90 research outputs found
The numerical solution of fractional differential equations: Speed versus accuracy
This paper discusses the development of efficient algorithms for a certain fractional differential equation.Manchester Centre for Computational Mathematic
Characterising small solutions in delay differential equations through numerical approximations
This paper discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.Manchester Centre for Computational Mathematic
Nodal bases for the serendipity family of finite elements
Using the notion of multivariate lower set interpolation, we construct nodal
basis functions for the serendipity family of finite elements, of any order and
any dimension. For the purpose of computation, we also show how to express
these functions as linear combinations of tensor-product polynomials.Comment: Pre-print of version that will appear in Foundations of Computational
Mathematic
Target Identification Using Dictionary Matching of Generalized Polarization Tensors
The aim of this paper is to provide a fast and efficient procedure for
(real-time) target identification in imaging based on matching on a dictionary
of precomputed generalized polarization tensors (GPTs). The approach is based
on some important properties of the GPTs and new invariants. A new shape
representation is given and numerically tested in the presence of measurement
noise. The stability and resolution of the proposed identification algorithm is
numerically quantified.Comment: Keywords: generalized polarization tensors, target identification,
shape representation, stability analysis. Submitted to Foundations of
Computational Mathematic
How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?
This paper discusses the numerical solution of delay differential equations undergoing a Hopf birufication. Three distinct and complementary approaches to the analysis are presented.Manchester Centre for Computational Mathematic
The Euclidean distance degree of an algebraic variety
The nearest point map of a real algebraic variety with respect to Euclidean
distance is an algebraic function. For instance, for varieties of low rank
matrices, the Eckart-Young Theorem states that this map is given by the
singular value decomposition. This article develops a theory of such nearest
point maps from the perspective of computational algebraic geometry. The
Euclidean distance degree of a variety is the number of critical points of the
squared distance to a generic point outside the variety. Focusing on varieties
seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic
Improved Complexity Bounds for Counting Points on Hyperelliptic Curves
We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a hyperelliptic curve of genus defined over . It
is based on the approaches by Schoof and Pila combined with a modeling of the
-torsion by structured polynomial systems. Our main result improves on
previously known complexity bounds by showing that there exists a constant
such that, for any fixed , this algorithm has expected time and space
complexity as grows and the characteristic is large
enough.Comment: To appear in Foundations of Computational Mathematic
Serendipity and Tensor Product Affine Pyramid Finite Elements
Using the language of finite element exterior calculus, we define two
families of -conforming finite element spaces over pyramids with a
parallelogram base. The first family has matching polynomial traces with tensor
product elements on the base while the second has matching polynomial traces
with serendipity elements on the base. The second family is new to the
literature and provides a robust approach for linking between Lagrange elements
on tetrahedra and serendipity elements on affinely-mapped cubes while
preserving continuity and approximation properties. We define shape functions
and degrees of freedom for each family and prove unisolvence and polynomial
reproduction results.Comment: Accepted to SMAI Journal of Computational Mathematic
Algebraic boundaries among typical ranks for real binary forms of arbitrary degree
We show that the algebraic boundaries of the regions of real binary forms
with fixed typical rank are always unions of dual varieties to suitable
coincident root loci.Comment: Accepted for publication in Foundations of Computational Mathematic
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