24 research outputs found
Lack of compactness in the 2D critical Sobolev embedding, the general case
This paper is devoted to the description of the lack of compactness of the
Sobolev embedding of in the critical Orlicz space {\cL}(\R^2). It
turns out that up to cores our result is expressed in terms of the
concentration-type examples derived by J. Moser in \cite{M} as in the radial
setting investigated in \cite{BMM}. However, the analysis we used in this work
is strikingly different from the one conducted in the radial case which is
based on an estimate far away from the origin and which is no
longer valid in the general framework. Within the general framework of
, the strategy we adopted to build the profile decomposition in
terms of examples by Moser concentrated around cores is based on capacity
arguments and relies on an extraction process of mass concentrations. The
essential ingredient to extract cores consists in proving by contradiction that
if the mass responsible for the lack of compactness of the Sobolev embedding in
the Orlicz space is scattered, then the energy used would exceed that of the
starting sequence.Comment: Submitte
Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs
Prediction error quantification in machine learning has been left out of most
methodological investigations of neural networks, for both purely data-driven
and physics-informed approaches. Beyond statistical investigations and generic
results on the approximation capabilities of neural networks, we present a
rigorous upper bound on the prediction error of physics-informed neural
networks. This bound can be calculated without the knowledge of the true
solution and only with a priori available information about the characteristics
of the underlying dynamical system governed by a partial differential equation.
We apply this a posteriori error bound exemplarily to four problems: the
transport equation, the heat equation, the Navier-Stokes equation and the
Klein-Gordon equation
[Book of abstracts]
USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos
Schrödinger operators in the twentieth century
This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics
Differentiable positive definite kernels on two-point homogeneous spaces
In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014