228 research outputs found
Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation
In this paper we derive new uniform convergence estimates for the V-cycle MGM
applied to symmetric positive definite Toeplitz block tridiagonal matrices, by
also discussing few connections with previous results. More concretely, the
contributions of this paper are as follows: (1) It tackles the Toeplitz systems
directly for the elliptic PDEs. (2) Simple (traditional) restriction operator
and prolongation operator are employed in order to handle general Toeplitz
systems at each level of the recursion. Such a technique is then applied to
systems of algebraic equations generated by the difference scheme of the
two-dimensional fractional Feynman-Kac equation, which describes the joint
probability density function of non-Brownian motion. In particular, we consider
the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and
Galerkin approach (algebraic MGM), which lead to the distinct coarsening
stiffness matrices in the general case: however, several numerical experiments
show that the two algorithms produce almost the same error behaviour.Comment: 26 page
On the Min-Max-Delay Problem: NP-completeness, Algorithm, and Integrality Gap
We study a delay-sensitive information flow problem where a source streams
information to a sink over a directed graph G(V,E) at a fixed rate R possibly
using multiple paths to minimize the maximum end-to-end delay, denoted as the
Min-Max-Delay problem. Transmission over an edge incurs a constant delay within
the capacity. We prove that Min-Max-Delay is weakly NP-complete, and
demonstrate that it becomes strongly NP-complete if we require integer flow
solution. We propose an optimal pseudo-polynomial time algorithm for
Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log
R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the
maximum edge delay. Besides, we show that the integrality gap, which is defined
as the ratio of the maximum delay of an optimal integer flow to the maximum
delay of an optimal fractional flow, could be arbitrarily large
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