8 research outputs found
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Advances in Trans-dimensional Geophysical Inference
This research presents a series of novel Bayesian
trans-dimensional
methods for geophysical inversion. A first example illustrates
how
Bayesian prior information obtained from theory and numerical
experiments can be used to better inform a difficult
multi-modal inversion of dispersion information from empirical
Greens
functions obtained from ambient noise cross-correlation. This
approach
is an extension of existing partition modeling schemes.
An entirely new class of trans-dimensional algorithm, called the
trans-dimensional tree method is introduced. This new method is
shown
to be more efficient at coupling to a forward model, more
efficient at
convergence, and more adaptable to different dimensions and
geometries
than existing approaches. The efficiency and flexibility of the
trans-dimensional tree method is demonstrated in two different
examples: (1) airborne electromagnetic tomography (AEM) in a 2D
transect inversion, and (2) a fully non-linear inversion of
ambient
noise tomography. In this latter example the resolution at depth
has
been significantly improved by inverting a contiguous band of
frequencies jointly rather than as independent phase velocity
maps,
allowing new insights into crustal architecture beneath Iceland.
In a first test case for even larger scale problems, an
application of
the trans-dimensional tree approach to large global data set is
presented. A global database of nearly 5 million multi-model
path
average Rayleigh wave phase velocity observations has been used
to
construct global phase velocity maps. Results are comparable to
existing published phase velocity maps, however, as the
trans-dimensional approach adapts the resolution appropriate to
the
data, rather than imposing damping or smoothing constraints to
stabilize the inversion, the recovered anomaly magnitudes are
generally higher with low uncertainties. While further
investigation is
needed, this early test case shows that trans-dimensional
sampling can
be applied to global scale seismology problems and that previous
analyses may, in some locales, under estimate the heterogeneity
of the
Earth.
Finally, in a further advancement of partition modelling with
variable
order polynomials, a new method has been developed called
trans-dimensional spectral elements. Previous applications
involving
variable order polynomials have used polynomials that are both
difficult to work with in a Bayesian framework and unstable at
higher orders. By using the orthogonal polynomials typically used
in
modern full-waveform solvers, the useful properties of this type
of
polynomial and its application in trans-dimensional inversion
are
demonstrated. Additionally, these polynomials can be directly
used in
complex differential solvers and an example of this for 1D
inversion
of surface wave dispersion curves is given