5,282 research outputs found
On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations
In this note we study the convergence of monotone P1 finite element methods
on unstructured meshes for fully non-linear Hamilton-Jacobi-Bellman equations
arising from stochastic optimal control problems with possibly degenerate,
isotropic diffusions. Using elliptic projection operators we treat
discretisations which violate the consistency conditions of the framework by
Barles and Souganidis. We obtain strong uniform convergence of the numerical
solutions and, under non-degeneracy assumptions, strong L2 convergence of the
gradients.Comment: Keywords: Bellman equations, finite element methods, viscosity
solutions, fully nonlinear operators; 18 pages, 1 figur
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
Quenched invariance principle for random walks in balanced random environment
We consider random walks in a balanced random environment in ,
. We first prove an invariance principle (for ) and the
transience of the random walks when (recurrence when ) in an
ergodic environment which is not uniformly elliptic but satisfies certain
moment condition. Then, using percolation arguments, we show that under mere
ellipticity, the above results hold for random walks in i.i.d. balanced
environments.Comment: Published online in Probab. Theory Relat. Fields, 05 Oct 2010. Typo
(in journal version) corrected in (26
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
Positive approximations of the inverse of fractional powers of SPD M-matrices
This study is motivated by the recent development in the fractional calculus
and its applications. During last few years, several different techniques are
proposed to localize the nonlocal fractional diffusion operator. They are based
on transformation of the original problem to a local elliptic or
pseudoparabolic problem, or to an integral representation of the solution, thus
increasing the dimension of the computational domain. More recently, an
alternative approach aimed at reducing the computational complexity was
developed. The linear algebraic system , is considered, where is a properly normalized (scalded) symmetric
and positive definite matrix obtained from finite element or finite difference
approximation of second order elliptic problems in ,
. The method is based on best uniform rational approximations (BURA)
of the function for and natural .
The maximum principles are among the major qualitative properties of linear
elliptic operators/PDEs. In many studies and applications, it is important that
such properties are preserved by the selected numerical solution method. In
this paper we present and analyze the properties of positive approximations of
obtained by the BURA technique. Sufficient conditions for
positiveness are proven, complemented by sharp error estimates. The theoretical
results are supported by representative numerical tests
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