5,552 research outputs found
Approximating gradients with continuous piecewise polynomial functions
Motivated by conforming finite element methods for elliptic problems of
second order, we analyze the approximation of the gradient of a target function
by continuous piecewise polynomial functions over a simplicial mesh. The main
result is that the global best approximation error is equivalent to an
appropriate sum in terms of the local best approximations errors on elements.
Thus, requiring continuity does not downgrade local approximability and
discontinuous piecewise polynomials essentially do not offer additional
approximation power, even for a fixed mesh. This result implies error bounds in
terms of piecewise regularity over the whole admissible smoothness range.
Moreover, it allows for simple local error functionals in adaptive tree
approximation of gradients.Comment: 21 pages, 1 figur
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
- …