Randomize-Then-Optimize: A Method for Sampling from Posterior Distributions in Nonlinear Inverse Problems

Bardsley, Johnathan M.; Solonen, Antti; Haario, Heikki; Laine, Marko

doi:10.1137/140964023

Cited by 116 publications

(169 citation statements)

References 15 publications

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“…The Knothe-Rosenblatt rearrangement is also monotone increasing according to the lexicographic order on R n [64]. It turns out that we can further constrain (8) so that the Knothe-Rosenblatt rearrangement is the unique global minimizer of:…”

Section: Optimization Problemsmentioning

confidence: 99%

“…The constraint ∇T 0 suffices to enforce invertibility of a feasible triangular map. (9) is a far better behaved optimization problem than the original formulation (8). Hence, for the rest of this section we will focus on the computation of a Knothe-Rosenblatt rearrangement by solving (9).…”

Section: T ∈ Tmentioning

confidence: 99%

“…Another optimization-based sampling algorithm, similar in spirit to implicit sampling though different in construction, is the randomize-then-optimize (RTO) approach of [8]. This scheme is well-defined for target distributions whose log-densities can be written in a particular quadratic form following a transformation of the parameters, with some restrictions on the target's degree of nonGaussianity.…”

Section: Related Workmentioning

confidence: 99%

“…This problem is a popular and interesting test case for sampling methods (e.g., MCMC [67], RTO [8]). The time-dependent forward model is defined by…”

Section: Example: Biochemical Oxygen Demand Modelmentioning

confidence: 99%

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Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

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We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling-i.e., a transport map-between a complex "target" probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

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Section: Optimization Problemsmentioning

confidence: 99%

Section: T ∈ Tmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

“…This problem is a popular and interesting test case for sampling methods (e.g., MCMC [67], RTO [8]). The time-dependent forward model is defined by…”

Section: Example: Biochemical Oxygen Demand Modelmentioning

confidence: 99%

See 2 more Smart Citations

Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

View full text Add to dashboard Cite

show abstract

“…And, while 100 the present papers were finalized, the authors became aware of a paper by Bardsley et al (2014). These authors introduce a method, which they call Randomize-then-Optimize (RTO), which produces an ensemble of estimates through independent minimizations that take nonlinearity into account through the Jacobian of the model.…”

mentioning

confidence: 99%

Ensemble Variational Assimilation as a Probabilistic Estimator. Part I: The linear and weak non-linear case

Jardak¹,

Talagrand²

2018

Preprint

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Abstract.Data assimilation is considered as a problem in Bayesian estimation, viz. determine the probability distribution for the state of the observed system, conditioned by the available data. In the linear and additive Gaussian case, a Monte-Carlo sample of the Bayesian probability distribution (which is Gaussian and known explicitly) can be obtained by a simple procedure : perturb the data according to the probability 5 distribution of their own errors, and perform an assimilation on the perturbed data. The performance of that approach, called Ensemble Variational Assimilation (EnsVAR), is studied in the two parts of the paper on the non-linear low-dimensional Lorenz-96 chaotic system, the assimilation being performed by the standard variational procedure. In Part I, EnsVAR is implemented first, for reference, in a linear and Gaussian case, and then in a weakly non-linear case (assimilation over 5 days of the system). The perfor-10 mances of the algorithm, considered as a statistical estimator, are very similar in the two cases. Additional comparison shows that the performance of EnsVAR is better, both in the assimilation and forecast phases, than that of standard algorithms for Ensemble Kalman Filter and Particle Filter (although at a higher cost). Globally similar results are obtained with the Kuramoto-Sivashinsky equation.

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Graben formation and dike arrest during the 2009 Harrat Lunayyir dike intrusion in Saudi Arabia: Insights from InSAR, stress calculations and analog experiments

Jónsson

Corbi

et al. 2016

JGR Solid Earth

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Detailed spatial and temporal accounts of propagating dikes from crustal deformation data, including their interplay with faulting, are rare, leaving many questions about how this interplay affects graben formation and the arrest of dikes unanswered. Here we use interferometric synthetic aperture radar (InSAR) observations, stress calculations, and analog experiments to investigate the interaction between an intruding dike and normal faulting during the 2009 Harrat Lunayyir dike intrusion in western Saudi Arabia. We generated five displacement maps from InSAR data to unravel the temporal evolution of deformation covering the majority of the intrusion. We find that the observed surface displacements can be modeled by a ~2 m thick dike with an upper tip ~2 km below the surface on 16 May 2009, 4 weeks after the onset of seismic unrest. In the following three days, the dike propagated to within ~1 km of the surface with graben‐bounding normal faulting dominating the near‐field deformation. The volume of the dike doubled between mid‐May and mid‐June. We carried out analog experiments that indicate that the wedge‐shaped graben grew outward with the faulting style changing progressively from normal faulting to oblique. Coulomb failure stress change calculations show that the intruding dike caused two zones of shallow horizontal tension on both sides of the dike, producing two zones of fissuring and normal faulting at the surface. In return, the faulting provoked compression around the upper tip of the dike, holding back its vertical propagation.

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Randomize-Then-Optimize: A Method for Sampling from Posterior Distributions in Nonlinear Inverse Problems

Cited by 116 publications

References 15 publications

Sampling via Measure Transport: An Introduction

Sampling via Measure Transport: An Introduction

Ensemble Variational Assimilation as a Probabilistic Estimator. Part I: The linear and weak non-linear case

Graben formation and dike arrest during the 2009 Harrat Lunayyir dike intrusion in Saudi Arabia: Insights from InSAR, stress calculations and analog experiments

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