Monte Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

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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Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors

Inverse Problems ◽

10.1088/0266-5611/28/12/125012 ◽

2012 ◽

Vol 28 (12) ◽

pp. 125012 ◽

Cited By ~ 10

Author(s):

Felix Lucka

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Bayesian Inference ◽

Inverse Problems ◽

Monte Carlo Sampling ◽

High Dimensional

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Motion Redirection Based on Markov Chain Monte Carlo Particle Filter

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.668-669.1086 ◽

2014 ◽

Vol 668-669 ◽

pp. 1086-1089

Author(s):

Jin Bao Song ◽

Long Ye ◽

Qin Zhang ◽

Jian Ping Chai

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Particle Filter ◽

Monte Carlo Sampling ◽

Space Time ◽

High Dimensional ◽

Observation Model ◽

Key Frame ◽

The Subject

This paper aims at the difficulty that lack of observation model and high-dimensional sampling in video tooning, proposes a method based on key frame matching and dual-directional Markov chain Monte Carlo sampling of video motion redirection. At first, after extracting the key frame of a given video, By affine transformation and linear superposition, the subject initializes the video’s space-time parameters and forms the observation model; Secondly, in each space-time, based on the bi-directional Markov property of each frame, This paper proposed a dual-directional Markov chain Monte Carlo sampling particle filter structure and takes full advantage of the relationship of the front and back frame of the parameters to estimate motion redirection parameters. At the same time, for high-dimensional sampling problem, the subject according to the directional parameters’ correlation implements classification of skeleton parameters-morphological parameters-physical parameters, proposes a hierarchical genetic strategy to optimize the output parameters and improves the efficiency of the algorithm. The research of this paper will produce an efficient and prominent animation expressive video motion redirection method and play an important role on video animation of the development.

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

Geoscientific Model Development ◽

10.5194/gmd-12-2941-2019 ◽

2019 ◽

Vol 12 (7) ◽

pp. 2941-2960 ◽

Cited By ~ 15

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 multi-modal; 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 a 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–Nicolson proposal outperforms more conventional forms of random walk. Aspects of the problem setup, such as priors on petrophysics and on 3-D geological structure, affect the shape and separation of posterior modes, influencing sampling performance as well as the inversion results. The use of uninformative priors on sensor noise enables optimal weighting among multiple sensors even if noise levels are uncertain.

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QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND

The ANZIAM Journal ◽

10.1017/s1446181112000077 ◽

2011 ◽

Vol 53 (1) ◽

pp. 1-37 ◽

Cited By ~ 46

Author(s):

F. Y. KUO ◽

CH. SCHWAB ◽

I. H. SLOAN

Keyword(s):

Monte Carlo ◽

Hilbert Spaces ◽

Unit Cube ◽

High Dimensional ◽

Equal Weight ◽

Lattice Rules ◽

Quasi Monte Carlo ◽

Strong Focus ◽

The Family ◽

Alternative Settings

AbstractThis paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

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