174 research outputs found
A surrogate accelerated multicanonical Monte Carlo method for uncertainty quantification
In this work we consider a class of uncertainty quantification problems where
the system performance or reliability is characterized by a scalar parameter
. The performance parameter is random due to the presence of various
sources of uncertainty in the system, and our goal is to estimate the
probability density function (PDF) of . We propose to use the multicanonical
Monte Carlo (MMC) method, a special type of adaptive importance sampling
algorithm, to compute the PDF of interest. Moreover, we develop an adaptive
algorithm to construct local Gaussian process surrogates to further accelerate
the MMC iterations. With numerical examples we demonstrate that the proposed
method can achieve several orders of magnitudes of speedup over the standard
Monte Carlo method
A subset multicanonical Monte Carlo method for simulating rare failure events
Estimating failure probabilities of engineering systems is an important
problem in many engineering fields. In this work we consider such problems
where the failure probability is extremely small (e.g ). In this
case, standard Monte Carlo methods are not feasible due to the extraordinarily
large number of samples required. To address these problems, we propose an
algorithm that combines the main ideas of two very powerful failure probability
estimation approaches: the subset simulation (SS) and the multicanonical Monte
Carlo (MMC) methods. Unlike the standard MMC which samples in the entire domain
of the input parameter in each iteration, the proposed subset MMC algorithm
adaptively performs MMC simulations in a subset of the state space and thus
improves the sampling efficiency. With numerical examples we demonstrate that
the proposed method is significantly more efficient than both of the SS and the
MMC methods. Moreover, the proposed algorithm can reconstruct the complete
distribution function of the parameter of interest and thus can provide more
information than just the failure probabilities of the systems
A Derivative-Free Trust-Region Algorithm for Reliability-Based Optimization
In this note, we present a derivative-free trust-region (TR) algorithm for
reliability based optimization (RBO) problems. The proposed algorithm consists
of solving a set of subproblems, in which simple surrogate models of the
reliability constraints are constructed and used in solving the subproblems.
Taking advantage of the special structure of the RBO problems, we employ a
sample reweighting method to evaluate the failure probabilities, which
constructs the surrogate for the reliability constraints by performing only a
single full reliability evaluation in each iteration. With numerical
experiments, we illustrate that the proposed algorithm is competitive against
existing methods
Gaussian process surrogates for failure detection: a Bayesian experimental design approach
An important task of uncertainty quantification is to identify {the
probability of} undesired events, in particular, system failures, caused by
various sources of uncertainties. In this work we consider the construction of
Gaussian {process} surrogates for failure detection and failure probability
estimation. In particular, we consider the situation that the underlying
computer models are extremely expensive, and in this setting, determining the
sampling points in the state space is of essential importance. We formulate the
problem as an optimal experimental design for Bayesian inferences of the limit
state (i.e., the failure boundary) and propose an efficient numerical scheme to
solve the resulting optimization problem. In particular, the proposed
limit-state inference method is capable of determining multiple sampling points
at a time, and thus it is well suited for problems where multiple computer
simulations can be performed in parallel. The accuracy and performance of the
proposed method is demonstrated by both academic and practical examples
On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences
Many scientific and engineering problems require to perform Bayesian
inferences for unknowns of infinite dimension. In such problems, many standard
Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh
refinement, which is referred to as being dimension dependent. To this end, a
family of dimensional independent MCMC algorithms, known as the preconditioned
Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional
parameters. In this work we develop an adaptive version of the pCN algorithm,
where the covariance operator of the proposal distribution is adjusted based on
sampling history to improve the simulation efficiency. We show that the
proposed algorithm satisfies an important ergodicity condition under some mild
assumptions. Finally we provide numerical examples to demonstrate the
performance of the proposed method
A hybrid adaptive MCMC algorithm in function spaces
The preconditioned Crank-Nicolson (pCN) method is a Markov Chain Monte Carlo
(MCMC) scheme, specifically designed to perform Bayesian inferences in function
spaces. Unlike many standard MCMC algorithms, the pCN method can preserve the
sampling efficiency under the mesh refinement, a property referred to as being
dimension independent. In this work we consider an adaptive strategy to further
improve the efficiency of pCN. In particular we develop a hybrid adaptive MCMC
method: the algorithm performs an adaptive Metropolis scheme in a chosen finite
dimensional subspace, and a standard pCN algorithm in the complement space of
the chosen subspace. We show that the proposed algorithm satisfies certain
important ergodicity conditions. Finally with numerical examples we demonstrate
that the proposed method has competitive performance with existing adaptive
algorithms.Comment: arXiv admin note: text overlap with arXiv:1511.0583
On Estimating the Gradient of the Expected Information Gain in Bayesian Experimental Design
Bayesian Experimental Design (BED), which aims to find the optimal
experimental conditions for Bayesian inference, is usually posed as to optimize
the expected information gain (EIG). The gradient information is often needed
for efficient EIG optimization, and as a result the ability to estimate the
gradient of EIG is essential for BED problems. The primary goal of this work is
to develop methods for estimating the gradient of EIG, which, combined with the
stochastic gradient descent algorithms, result in efficient optimization of
EIG. Specifically, we first introduce a posterior expected representation of
the EIG gradient with respect to the design variables. Based on this, we
propose two methods for estimating the EIG gradient, UEEG-MCMC that leverages
posterior samples generated through Markov Chain Monte Carlo (MCMC) to estimate
the EIG gradient, and BEEG-AP that focuses on achieving high simulation
efficiency by repeatedly using parameter samples. Theoretical analysis and
numerical studies illustrate that UEEG-MCMC is robust agains the actual EIG
value, while BEEG-AP is more efficient when the EIG value to be optimized is
small. Moreover, both methods show superior performance compared to several
popular benchmarks in our numerical experiments
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