Introduction to Hastings (1970) Monte Carlo Sampling Methods Using Markov Chains and Their Applications

1997 ◽  
pp. 235-256 ◽  
Author(s):  
T. L. Lai
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Sampling Methods ◽  
Monte Carlo Sampling

Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography

2000 ◽  
Vol 16 (5) ◽  
pp. 1487-1522 ◽  
Author(s):  
Jari P Kaipio ◽  
Ville Kolehmainen ◽  
Erkki Somersalo ◽  
Marko Vauhkonen
Keyword(s):  
Monte Carlo ◽  
Electrical Impedance Tomography ◽  
Electrical Impedance ◽  
Sampling Methods ◽  
Monte Carlo Sampling ◽  
Impedance Tomography

Examining improved experimental designs for wind tunnel testing using Monte Carlo sampling methods

2010 ◽  
Vol 27 (6) ◽  
pp. 795-803 ◽  
Author(s):  
Raymond R. Hill ◽  
Derek A. Leggio ◽  
Shay R. Capehart ◽  
August G. Roesener
Keyword(s):  
Monte Carlo ◽  
Wind Tunnel ◽  
Sampling Methods ◽  
Monte Carlo Sampling ◽  
Experimental Designs ◽  
Wind Tunnel Testing ◽  
Tunnel Testing

scholarly journals A comparison of Monte Carlo sampling methods for metabolic network models

PLoS ONE ◽  
2020 ◽  
Vol 15 (7) ◽  
pp. e0235393
Author(s):  
Shirin Fallahi ◽  
Hans J. Skaug ◽  
Guttorm Alendal
Keyword(s):  
Monte Carlo ◽  
Metabolic Network ◽  
Sampling Methods ◽  
Network Models ◽  
Monte Carlo Sampling

scholarly journals Seismic Tomography Using Variational Inference Methods

2020 ◽  
Author(s):  
Xin Zhang ◽  
Andrew Curtis
Keyword(s):  
Monte Carlo ◽  
Probability Distribution ◽  
Seismic Tomography ◽  
Posterior Probability ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Monte Carlo Sampling ◽  
Variational Inference ◽  
Posterior Probability Distribution ◽  
Inference Methods

<p><span>In a variety of geoscientific applications we require maps of subsurface properties together with the corresponding maps of uncertainties to assess their reliability. Seismic tomography is a method that is widely used to generate those maps. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large data sets and high-dimensionality parameter spaces. To extend uncertainty analysis to larger systems, we introduce variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem yet still provide fully nonlinear, probabilistic results. This is achieved by minimizing the Kullback-Leibler (KL) divergence between approximate and target probability distributions within a predefined family of probability distributions.</span></p><p><span>We introduce two variational inference methods: automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD). In ADVI a Gaussian probability distribution is assumed and optimized to approximate the posterior probability distribution. In SVGD a smooth transform is iteratively applied to an initial probability distribution to obtain an approximation to the posterior probability distribution. At each iteration the transform is determined by seeking the steepest descent direction that minimizes the KL-divergence. </span></p><p><span>We apply the two variational inference methods to 2D travel time tomography using both synthetic and real data, and compare the results to those obtained from two different Monte Carlo sampling methods: Metropolis-Hastings Markov chain Monte Carlo (MH-McMC) and reversible jump Markov chain Monte Carlo (rj-McMC). The results show that ADVI provides a biased approximation because of its Gaussian approximation, whereas SVGD produces more accurate approximations to the results of MH-McMC. In comparison rj-McMC produces smoother mean velocity models and lower standard deviations because the parameterization used in rj-McMC (Voronoi cells) imposes prior restrictions on the pixelated form of models: all pixels within each Voronoi cell have identical velocities. This suggests that the results of rj-McMC need to be interpreted in the light of the specific prior information imposed by the parameterization. Both variational methods estimate the posterior distribution at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We therefore expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.</span></p>

scholarly journals Efficiency and robustness in Monte Carlo sampling of 3-D geophysical inversions with Obsidian v0.1.2: Setting up for success

2019 ◽  
Author(s):  
Richard Scalzo ◽  
David Kohn ◽  
Hugo Olierook ◽  
Gregory Houseman ◽  
Rohitash Chandra ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Adaptive Sampling ◽  
Sampling Methods ◽  
South Australia ◽  
Monte Carlo Sampling ◽  
Dimensional Structure ◽  
High Dimensional ◽  
Dimensional Parameter

Abstract. The rigorous quantification of uncertainty in geophysical inversions is a challenging problem. Inversions are often ill-posed and the likelihood surface may be multimodal; properties of any single mode become inadequate uncertainty measures, and sampling methods become inefficient for irregular posteriors or high-dimensional parameter spaces. We explore the influences of different choices made by the practitioner on the efficiency and accuracy of Bayesian geophysical inversion methods that rely on Markov chain Monte Carlo sampling to assess uncertainty, using a multi-sensor inversion of the three-dimensional structure and composition of a region in the Cooper Basin of South Australia as a case study. The inversion is performed using an updated version of the Obsidian distributed inversion software. We find that the posterior for this inversion has complex local covariance structure, hindering the efficiency of adaptive sampling methods that adjust the proposal based on the chain history. Within the context of a parallel-tempered Markov chain Monte Carlo scheme for exploring high-dimensional multi-modal posteriors, a preconditioned Crank-Nicholson proposal outperforms more conventional forms of random walk. Aspects of the problem setup, such as priors on petrophysics or on 3-D geological structure, affect the shape and separation of posterior modes, influencing sampling performance as well as the inversion results. Use of uninformative priors on sensor noise can improve inversion results by enabling optimal weighting among multiple sensors even if noise levels are uncertain. Efficiency could be further increased by using posterior gradient information within proposals, which Obsidian does not currently support, but which could be emulated using posterior surrogates.

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