4,756 research outputs found
A boundary integral method for an inverse problem in thermal imaging
An inverse problem in thermal imaging involving the recovery of a void in a material from its surface temperature response to external heating is examined. Uniqueness and continuous dependence results for the inverse problem are demonstrated, and a numerical method for its solution is developed. This method is based on an optimization approach, coupled with a boundary integral equation formulation of the forward heat conduction problem. Some convergence results for the method are proved, and several examples are presented using computationally generated data
A sequential sampling strategy for extreme event statistics in nonlinear dynamical systems
We develop a method for the evaluation of extreme event statistics associated
with nonlinear dynamical systems, using a small number of samples. From an
initial dataset of design points, we formulate a sequential strategy that
provides the 'next-best' data point (set of parameters) that when evaluated
results in improved estimates of the probability density function (pdf) for a
scalar quantity of interest. The approach utilizes Gaussian process regression
to perform Bayesian inference on the parameter-to-observation map describing
the quantity of interest. We then approximate the desired pdf along with
uncertainty bounds utilizing the posterior distribution of the inferred map.
The 'next-best' design point is sequentially determined through an optimization
procedure that selects the point in parameter space that maximally reduces
uncertainty between the estimated bounds of the pdf prediction. Since the
optimization process utilizes only information from the inferred map it has
minimal computational cost. Moreover, the special form of the metric emphasizes
the tails of the pdf. The method is practical for systems where the
dimensionality of the parameter space is of moderate size, i.e. order O(10). We
apply the method to estimate the extreme event statistics for a very
high-dimensional system with millions of degrees of freedom: an offshore
platform subjected to three-dimensional irregular waves. It is demonstrated
that the developed approach can accurately determine the extreme event
statistics using limited number of samples
The ROMES method for statistical modeling of reduced-order-model error
This work presents a technique for statistically modeling errors introduced
by reduced-order models. The method employs Gaussian-process regression to
construct a mapping from a small number of computationally inexpensive `error
indicators' to a distribution over the true error. The variance of this
distribution can be interpreted as the (epistemic) uncertainty introduced by
the reduced-order model. To model normed errors, the method employs existing
rigorous error bounds and residual norms as indicators; numerical experiments
show that the method leads to a near-optimal expected effectivity in contrast
to typical error bounds. To model errors in general outputs, the method uses
dual-weighted residuals---which are amenable to uncertainty control---as
indicators. Experiments illustrate that correcting the reduced-order-model
output with this surrogate can improve prediction accuracy by an order of
magnitude; this contrasts with existing `multifidelity correction' approaches,
which often fail for reduced-order models and suffer from the curse of
dimensionality. The proposed error surrogates also lead to a notion of
`probabilistic rigor', i.e., the surrogate bounds the error with specified
probability
Solving a Set of Truncated Dyson-Schwinger Equations with a Globally Converging Method
A globally converging numerical method to solve coupled sets of non-linear
integral equations is presented. Such systems occur e.g. in the study of
Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on
the knowledge of the qualitative properties of the solution functions in the
far infrared and ultraviolet. Using this input, the full solutions are
constructed using a globally convergent modified Newton iteration. Two
different systems will be treated as examples: The Dyson-Schwinger equations of
3-dimensional Yang-Mills-Higgs theory provide a system of finite integrals,
while those of 4-dimensional Yang-Mills theory at high temperatures are only
finite after renormalization.Comment: 23 pages, 3 figures, 4 tables, submitted to Comput. Phys. Commun; one
subsection expanded with additional technical details, a few other minor
modifications and updates, version to appear in Comput. Phys. Commu
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