100 research outputs found
Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
For big data analysis, high computational cost for Bayesian methods often
limits their applications in practice. In recent years, there have been many
attempts to improve computational efficiency of Bayesian inference. Here we
propose an efficient and scalable computational technique for a
state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian
Monte Carlo (HMC). The key idea is to explore and exploit the structure and
regularity in parameter space for the underlying probabilistic model to
construct an effective approximation of its geometric properties. To this end,
we build a surrogate function to approximate the target distribution using
properly chosen random bases and an efficient optimization process. The
resulting method provides a flexible, scalable, and efficient sampling
algorithm, which converges to the correct target distribution. We show that by
choosing the basis functions and optimization process differently, our method
can be related to other approaches for the construction of surrogate functions
such as generalized additive models or Gaussian process models. Experiments
based on simulated and real data show that our approach leads to substantially
more efficient sampling algorithms compared to existing state-of-the art
methods
A surrogate modelling approach based on nonlinear dimension reduction for uncertainty quantification in groundwater flow models
In this paper, we develop a surrogate modelling approach for capturing the output field (e.g., the pressure head) from groundwater flow models involving a stochastic input field (e.g., the hy- draulic conductivity). We use a Karhunen-Lo`eve expansion for a log-normally distributed input field, and apply manifold learning (local tangent space alignment) to perform Gaussian process Bayesian inference using Hamiltonian Monte Carlo in an abstract feature space, yielding outputs for arbitrary unseen inputs. We also develop a framework for forward uncertainty quantification in such problems, including analytical approximations of the mean of the marginalized distri- bution (with respect to the inputs). To sample from the distribution we present Monte Carlo approach. Two examples are presented to demonstrate the accuracy of our approach: a Darcy flow model with contaminant transport in 2-d and a Richards equation model in 3-d
The Bayesian Formulation of EIT: Analysis and Algorithms
We provide a rigorous Bayesian formulation of the EIT problem in an infinite
dimensional setting, leading to well-posedness in the Hellinger metric with
respect to the data. We focus particularly on the reconstruction of binary
fields where the interface between different media is the primary unknown. We
consider three different prior models - log-Gaussian, star-shaped and level
set. Numerical simulations based on the implementation of MCMC are performed,
illustrating the advantages and disadvantages of each type of prior in the
reconstruction, in the case where the true conductivity is a binary field, and
exhibiting the properties of the resulting posterior distribution.Comment: 30 pages, 10 figure
Markov chain Monte Carlo with Gaussian processes for fast parameter estimation and uncertainty quantification in a 1D fluidâdynamics model of the pulmonary circulation
The past few decades have witnessed an explosive synergy between physics and the life sciences. In particular, physical modelling in medicine and physiology is a topical research area. The present work focuses on parameter inference and uncertainty quantification in a 1D fluidâdynamics model for quantitative physiology: the pulmonary blood circulation. The practical challenge is the estimation of the patientâspecific biophysical model parameters, which cannot be measured directly. In principle this can be achieved based on a comparison between measured and predicted data. However, predicting data requires solving a system of partial differential equations (PDEs), which usually have no closedâform solution, and repeated numerical integrations as part of an adaptive estimation procedure are computationally expensive. In the present article, we demonstrate how fast parameter estimation combined with sound uncertainty quantification can be achieved by a combination of statistical emulation and Markov chain Monte Carlo (MCMC) sampling. We compare a range of stateâofâtheâart MCMC algorithms and emulation strategies, and assess their performance in terms of their accuracy and computational efficiency. The longâterm goal is to develop a method for reliable disease prognostication in real time, and our work is an important step towards an automatic clinical decision support system
Inverse Problems in a Bayesian Setting
In a Bayesian setting, inverse problems and uncertainty quantification (UQ)
--- the propagation of uncertainty through a computational (forward) model ---
are strongly connected. In the form of conditional expectation the Bayesian
update becomes computationally attractive. We give a detailed account of this
approach via conditional approximation, various approximations, and the
construction of filters. Together with a functional or spectral approach for
the forward UQ there is no need for time-consuming and slowly convergent Monte
Carlo sampling. The developed sampling-free non-linear Bayesian update in form
of a filter is derived from the variational problem associated with conditional
expectation. This formulation in general calls for further discretisation to
make the computation possible, and we choose a polynomial approximation. After
giving details on the actual computation in the framework of functional or
spectral approximations, we demonstrate the workings of the algorithm on a
number of examples of increasing complexity. At last, we compare the linear and
nonlinear Bayesian update in form of a filter on some examples.Comment: arXiv admin note: substantial text overlap with arXiv:1312.504
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