154 research outputs found
The derivation of continuum limits of neuronal networks with gap-junction couplings
We consider an idealized network, formed by N neurons individually described
by the FitzHugh-Nagumo equations and connected by electrical synapses. The
limit for N to infinity of the resulting discrete model is thoroughly
investigated, with the aim of identifying a model for a continuum of neurons
having an equivalent behaviour. Two strategies for passing to the limit are
analysed: i) a more conventional approach, based on a fixed nearest-neighbour
connection topology accompanied by a suitable scaling of the diffusion
coefficients; ii) a new approach, in which the number of connections to any
given neuron varies with N according to a precise law, which simultaneously
guarantees the non-triviality of the limit and the locality of neuronal
interactions. Both approaches yield in the limit a pde-based model, in which
the distribution of action potential obeys a nonlinear
reaction-convection-diffusion equation; convection accounts for the possible
lack of symmetry in the connection topology. Several convergence issues are
discussed, both theoretically and numerically
On the decay of the inverse of matrices that are sum of Kronecker products
Decay patterns of matrix inverses have recently attracted considerable
interest, due to their relevance in numerical analysis, and in applications
requiring matrix function approximations. In this paper we analyze the decay
pattern of the inverse of banded matrices in the form where is tridiagonal, symmetric and positive definite, is
the identity matrix, and stands for the Kronecker product. It is well
known that the inverses of banded matrices exhibit an exponential decay pattern
away from the main diagonal. However, the entries in show a
non-monotonic decay, which is not caught by classical bounds. By using an
alternative expression for , we derive computable upper bounds that
closely capture the actual behavior of its entries. We also show that similar
estimates can be obtained when has a larger bandwidth, or when the sum of
Kronecker products involves two different matrices. Numerical experiments
illustrating the new bounds are also reported
Legendre-Gauss-Lobatto grids and associated nested dyadic grids
Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral
methods for the numerical solution of partial differential equations. They not
only provide efficient high-order quadrature rules, but give also rise to norm
equivalences that could eventually lead to efficient preconditioning techniques
in high-order methods. Unfortunately, a serious obstruction to fully exploiting
the potential of such concepts is the fact that LGL grids of different degree
are not nested. This affects, on the one hand, the choice and analysis of
suitable auxiliary spaces, when applying the auxiliary space method as a
principal preconditioning paradigm, and, on the other hand, the efficient
solution of the auxiliary problems. As a central remedy, we consider certain
nested hierarchies of dyadic grids of locally comparable mesh size, that are in
a certain sense properly associated with the LGL grids. Their actual
suitability requires a subtle analysis of such grids which, in turn, relies on
a number of refined properties of LGL grids. The central objective of this
paper is to derive just these properties. This requires first revisiting
properties of close relatives to LGL grids which are subsequently used to
develop a refined analysis of LGL grids. These results allow us then to derive
the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords:
Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid
Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case
We analyze the theoretical properties of an adaptive Legendre-Galerkin method
in the multidimensional case. After the recent investigations for
Fourier-Galerkin methods in a periodic box and for Legendre-Galerkin methods in
the one dimensional setting, the present study represents a further step
towards a mathematically rigorous understanding of adaptive spectral/
discretizations of elliptic boundary-value problems. The main contribution of
the paper is a careful construction of a multidimensional Riesz basis in ,
based on a quasi-orthonormalization procedure. This allows us to design an
adaptive algorithm, to prove its convergence by a contraction argument, and to
discuss its optimality properties (in the sense of non-linear approximation
theory) in certain sparsity classes of Gevrey type
Adaptive Fourier-Galerkin Methods
We study the performance of adaptive Fourier-Galerkin methods in a periodic
box in with dimension . These methods offer unlimited
approximation power only restricted by solution and data regularity. They are
of intrinsic interest but are also a first step towards understanding
adaptivity for the -FEM. We examine two nonlinear approximation classes,
one classical corresponding to algebraic decay of Fourier coefficients and
another associated with exponential decay. We study the sparsity classes of the
residual and show that they are the same as the solution for the algebraic
class but not for the exponential one. This possible sparsity degradation for
the exponential class can be compensated with coarsening, which we discuss in
detail. We present several adaptive Fourier algorithms, and prove their
contraction and optimal cardinality properties.Comment: 48 page
Adaptive Spectral Galerkin Methods with Dynamic Marking
The convergence and optimality theory of adaptive Galerkin methods is almost
exclusively based on the D\"orfler marking. This entails a fixed parameter and
leads to a contraction constant bounded below away from zero. For spectral
Galerkin methods this is a severe limitation which affects performance. We
present a dynamic marking strategy that allows for a super-linear relation
between consecutive discretization errors, and show exponential convergence
with linear computational complexity whenever the solution belongs to a Gevrey
approximation class.Comment: 20 page
On p-Robust Saturation for hp-AFEM
We consider the standard adaptive finite element loop SOLVE, ESTIMATE, MARK,
REFINE, with ESTIMATE being implemented using the -robust equilibrated flux
estimator, and MARK being D\"orfler marking. As a refinement strategy we employ
-refinement. We investigate the question by which amount the local
polynomial degree on any marked patch has to be increase in order to achieve a
-independent error reduction. The resulting adaptive method can be turned
into an instance optimal -adaptive method by the addition of a coarsening
routine
A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square
Both practice and analysis of adaptive -FEMs and -FEMs raise the
question what increment in the current polynomial degree guarantees a
-independent reduction of the Galerkin error. We answer this question for
the -FEM in the simplified context of homogeneous Dirichlet problems for the
Poisson equation in the two dimensional unit square with polynomial data of
degree . We show that an increment proportional to yields a -robust
error reduction and provide computational evidence that a constant increment
does not
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