performance and cost.

5.1. Gaussian sine-series inverse problem

  • The benchmark uses a linear-Gaussian sine-series inverse problem with an exact diagonal-Gaussian posterior, allowing direct accuracy assessment.Observations are generated at 31 equispaced interior points with a specified noise level.Source: §5.1 · pp. 21–22

5.1.1. Method comparison

  • Below 10−12 versus below 10−7: MF-BBVI (Newton) and BBVI (Newton) achieve these terminal median KL values; MF-BBVI (Adam) reaches 46.9 versus BBVI (Adam) at 130.MF-BBVI (Newton) closely approximates the correct posterior within a hundred iterations for all quantities; both Adam methods fail to reach it within the budget.Source: §5.1.1 · p. 22, Table 3 · p. 22

5.1.2. Sample-size study

  • Across the tested sample allocations, MF-BBVI (Newton) has the smallest terminal median KL, and increasing additional LF samples from 128 to 512 improves performance.Larger HF batches stabilize optimization but cost more per iteration, so they do not uniformly improve results under a fixed HF budget.Source: §5.1.2 · p. 25, Table 4 · p. 26

5.1.3. Multifidelity studies

  • Componentwise and scalar fitted gains outperform unit gains across HF batch sizes, while the fitted gain structures perform similarly.The authors suggest finite sampling error in gain computation can offset the benefits of greater gain complexity.Source: §5.1.3 · p. 26, Figure 8 · p. 26
  • Same-sample and delayed LF-model updates perform similarly, suggesting that bias from immediate retraining has minimal optimization impact.Source: §5.1.3 · p. 26, Figure 8 · p. 26
  • In gradient-gain sampling, biased same-sample gains slightly outperform LOO and independent sampling, while independent gain-fitting runs use approximately twice the total HF work.Thus, separating gain-fitting samples offers no practical advantage in this example.Source: §5.1.3 · p. 26, Figure 8 · p. 26

5.1.4. High-dimensional configuration

  • 6.73 × 10−6 versus 4.48: in the 30-term case, MF-BBVI (Newton) and BBVI (Newton) achieve these median terminal KL values, respectively.MF-BBVI (Newton) also has terminal median mean and variance errors of 0.147% and 0.0601%, versus 153% and 44.0% for BBVI (Newton).Source: §5.1.4 · p. 27, Table 5 · p. 27

5.2. Steady convection–diffusion–reaction inverse problem

The CDR inverse problem tests MF-BBVI against a numerical VI reference, with comparisons spanning method performance, sample allocation, and multifidelity design choices. MF-BBVI (Newton) performs best in the reported method and sample-size studies, while the multifidelity design tests show limited sensitivity to several choices.

5.2. Steady convection–diffusion–reaction inverse problem

  • The CDR study infers convection speed, angle, diffusivity, and reaction coefficient from 256 noisy observations of a 31 × 31 interior-grid solution.Because this benchmark lacks an analytic posterior, its comparisons use a numerical VI reference computed with 2,000 Newton updates and 1,024 HF samples per update.Source: §5.2 · p. 28

5.2.1. Method comparison

  • 5.36 × 10−3 terminal median KL for MF-BBVI (Newton), versus 7.79 for BBVI (Newton), the next-best method.The comparison used four HF samples per batch and a nominal budget of 5,000 HF evaluations across 10 runs.Source: Table 6 · p. 29, §5.2.1 · p. 29
  • MF-BBVI (Newton) best approximates the reference posterior within 1,000 FOM evaluations and has the smallest terminal median variance error.The reported order is MF-BBVI (Newton), BBVI (Newton), MF-BBVI (Adam), then BBVI (Adam) for both posterior approximation and terminal median variance error.Source: §5.2.1 · p. 29, Figure 13 · p. 31, Figure 14 · p. 32

5.2.2. Sample-size study

  • MF-BBVI (Newton) has the smallest terminal median KL in every tested HF batch and ROM sample-size configuration.Increasing additional ROM samples from 128 to 512 reduces its median in every configuration; MF estimators improve terminal median KL for both optimizers across the tested configurations.Source: §5.2.2 · p. 30, Table 7 · p. 30, Figure 15 · p. 33

5.2.3. Multifidelity studies

  • Unit-magnitude gains perform almost as well as scalar and componentwise gains, and more expressive gains do not necessarily improve convergence under finite sampling error.The comparison used same-sample gain estimation and delayed LF updates.Source: §5.2.3 · p. 32
  • Same-sample ROM updates yield lower terminal medians at HF batch sizes 4 and 16, while delayed updates do so at 8 and 32.The results suggest that bias from LF model updates has minimal impact on the overall optimization.Source: §5.2.3 · p. 32
  • Same-sample and LOO gradient-gain sampling show minimal statistical differences from independent sampling across HF batch sizes, while avoiding its added HF cost.Independent sampling requires λHF additional HF samples at each iteration.Source: §5.2.3 · p. 32

5.3. Premixed H2-air flame

The premixed H2-air flame example compares MF-BBVI and single-fidelity BBVI against a numerical VI reference. MF-BBVI (Newton) achieves the lowest reported KL and remains best across the tested sample-size configurations.

5.3. Premixed H2-air flame

  • The flame model is a two-dimensional premixed H2-air advection-diffusion-reaction system, with inference over diffusivity, activation energy, and two velocity components.The example uses a 64 × 32 grid and temperature observations sampled across space and time.Source: §5.3 · p. 34

5.3.1. Method comparison

  • MF-BBVI (Newton) well-approximates the reference posterior for all parameters within 200 iterations, with Newton outperforming Adam and multifidelity improving both optimizers.BBVI (Newton) follows MF-BBVI (Newton), then MF-BBVI (Adam), in the reported method comparison.Source: §5.3.1 · pp. 35, 37, Figure 20 · p. 38, Figure 21 · p. 39
  • Across four seeds, MF-BBVI (Newton) achieved the smallest terminal median KL, 6.97 × 10−3; BBVI (Newton) was next best at 0.285, as reported for the last recorded iteration.Source: §5.3.1 · p. 35, Table 8 · p. 36

5.3.2. Sample-size study

  • MF-BBVI (Newton) has the smallest terminal median in every tested configuration, averaging an order-of-magnitude improvement over BBVI (Newton).Increasing ROM sample size improves every configuration; MF sampling improves all but one Adam configuration, and Newton outperforms Adam.Source: §5.3.2 · p. 37, Table 9 · p. 37, Figure 22 · p. 39

5.4. Shallow-water bathymetry inverse problem

The shallow-water study infers nine bathymetry coefficients and compares posterior moments and KL against a numerical VI reference. MF-BBVI (Newton) has the lowest median KL and smaller aggregate moment errors than BBVI (Newton).

5.4. Shallow-water bathymetry inverse problem

  • The inverse problem represents bathymetry with nine Gaussian basis functions and compares against a VI Newton reference using 512 samples per iteration for 102 iterations.The 200 × 200 finite-volume model supplies 2,100 noisy free-surface observations from 100 spatial locations at 21 times.Source: §5.4 · p. 40

5.4.1. Method comparison

  • MF-BBVI (Newton) has the smallest median KL, followed by BBVI (Newton); multifidelity lowers median KL for both optimizers, and Newton outperforms Adam.The comparison used eight HF samples per batch, a nominal budget of 3,000 HF evaluations, and four runs.Source: Table 10 · p. 41, §5.4.1 · p. 41, Figure 24 · p. 41
  • Terminal median mean error is 0.448% for MF-BBVI (Newton) versus 0.627% for BBVI (Newton); median relative variance error is 4.92% versus 7.09%, respectively.For Adam, multifidelity lowers variance error and KL despite a larger mean error, so improved KL need not imply improved agreement of every moment.Source: §5.4.1 · p. 41, Figure 27 · p. 44

6. Conclusions

The paper concludes that multifidelity score-function BBVI with adaptive reduced-order models and second-order optimization improves on single-fidelity score-function BBVI across four benchmarks. It identifies richer variational families and further development of sampling, ROM, and Newton-like optimization strategies as future work.

6. Conclusions

  • Across four benchmarks with up to nine parameters, MF-BBVI substantially improves on single-fidelity score-function BBVI using the same optimizers.The framework combines score-function gradient and Hessian estimators, approximate control variates, adaptive reduced-order models, and a regularized diagonal-Newton strategy.Source: §6 · p. 44
  • The study primarily uses a mean-field variational family; richer families are a future direction for representing correlated and non-Gaussian posteriors.Suggested extensions include multivariate Gaussian and neural-network-based variational densities; the authors also identify adaptive sampling, ROM substeps, and improved Newton-like optimization as future work.Source: §6 · p. 44

Appendix A. Derivation of optimal control-variate gains

Appendix A derives control-variate gains by minimizing the trace of estimator covariance, treating full-matrix, diagonal, scalar, and multifidelity approximate gains.

Standard control variates

  • The gain objective is the trace of the estimator covariance, with covariance matrices defined for the high-fidelity and control-variate quantities.The appendix expands this objective and uses the trace identity tr(AΣ_YX) = tr(Σ_XY A⊤).Source: Appendix A · p. 45
  • For a full-matrix gain, the appendix sets the objective's derivative to zero to obtain the optimal gain.Source: Appendix A · p. 45
  • Restricting the gain to a diagonal matrix makes the objective components decouple, yielding a componentwise gain condition.Source: Appendix A · pp. 45–46

Approximate control variates

  • The multifidelity approximate-control-variate estimator uses nested high- and low-fidelity sample sets, with the low-fidelity mean decomposed using samples outside the high-fidelity set.The outside-set average is independent of the high-fidelity means under the stated i.i.d. sampling assumption, and the estimator covariance is then derived.Source: Appendix A · p. 46

Standard control variates

  • For a scalar gain A = αI, the derivation applies separately to mean and log-standard-deviation gradient blocks and to the corresponding same-type Hessian blocks.The appendix obtains the scalar optimum by setting the derivative to zero.Source: Appendix A · p. 47

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