Importance Sampling: Intrinsic Dimension and Computational Cost

Agapiou, Sergios; Papaspiliopoulos, Omiros; Sanz-Alonso, Daniel; Stuart, Andrew M.

doi:10.1214/17-sts611

Cited by 125 publications

(182 citation statements)

References 100 publications

(162 reference statements)

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“…The physical interpretation of (49) is that the scaling of the error is controlled by the variance of the weightŝ W . There are two contributing factors to this variance, which are the randomness in the Jarzynski integration and the fact that C i is distributed as p c .…”

Section: Mean Square Errormentioning

confidence: 99%

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model

Kobayashi

Rohrbach

Scheichl

et al. 2019

The Journal of Chemical Physics

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We present a method that exploits self-consistent simulation of coarse-grained and fine-grained models, in order to analyse properties of physical systems. The method uses the coarse-grained model to obtain a first estimate of the quantity of interest, before computing a correction by analysing properties of the fine system. We illustrate the method by applying it to the Asakura-Oosawa (AO) model of colloid-polymer mixtures. We show that the liquid-vapour critical point in that system is affected by three-body interactions which are neglected in the corresponding coarse-grained model. We analyse the size of this effect and the nature of the three-body interactions. We also analyse the accuracy of the method, as a function of the associated computational effort. arXiv:1907.07912v1 [cond-mat.stat-mech]

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Section: Mean Square Errormentioning

confidence: 99%

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model

Kobayashi

Rohrbach

Scheichl

et al. 2019

The Journal of Chemical Physics

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“…In [14], Novak and Mathé prove that it is optimal on a certain class of tuples (f, u). However, recently this Monte Carlo approach attracted considerable attention, let us mention here [1,4]. In particular, in [1] upper error bounds not only for bounded functions f are provided and the relevance of the method for inverse problems is presented.…”

Section: Computementioning

confidence: 99%

A weighted discrepancy bound of quasi-Monte Carlo importance sampling

Dick

Rudolf

Zhu

2019

Statistics & Probability Letters

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“…However, since this is not illustrative for the reader we consider the random field at two fixed points in the spatial domain; these points are x (1) = (0.5, 0.5) and x (2) = (0.75, 0.25). Before looking at the KS distances of the distributions of θ N sto (x (1) ) and θ N sto (x (2) ) we assess their posterior mean estimates. The relative error of the posterior means in these points compared to the true values θ true (x (1) ) and θ true (x (2) ) is given in Figure 5.13.…”

Section: Posterior Approximation In High Dimensionsmentioning

confidence: 99%

“…Before looking at the KS distances of the distributions of θ N sto (x (1) ) and θ N sto (x (2) ) we assess their posterior mean estimates. The relative error of the posterior means in these points compared to the true values θ true (x (1) ) and θ true (x (2) ) is given in Figure 5.13. While the estimate of θ N sto (x (1) ) is quite accurate, the estimate of θ N sto (x (2) ) is very inaccurate -consistently in both methods.…”

Section: Posterior Approximation In High Dimensionsmentioning

confidence: 99%

“…The relative error of the posterior means in these points compared to the true values θ true (x (1) ) and θ true (x (2) ) is given in Figure 5.13. While the estimate of θ N sto (x (1) ) is quite accurate, the estimate of θ N sto (x (2) ) is very inaccurate -consistently in both methods. This is consistent with the plots of the posterior means in Figure 5.12.…”

Section: Posterior Approximation In High Dimensionsmentioning

confidence: 99%

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Multilevel Sequential2 Monte Carlo for Bayesian inverse problems

Latz

Papaioannou²,

Ullmann

2018

Journal of Computational Physics

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The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis, J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.In contrast to deterministic regularisation techniques, the Bayesian approach to inverse problems uses the probabilistic framework of Bayesian inference. Bayesian inference is built on Bayes' Formula in the formulation given by Laplace [34, II.1]. We remark that other formulations are possible, see e.g. the work by Matthies et al. [38]. We make use of the mathematical framework for Bayesian Inverse Problems (BIPs) given by Stuart [48]. Under weak assumptions -which we will give below -one can show that the BIP is well-posed. The solution of the BIP is the conditional probability measure of the unknown parameter given the observations.The Bayesian framework is very general and can handle different types of forward models. However, in this work we consider PDE-based forward models, and in particular an elliptic PDE. The exact solution of the associated BIP is often inaccessible for two reasons: (i) there is no closed form expression for the posterior measure, and (ii) the underlying PDE cannot be solved analytically. We focus on (i), and study efficient approximations to the full posterior measure. Alternatively, one could also only approximate the expectation of output quantities of interest with respect to the posterior measure, or estimate the model evidence, the normalization constant of the posterior measure.Typically, BIPs are approached with sampling based methods, such as Markov Chain Monte Carlo (M...

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Importance Sampling: Intrinsic Dimension and Computational Cost

Cited by 125 publications

References 100 publications

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model

A weighted discrepancy bound of quasi-Monte Carlo importance sampling

Multilevel Sequential2 Monte Carlo for Bayesian inverse problems

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