4,606 research outputs found
Fast and stable contour integration for high order divided differences via elliptic functions
In this paper, we will present a new method for evaluating high order divided
differences for certain classes of analytic, possibly, operator valued
functions. This is a classical problem in numerical mathematics but also
arises in new applications such as, e.g., the use of generalized convolution
quadrature to solve retarded potential integral equations. The functions which
we will consider are allowed to grow exponentially to the left complex half
plane, polynomially to the right half plane and have an oscillatory behaviour
with increasing imaginary part. The interpolation points are scattered in a
large real interval. Our approach is based on the representation of divided
differences as contour integral and we will employ a subtle parameterization
of the contour in combination with a quadrature approximation by the
trapezoidal rule
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Development and application of the GIM code for the Cyber 203 computer
The GIM computer code for fluid dynamics research was developed. Enhancement of the computer code, implicit algorithm development, turbulence model implementation, chemistry model development, interactive input module coding and wing/body flowfield computation are described. The GIM quasi-parabolic code development was completed, and the code used to compute a number of example cases. Turbulence models, algebraic and differential equations, were added to the basic viscous code. An equilibrium reacting chemistry model and implicit finite difference scheme were also added. Development was completed on the interactive module for generating the input data for GIM. Solutions for inviscid hypersonic flow over a wing/body configuration are also presented
Generalized convolution quadrature with variable time stepping
In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equatio
Stability of Underwater Periodic Locomotion
Most aquatic vertebrates swim by lateral flapping of their bodies and caudal
fins. While much effort has been devoted to understanding the flapping
kinematics and its influence on the swimming efficiency, little is known about
the stability (or lack of) of periodic swimming. It is believed that stability
limits maneuverability and body designs/flapping motions that are adapted for
stable swimming are not suitable for high maneuverability and vice versa. In
this paper, we consider a simplified model of a planar elliptic body undergoing
prescribed periodic heaving and pitching in potential flow. We show that
periodic locomotion can be achieved due to the resulting hydrodynamic forces,
and its value depends on several parameters including the aspect ratio of the
body, the amplitudes and phases of the prescribed flapping. We obtain
closed-form solutions for the locomotion and efficiency for small flapping
amplitudes, and numerical results for finite flapping amplitudes. We then study
the stability of the (finite amplitude flapping) periodic locomotion using
Floquet theory. We find that stability depends nonlinearly on all parameters.
Interesting trends of switching between stable and unstable motions emerge and
evolve as we continuously vary the parameter values. This suggests that, for
live organisms that control their flapping motion, maneuverability and
stability need not be thought of as disjoint properties, rather the organism
may manipulate its motion in favor of one or the other depending on the task at
hand.Comment: 15 pages, 15 figure
On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initial data
The behavior of solutions of the finite-genus Whitham equations for the weak
dispersion limit of the defocusing nonlinear Schrodinger equation is
investigated analytically and numerically for piecewise-constant initial data.
In particular, the dynamics of constant-amplitude initial conditions with one
or more frequency jumps (i.e., piecewise linear phase) are considered. It is
shown analytically and numerically that, for finite times, regions of
arbitrarily high genus can be produced; asymptotically with time, however, the
solution can be divided into expanding regions which are either of genus-zero,
genus-one or genus-two type, their precise arrangement depending on the
specifics of the initial datum given. This behavior should be compared to that
of the Korteweg-deVries equation, where the solution is devided into the
regions which are either genus-zero or genus-one asymptotically. Finally, the
potential application of these results to the generation of short optical
pulses is discussed: the method proposed takes advantage of nonlinear
compression via appropriate frequency modulation, and allows control of both
the pulse amplitude and its width, as well as the distance along the fiber at
which the pulse is produced.Comment: 44 pages, 21 figure
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