7,416 research outputs found
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
We present an effective harmonic density interpolation method for the
numerical evaluation of singular and nearly singular Laplace boundary integral
operators and layer potentials in two and three spatial dimensions. The method
relies on the use of Green's third identity and local Taylor-like
interpolations of density functions in terms of harmonic polynomials. The
proposed technique effectively regularizes the singularities present in
boundary integral operators and layer potentials, and recasts the latter in
terms of integrands that are bounded or even more regular, depending on the
order of the density interpolation. The resulting boundary integrals can then
be easily, accurately, and inexpensively evaluated by means of standard
quadrature rules. A variety of numerical examples demonstrate the effectiveness
of the technique when used in conjunction with the classical trapezoidal rule
(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type
quadrature rule (to integrate over surfaces given as unions of non-overlapping
quadrilateral patches) in three-dimensions
Stabilized Lattice Boltzmann-Enskog method for compressible flows and its application to one and two-component fluids in nanochannels
A numerically stable method to solve the discretized Boltzmann-Enskog
equation describing the behavior of non ideal fluids under inhomogeneous
conditions is presented. The algorithm employed uses a Lagrangian
finite-difference scheme for the treatment of the convective term and a forcing
term to account for the molecular repulsion together with a
Bhatnagar-Gross-Krook relaxation term. In order to eliminate the spurious
currents induced by the numerical discretization procedure, we use a
trapezoidal rule for the time integration together with a version of the
two-distribution method of He et al. (J. Comp. Phys 152, 642 (1999)). Numerical
tests show that, in the case of one component fluid in the presence of a
spherical potential well, the proposed method reduces the numerical error by
several orders of magnitude. We conduct another test by considering the flow of
a two component fluid in a channel with a bottleneck and provide information
about the density and velocity field in this structured geometry.Comment: to appear in Physical Review
A Matrix Approach to Numerical Solution of the DGLAP Evolution Equations
A matrix-based approach to numerical integration of the DGLAP evolution
equations is presented. The method arises naturally on discretisation of the
Bjorken x variable, a necessary procedure for numerical integration. Owing to
peculiar properties of the matrices involved, the resulting equations take on a
particularly simple form and may be solved in closed analytical form in the
variable t=ln(alpha_0/alpha). Such an approach affords parametrisation via data
x bins, rather than fixed functional forms. Thus, with the aid of the full
correlation matrix, appraisal of the behaviour in different x regions is
rendered more transparent and free of pollution from unphysical
cross-correlations inherent to functional parametrisations. Computationally,
the entire programme results in greater speed and stability; the matrix
representation developed is extremely compact. Moreover, since the parameter
dependence is linear, fitting is very stable and may be performed analytically
in a single pass over the data values.Comment: 13 pages, no figures, typeset with revtex4 and uses packages:
acromake, amssym
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