17,224 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
A new neural network technique for the design of multilayered microwave shielded bandpass filters
In this work, we propose a novel technique based on neural networks, for the design of microwave filters in shielded printed technology. The technique uses radial basis function neural networks to represent the non linear relations between the quality factors and coupling coefficients, with the geometrical dimensions of the resonators. The radial basis function neural networks are employed for the first time in the design task of shielded printed filters, and permit a fast and precise operation with only a limited set of training data. Thanks to a new cascade configuration, a set of two neural networks provide the dimensions of the complete filter in a fast and accurate way. To improve the calculation of the geometrical dimensions, the neural networks can take as inputs both electrical parameters and physical dimensions computed by other neural networks. The neural network technique is combined with gradient based optimization methods to further improve the response of the filters. Results are presented to demonstrate the usefulness of the proposed technique for the design of practical microwave printed coupled line and hairpin filters
A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation
In this paper we develop a discrete Hierarchical Basis (HB) to efficiently
solve the Radial Basis Function (RBF) interpolation problem with variable
polynomial order. The HB forms an orthogonal set and is adapted to the kernel
seed function and the placement of the interpolation nodes. Moreover, this
basis is orthogonal to a set of polynomials up to a given order defined on the
interpolating nodes. We are thus able to decouple the RBF interpolation problem
for any order of the polynomial interpolation and solve it in two steps: (1)
The polynomial orthogonal RBF interpolation problem is efficiently solved in
the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR
preconditioner. (2) The residual is then projected onto an orthonormal
polynomial basis. We apply our approach on several test cases to study its
effectiveness, including an application to the Best Linear Unbiased Estimator
regression problem
Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities
In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns.
With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem
Density fields for branching, stiff networks in rigid confining regions
We develop a formalism to describe the equilibrium distributions for segments
of confined branched networks consisting of stiff filaments. This is applicable
to certain situations of cytoskeleton in cells, such as for example actin
filaments with branching due to the Arp2/3 complex. We develop a grand ensemble
formalism that enables the computation of segment density and polarisation
profiles within the confines of the cell. This is expressed in terms of the
solution to nonlinear integral equations for auxiliary functions. We find three
specific classes of behaviour depending on filament length, degree of branching
and the ratio of persistence length to the dimensions of the geometry. Our
method allows a numerical approach for semi-flexible filaments that are
networked.Comment: 15 pages, revise
Electric-field-induced displacement of a charged spherical colloid embedded in an elastic Brinkman medium
When an electric field is applied to an electrolyte-saturated polymer gel
embedded with charged colloidal particles, the force that must be exerted by
the hydrogel on each particle reflects a delicate balance of electrical,
hydrodynamic and elastic stresses. This paper examines the displacement of a
single charged spherical inclusion embedded in an uncharged hydrogel. We
present numerically exact solutions of coupled electrokinetic transport and
elastic-deformation equations, where the gel is treated as an incompressible,
elastic Brinkman medium. This model problem demonstrates how the displacement
depends on the particle size and charge, the electrolyte ionic strength, and
Young's modulus of the polymer skeleton. The numerics are verified, in part,
with an analytical (boundary-layer) theory valid when the Debye length is much
smaller than the particle radius. Further, we identify a close connection
between the displacement when a colloid is immobilized in a gel and its
velocity when dispersed in a Newtonian electrolyte. Finally, we describe an
experiment where nanometer-scale displacements might be accurately measured
using back-focal-plane interferometry. The purpose of such an experiment is to
probe physicochemical and rheological characteristics of hydrogel composites,
possibly during gelation
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