Sampling from rough energy landscapes

Plecháč, Petr; Simpson, Gideon

doi:10.4310/cms.2020.v18.n8.a9

Cited by 3 publications

(4 citation statements)

References 22 publications

(38 reference statements)

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“…The efficiency of sampling algorithms depends strongly on the landscape of the pdf to be sampled and on the goal of the sampling. If the objective is simply to sample in the neighborhood of the maximum a posteriori point, then using exact gradients are not always beneficial, especially if the log posterior is characterized by multiple scales -a smooth, long range feature that is approximately quadratic and shorter range fluctuations to the surface [30]. If the posterior pdf is characterized, however, by a small number of nearly equivalent modes, then ensemble methods may fail to converge [25,13].…”

Section: Monotonic Log-permeability Transformmentioning

confidence: 99%

Weighted RML using ensemble-methods for data assimilation

Ba¹,

Oliver²

2023

Preprint

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The weighting of critical-point samples in the weighted randomized maximum likelihood method depend on the magnitude of the data mismatch at the critical points and on the Jacobian of the transformation from the prior density to the proposal density. When standard iterative ensemble smoothers are applied for data assimilation, the Jacobian is identical for all samples. If a hybrid data assimilation method is applied, however, there is the possibility that each ensemble member can have a distinct Jacobian and that the posterior density can be multimodal. In order to apply a hybrid method iterative ensemble smoother, it is necessary that a part of the transformation from the prior Gaussian random variable to the data be analytic. Examples might include analytic transformation from a latent Gaussian random variable to permeability followed by a black-box transformation from permeability to state variables in porous media flow, or a Gaussian hierarchical model for variables followed by a similar black-box transformation from permeability to state variables.In this paper, we investigate the application of weighting to both types of examples.

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Section: Monotonic Log-permeability Transformmentioning

confidence: 99%

Weighted RML using ensemble-methods for data assimilation

Ba¹,

Oliver²

2023

Preprint

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show abstract

“…(The last expression above is obtained by using the substitution u = sx.) It is convenient to focus on potentials (the log-marginal target probability densities), which are given by (26) V x|B (H ) ; c = log π x|B (H ) ; c = − log R ξ u|B (H ) ; c du −…”

Section: Application To Metropolis Adjusted Langevin Algorithmsmentioning

confidence: 99%

Section: Proof Property (I) Holds By Definition Sincementioning

confidence: 99%

“…In light of our theoretical and numerical results it is sensible to argue that classic MCMC algorithms do not really work well for rough or oscillatory targets, and that one should seek appropriate modifications. Suggestions for such modifications also already exist [26]. Such modifications could usefully be assessed in terms of optimally tuned RWM and MALA algorithms providing benchmark MCMC methods.…”

Section: Further Work and Open Questionsmentioning

confidence: 99%

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Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities

Vogrinc

Kendall

2021

Ann. Appl. Probab.

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For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n −1 and as n − 1 3 with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of C 1 (R) for RWM and C 3 (R) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H . For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as n − 1 H and as n − 1 2+H and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis-Hastings algorithms (possibly depending on a random environment).

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Importance Weighting in Hybrid Iterative Ensemble Smoothers for Data Assimilation

Ba,

Oliver

2024

Math Geosci

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Because it is generally impossible to completely characterize the uncertainty in complex model variables after assimilation of data, it is common to approximate the uncertainty by sampling from approximations of the posterior distribution for model variables. When minimization methods are used for the sampling, the weights on each of the samples depend on the magnitude of the data mismatch at the critical points and on the Jacobian of the transformation from the prior density to the sample proposal density. For standard iterative ensemble smoothers, the Jacobian is identical for all samples, and the weights depend only on the data mismatch. In this paper, a hybrid data assimilation method is proposed which makes it possible for each ensemble member to have a distinct Jacobian and for the approximation to the posterior density to be multimodal. For the proposed hybrid iterative ensemble smoother, it is necessary that a part of the mapping from the prior Gaussian random variable to the data be analytic. Examples might include analytic transformation from a latent Gaussian random variable to permeability followed by a black-box transformation from permeability to state variables in porous media flow, or a Gaussian hierarchical model for variables followed by a similar black-box transformation from permeability to state variables. In this paper, the application of weighting to both hybrid and standard iterative ensemble smoothers is investigated using a two-dimensional, two-phase flow problem in porous media with various degrees of nonlinearity. As expected, the weights in a standard iterative ensemble smoother become degenerate for problems with large amounts of data. In the examples, however, the weights for the hybrid iterative ensemble smoother were useful for improving forecast reliability.

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Sampling from rough energy landscapes

Cited by 3 publications

References 22 publications

Weighted RML using ensemble-methods for data assimilation

Weighted RML using ensemble-methods for data assimilation

Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities

Importance Weighting in Hybrid Iterative Ensemble Smoothers for Data Assimilation

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