8 research outputs found
Credible Intervals for Probability of Failure with Gaussian Processes
Efficiently approximating the probability of system failure has gained
increasing importance as expensive simulations begin to play a larger role in
reliability quantification tasks in areas such as structural design, power grid
design, and safety certification among others. This work derives credible
intervals on the probability of failure for a simulation which we assume is a
realizations of a Gaussian process. We connect these intervals to binary
classification error and comment on their applicability to a broad class of
iterative schemes proposed throughout the literature. A novel iterative
sampling scheme is proposed which can suggest multiple samples per batch for
simulations with parallel implementations. We empirically test our scalable,
open-source implementation on a variety simulations including a Tsunami model
where failure is quantified in terms of maximum wave hight
On Bounding and Approximating Functions of Multiple Expectations using Quasi-Monte Carlo
Monte Carlo and Quasi-Monte Carlo methods present a convenient approach for
approximating the expected value of a random variable. Algorithms exist to
adaptively sample the random variable until a user defined absolute error
tolerance is satisfied with high probability. This work describes an extension
of such methods which supports adaptive sampling to satisfy general error
criteria for functions of a common array of expectations. Although several
functions involving multiple expectations are being evaluated, only one random
sequence is required, albeit sometimes of larger dimension than the underlying
randomness. These enhanced Monte Carlo and Quasi-Monte Carlo algorithms are
implemented in the QMCPy Python package with support for economic and parallel
function evaluation. We exemplify these capabilities on problems from machine
learning and global sensitivity analysis
Challenges in Developing Great Quasi-Monte Carlo Software
Quasi-Monte Carlo (QMC) methods have developed over several decades. With the
explosion in computational science, there is a need for great software that
implements QMC algorithms. We summarize the QMC software that has been
developed to date, propose some criteria for developing great QMC software, and
suggest some steps toward achieving great software. We illustrate these
criteria and steps with the Quasi-Monte Carlo Python library (QMCPy), an
open-source community software framework, extensible by design with common
programming interfaces to an increasing number of existing or emerging QMC
libraries developed by the greater community of QMC researchers
Computationally Efficient and Error Aware Surrogate Construction for Numerical Solutions of Subsurface Flow Through Porous Media
Limiting the injection rate to restrict the pressure below a threshold at a
critical location can be an important goal of simulations that model the
subsurface pressure between injection and extraction wells. The pressure is
approximated by the solution of Darcy's partial differential equation (PDE) for
a given permeability field. The subsurface permeability is modeled as a random
field since it is known only up to statistical properties. This induces
uncertainty in the computed pressure. Solving the PDE for an ensemble of random
permeability simulations enables estimating a probability distribution for the
pressure at the critical location. These simulations are computationally
expensive, and practitioners often need rapid online guidance for real-time
pressure management. An ensemble of numerical PDE solutions is used to
construct a Gaussian process regression model that can quickly predict the
pressure at the critical location as a function of the extraction rate and
permeability realization.
Our first novel contribution is to identify a sampling methodology for the
random environment and matching kernel technology for which fitting the
Gaussian process regression model scales as O(n log n) instead of the typical
O(n^3) rate in the number of samples n used to fit the surrogate. The surrogate
model allows almost instantaneous predictions for the pressure at the critical
location as a function of the extraction rate and permeability realization. Our
second contribution is a novel algorithm to calibrate the uncertainty in the
surrogate model to the discrepancy between the true pressure solution of
Darcy's equation and the numerical solution. Although our method is derived for
building a surrogate for the solution of Darcy's equation with a random
permeability field, the framework broadly applies to solutions of other PDE
with random coefficients.Comment: 20 pages, 8 figures, 1 tabl
Covariant Action for the Super-Five-Brane of M-Theory
We propose a complete, d=6 covariant and kappa-symmetric, action for an
M-theory five-brane propagating in D=11 supergravity background.Comment: LaTeX file, 5 pages, misprints corrected, comments and references
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