140 research outputs found
Constructive proofs for localized radial solutions of semilinear elliptic systems on
Ground state solutions of elliptic problems have been analyzed extensively in
the theory of partial differential equations, as they represent fundamental
spatial patterns in many model equations. While the results for scalar
equations, as well as certain specific classes of elliptic systems, are
comprehensive, much less is known about these localized solutions in generic
systems of nonlinear elliptic equations. In this paper we present a general
method to prove constructively the existence of localized radially symmetric
solutions of elliptic systems on . Such solutions are essentially
described by systems of non-autonomous ordinary differential equations. We
study these systems using dynamical systems theory and computer-assisted proof
techniques, combining a suitably chosen Lyapunov-Perron operator with a
Newton-Kantorovich type theorem. We demonstrate the power of this methodology
by proving specific localized radial solutions of the cubic Klein-Gordon
equation on , the Swift-Hohenberg equation on , and
a three-component FitzHugh-Nagumo system on . These results
illustrate that ground state solutions in a wide range of elliptic systems are
tractable through constructive proofs
Null Spaces of Radon Transforms
We obtain new descriptions of the null spaces of several projectively
equivalent transforms in integral geometry. The paper deals with the hyperplane
Radon transform, the totally geodesic transforms on the sphere and the
hyperbolic space, the spherical slice transform, and the Cormack-Quinto
spherical mean transform for spheres through the origin. The consideration
extends to the corresponding dual transforms and the relevant exterior/interior
modifications. The method relies on new results for the Gegenbauer-Chebyshev
integrals, which generalize Abel type fractional integrals on the positive
half-line.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1410.411
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
A biconjugate gradient type algorithm on massively parallel architectures
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine
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