63,139 research outputs found
An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
We propose an adaptive finite element method for the solution of a
coefficient inverse problem of simultaneous reconstruction of the dielectric
permittivity and magnetic permeability functions in the Maxwell's system using
limited boundary observations of the electric field in 3D. We derive a
posteriori error estimates in the Tikhonov functional to be minimized and in
the regularized solution of this functional, as well as formulate corresponding
adaptive algorithm. Our numerical experiments justify the efficiency of our a
posteriori estimates and show significant improvement of the reconstructions
obtained on locally adaptively refined meshes.Comment: Corrected typo
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Effective results for hyper- and superelliptic equations over number fields
We consider hyper- and superelliptic equations with unknowns x,y
from the ring of S-integers of a given number field K. Here, f is a polynomial
with S-integral coefficients of degree n with non-zero discriminant and b is a
non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely
explicit upper bounds for the heights of the solutions x,y in terms of the
heights of f and b, the discriminant of K, and the norms of the prime ideals in
S. Further, we give a completely explicit bound C such that has no
solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We
will apply these results in another paper where we consider hyper- and
superelliptic equations with unknowns taken from an arbitrary finitely
generated integral domain of characteristic 0.Comment: 31 page
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
Multiplicity Estimates: a Morse-theoretic approach
The problem of estimating the multiplicity of the zero of a polynomial when
restricted to the trajectory of a non-singular polynomial vector field, at one
or several points, has been considered by authors in several different fields.
The two best (incomparable) estimates are due to Gabrielov and Nesterenko.
In this paper we present a refinement of Gabrielov's method which
simultaneously improves these two estimates. Moreover, we give a geometric
description of the multiplicity function in terms certain naturally associated
polar varieties, giving a topological explanation for an asymptotic phenomenon
that was previously obtained by elimination theoretic methods in the works of
Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton
polytopes, strongly generalizing the classical estimates.Comment: Minor revision; To appear in Duke Math. Journa
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