2,905 research outputs found
On the spectral distribution of kernel matrices related to\ud radial basis functions
This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants
Coherent Matter Wave Transport in Speckle Potentials
This article studies multiple scattering of matter waves by a disordered
optical potential in two and in three dimensions. We calculate fundamental
transport quantities such as the scattering mean free path , the
Boltzmann transport mean free path \elltrb, and the Boltzmann diffusion
constant , using a diagrammatic Green functions approach. Coherent
multiple scattering induces interference corrections known as weak localization
which entail a reduced diffusion constant. We derive the corresponding
expressions for matter wave transport in an correlated speckle potential and
provide the relevant parameter values for a possible experimental study of this
coherent transport regime, including the critical crossover to the regime of
strong or Anderson localization.Comment: 33 pages, minor corrections, published versio
Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives
A time-domain numerical modeling of Biot poroelastic waves is presented. The
viscous dissipation occurring in the pores is described using the dynamic
permeability model developed by Johnson-Koplik-Dashen (JKD). Some of the
coefficients in the Biot-JKD model are proportional to the square root of the
frequency: in the time-domain, these coefficients introduce order 1/2 shifted
fractional derivatives involving a convolution product. Based on a diffusive
representation, the convolution kernel is replaced by a finite number of memory
variables that satisfy local-in-time ordinary differential equations. Thanks to
the dispersion relation, the coefficients in the diffusive representation are
obtained by performing an optimization procedure in the frequency range of
interest. A splitting strategy is then applied numerically: the propagative
part of Biot-JKD equations is discretized using a fourth-order ADER scheme on a
Cartesian grid, whereas the diffusive part is solved exactly. Comparisons with
analytical solutions show the efficiency and the accuracy of this approach.Comment: arXiv admin note: substantial text overlap with arXiv:1210.036
Tomographic errors from wavefront healing: more than just a fast bias
Wave front healing, in which diffractions interfere with directly travelling waves causing a reduction in recorded traveltime delays, has been postulated to cause a bias towards faster estimated earth models. This paper reviews the theory from the mathematical physics community that explains the properties of diffractions and applies it to a suite of increasingly complicated numerical examples. We focus in particular on the elastic case and on the differences between P and S healing. We find that rather than introducing a systemic fast bias, wave front healing gives a more complicated bias in the results of traveltime tomography, with fast anomalies even manifesting themselves as slow anomalies in some situations. Of particular interest, we find that a negative correlation between the bulk and shear or compressional velocities may result to a large extend from healing.Netherlands Organization for Scientific Research (NWO:VICI865.03.007
Interpretable statistics for complex modelling: quantile and topological learning
As the complexity of our data increased exponentially in the last decades, so has our
need for interpretable features. This thesis revolves around two paradigms to approach
this quest for insights.
In the first part we focus on parametric models, where the problem of interpretability
can be seen as a “parametrization selection”. We introduce a quantile-centric
parametrization and we show the advantages of our proposal in the context of regression,
where it allows to bridge the gap between classical generalized linear (mixed)
models and increasingly popular quantile methods.
The second part of the thesis, concerned with topological learning, tackles the
problem from a non-parametric perspective. As topology can be thought of as a way
of characterizing data in terms of their connectivity structure, it allows to represent
complex and possibly high dimensional through few features, such as the number of
connected components, loops and voids. We illustrate how the emerging branch of
statistics devoted to recovering topological structures in the data, Topological Data
Analysis, can be exploited both for exploratory and inferential purposes with a special
emphasis on kernels that preserve the topological information in the data.
Finally, we show with an application how these two approaches can borrow strength
from one another in the identification and description of brain activity through fMRI
data from the ABIDE project
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