99,087 research outputs found
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
A sparse decomposition of low rank symmetric positive semi-definite matrices
Suppose that is symmetric positive
semidefinite with rank . Our goal is to decompose into
rank-one matrices where the modes
are required to be as sparse as possible. In contrast to eigen decomposition,
these sparse modes are not required to be orthogonal. Such a problem arises in
random field parametrization where is the covariance function and is
intractable to solve in general. In this paper, we partition the indices from 1
to into several patches and propose to quantify the sparseness of a vector
by the number of patches on which it is nonzero, which is called patch-wise
sparseness. Our aim is to find the decomposition which minimizes the total
patch-wise sparseness of the decomposed modes. We propose a
domain-decomposition type method, called intrinsic sparse mode decomposition
(ISMD), which follows the "local-modes-construction + patching-up" procedure.
The key step in the ISMD is to construct local pieces of the intrinsic sparse
modes by a joint diagonalization problem. Thereafter a pivoted Cholesky
decomposition is utilized to glue these local pieces together. Optimal sparse
decomposition, consistency with different domain decomposition and robustness
to small perturbation are proved under the so called regular-sparse assumption
(see Definition 1.2). We provide simulation results to show the efficiency and
robustness of the ISMD. We also compare the ISMD to other existing methods,
e.g., eigen decomposition, pivoted Cholesky decomposition and convex relaxation
of sparse principal component analysis [25] and [40]
An Optimal Design for Universal Multiport Interferometers
Universal multiport interferometers, which can be programmed to implement any
linear transformation between multiple channels, are emerging as a powerful
tool for both classical and quantum photonics. These interferometers are
typically composed of a regular mesh of beam splitters and phase shifters,
allowing for straightforward fabrication using integrated photonic
architectures and ready scalability. The current, standard design for universal
multiport interferometers is based on work by Reck et al (Phys. Rev. Lett. 73,
58, 1994). We demonstrate a new design for universal multiport interferometers
based on an alternative arrangement of beam splitters and phase shifters, which
outperforms that by Reck et al. Our design occupies half the physical footprint
of the Reck design and is significantly more robust to optical losses.Comment: 8 pages, 4 figure
The BEM with graded meshes for the electric field integral equation on polyhedral surfaces
We consider the variational formulation of the electric field integral
equation on a Lipschitz polyhedral surface . We study the Galerkin
boundary element discretisations based on the lowest-order Raviart-Thomas
surface elements on a sequence of anisotropic meshes algebraically graded
towards the edges of . We establish quasi-optimal convergence of
Galerkin solutions under a mild restriction on the strength of grading. The key
ingredient of our convergence analysis are new componentwise stability
properties of the Raviart-Thomas interpolant on anisotropic elements
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