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by Juan Chiachio-Ruano, Manuel Chiachio-Ruano, Shankar Sankararaman
Bayesian Inverse Problems
Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
List of Figures
List of Tables
Contributors
Part I Fundamentals
Part II Engineering Applications
Appendices
Bibliography
Index
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List of Tables
List of Figures
1.1 Illustration of the stochastic embedding process represented in Eq. (1.17).
1.2 Illustrative example of different model classes consistent with the data.
1.3 Illustration of the prior and posterior information of model parameters.
1.4 Illustration of stochastic simulation for Bayesian inference.
1.5 Example of relative prior and posterior probabilities for model classes.
2.1 Scheme of structure used for Example 4.
2.2 Output of the ABC rejection algorithm in application to Example 4.
2.3 Illustration of Subset Simulation method.
2.4 Output of the ABC-SubSim algorithm in application to Example 5.
3.1 Illustrations of PH and
α
–
λ
prognostics metrics.
3.2 Conceptual scheme for RUL and EOL calculation.
3.3 Conceptual illustration of Monte Carlo approximation.
3.4 Failure time calculation using a random walk (Monte Carlo simulations) approach.
4.1 MMSE estimation with increasing polynomial degrees.
4.2 MMSE estimation with different configuration parameters.
4.3 Approximation of the conditional expectation for different polynomial degrees.
4.4 MMSE filter with different polynomial orders.
4.5 Comparison of the MMSE filter with with a Bayes posterior and an MCMC simulation.
4.6 Continuation of Fig. 4.5.
5.1 Information flow in the hierarchical sparse Bayesian model.
5.2 A 12-storey shear building structure.
5.3 Iteration histories for the MAP values of the twelve stiffness scaling parameter.
5.4 Scheme of the investigated structure.
5.5 Probability of substructure damage exceeding f for different damage patterns.
6.1 Likelihood functions derived from each time-of-flight (
d̂
j
(
k
)
). The standard deviation of the proposed model ranking is expected to have different values in each model class. The time-of-flight data are then substituted in the likelihood function
p
(
d
^
(
k
)
|
σ
ϵ
,
M
j
(
k
)
)
.
6.2 Flowchart describing the ultrasonic guided-waves-based model class assessment problem for one arbitrary scattered signal.
6.3 Flat aluminium plate along with the sensor layout.
6.4 Example of the outputs for the different TF models considered in this example. The time represented in each caption corresponds to the first energy peak (time of flight), which is used later for damage localisation purposes.
6.5 Posterior probability of each TF model for every sensor.
6.6 Flowchart describing the ultrasonic guided-waves based Bayesian damage localisation problem.
6.7 Panel (a): Posterior PDF of the damage localisation variable and comparison with the ground truth about the enclosed damaged area. Panel (b): Comparison of prior and posterior PDFs of the velocity parameter.
6.8
Optimal sensor layouts for different prior PDFs.
6.9 Damage location reconstruction using the optimal sensor configurations.
7.1 Multiple-level hierarchical architecture of modal analysis.
7.2 CDFs of (
S
k
sum
) at
f
k
= 0.806 Hz (left) and
f
k
=3.988 Hz (right) with different
n
s
.
7.3 Conditional PDFs of
f
s
(left) and
ς
s
(right) for the shear building.
7.4 Iteration histories of
θ
i
for the shear building.
7.5 Shear building used for laboratory testing.
7.6 Acceleration of the top story and the trace of PSD matrix.
7.7 Iteration histories of model updating for four scenarios.
7.8 Identified optimal 'system mode shapes' for different scenarios.
8.1 FEM grid used for the geomechanical simulations.
8.2 Schematic flowchart of the deterministic solver, the computation of subsidence, and the measurable expression.
8.3 Lognormal prior probability distribution of
f
cm
.
8.4 Replacing the deterministic solver by a PCE surrogate model.
8.5 KLE mesh details.
8.6 Relative variance
ρ
L
for different eigenfunctions.
8.7 Realisations of the random field
f
cM
,
j
.
8.8 Posterior samples from the MCMC random walk chain.
8.9 MCMC update results: Prior and posterior for FcM.
8.10 Random chain of the first four elements of
q
(right).
8.11 MCMC update results: Scatter plot.
8.12 2D and 3D view of the ‘true’
f
cM
’ field.
8.13 PCE-based Kalman filter update results.
8.14 2D and 3D view of the scaling factor.
8.15 Testing low-rank approximation on MCMC update.
8.16 Prior and posterior of
F
cM
.
8.17 MMSE field update using linear estimator and a low rank approximation of the measurement model.
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