2,045 research outputs found
APPROXIMATION OF LIMIT STATE SURFACES IN MONOTONIC MONTE CARLO SETTINGS
International audienceThis article investigates the theoretical convergence properties of the estimators produced by a numerical exploration of a monotonic function with multivariate random inputs in a structural reliability framework.The quantity to be estimated is a probability typically associated to an undesirable (unsafe) event and the function is usually implemented as a computer model. The estimators produced by a Monte Carlo numerical design are two subsets of inputs leading to safe and unsafe situations, the measures of which can be traduced as deterministic bounds for the probability. Several situations are considered, when the design is independent, identically distributed or not, or sequential. As a major consequence, a consistent estimator of the (limit state) surface separating the subsets under isotonicity and regularity arguments can be built, and its convergence speed can be exhibited. This estimator is built by aggregating semi-supervized binary classifiers chosen as constrained Support Vector Machines. Numerical experiments conducted on toy examples highlight that they work faster than recently developed monotonic neural networks with comparable predictable power. They are therefore more adapted when the computational time is a key issue
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Sequential Design for Gaussian Process Surrogates in Noisy Level Set Estimation
We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels and sequential design frameworks. Our focus is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. We introduce the use of four GP-based metamodels in level set estimation that are robust to noise misspecification, and evaluate the performance of them. In conjunction with these metamodels, we develop several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions for the proposed metamodels. To expedite sequential design in stochastic experiments, we also develop adaptive batching designs, which are natural extensions of sequential design heuristics with the benefit of replication growing as response features are learned, inputs concentrate, and the metamodeling overhead rises. We develop four novel schemes that simultaneously or sequentially determine the sequential design inputs and the respective number of replicates. Our schemes are benchmarked by using synthetic examples and an application in quantitative finance (Bermudan option pricing)
Failure Probability Estimation and Detection of Failure Surfaces via Adaptive Sequential Decomposition of the Design Domain
We propose an algorithm for an optimal adaptive selection of points from the
design domain of input random variables that are needed for an accurate
estimation of failure probability and the determination of the boundary between
safe and failure domains. The method is particularly useful when each
evaluation of the performance function g(x) is very expensive and the function
can be characterized as either highly nonlinear, noisy, or even discrete-state
(e.g., binary). In such cases, only a limited number of calls is feasible, and
gradients of g(x) cannot be used. The input design domain is progressively
segmented by expanding and adaptively refining mesh-like lock-free geometrical
structure. The proposed triangulation-based approach effectively combines the
features of simulation and approximation methods. The algorithm performs two
independent tasks: (i) the estimation of probabilities through an ingenious
combination of deterministic cubature rules and the application of the
divergence theorem and (ii) the sequential extension of the experimental design
with new points. The sequential selection of points from the design domain for
future evaluation of g(x) is carried out through a new learning function, which
maximizes instantaneous information gain in terms of the probability
classification that corresponds to the local region. The extension may be
halted at any time, e.g., when sufficiently accurate estimations are obtained.
Due to the use of the exact geometric representation in the input domain, the
algorithm is most effective for problems of a low dimension, not exceeding
eight. The method can handle random vectors with correlated non-Gaussian
marginals. The estimation accuracy can be improved by employing a smooth
surrogate model. Finally, we define new factors of global sensitivity to
failure based on the entire failure surface weighted by the density of the
input random vector.Comment: 42 pages, 24 figure
Active Mean Fields for Probabilistic Image Segmentation: Connections with Chan-Vese and Rudin-Osher-Fatemi Models
Segmentation is a fundamental task for extracting semantically meaningful
regions from an image. The goal of segmentation algorithms is to accurately
assign object labels to each image location. However, image-noise, shortcomings
of algorithms, and image ambiguities cause uncertainty in label assignment.
Estimating the uncertainty in label assignment is important in multiple
application domains, such as segmenting tumors from medical images for
radiation treatment planning. One way to estimate these uncertainties is
through the computation of posteriors of Bayesian models, which is
computationally prohibitive for many practical applications. On the other hand,
most computationally efficient methods fail to estimate label uncertainty. We
therefore propose in this paper the Active Mean Fields (AMF) approach, a
technique based on Bayesian modeling that uses a mean-field approximation to
efficiently compute a segmentation and its corresponding uncertainty. Based on
a variational formulation, the resulting convex model combines any
label-likelihood measure with a prior on the length of the segmentation
boundary. A specific implementation of that model is the Chan-Vese segmentation
model (CV), in which the binary segmentation task is defined by a Gaussian
likelihood and a prior regularizing the length of the segmentation boundary.
Furthermore, the Euler-Lagrange equations derived from the AMF model are
equivalent to those of the popular Rudin-Osher-Fatemi (ROF) model for image
denoising. Solutions to the AMF model can thus be implemented by directly
utilizing highly-efficient ROF solvers on log-likelihood ratio fields. We
qualitatively assess the approach on synthetic data as well as on real natural
and medical images. For a quantitative evaluation, we apply our approach to the
icgbench dataset
Bayesian Modelling Approaches for Quantum States -- The Ultimate Gaussian Process States Handbook
Capturing the correlation emerging between constituents of many-body systems
accurately is one of the key challenges for the appropriate description of
various systems whose properties are underpinned by quantum mechanical
fundamentals. This thesis discusses novel tools and techniques for the
(classical) modelling of quantum many-body wavefunctions with the ultimate goal
to introduce a universal framework for finding accurate representations from
which system properties can be extracted efficiently. It is outlined how
synergies with standard machine learning approaches can be exploited to enable
an automated inference of the most relevant intrinsic characteristics through
rigorous Bayesian regression techniques. Based on the probabilistic framework
forming the foundation of the introduced ansatz, coined the Gaussian Process
State, different compression techniques are explored to extract numerically
feasible representations of relevant target states within stochastic schemes.
By following intuitively motivated design principles, the resulting model
carries a high degree of interpretability and offers an easily applicable tool
for the numerical study of quantum systems, including ones which are
notoriously difficult to simulate due to a strong intrinsic correlation. The
practical applicability of the Gaussian Process States framework is
demonstrated within several benchmark applications, in particular, ground state
approximations for prototypical quantum lattice models, Fermi-Hubbard models
and models, as well as simple ab-initio quantum chemical systems.Comment: PhD Thesis, King's College London, 202 page
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