120,441 research outputs found
High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: Sampling Cost via Incident-Field Windowing and Recentering
This paper proposes a frequency/time hybrid integral-equation method for the
time dependent wave equation in two and three-dimensional spatial domains.
Relying on Fourier Transformation in time, the method utilizes a fixed
(time-independent) number of frequency-domain integral-equation solutions to
evaluate, with superalgebraically-small errors, time domain solutions for
arbitrarily long times. The approach relies on two main elements, namely, 1) A
smooth time-windowing methodology that enables accurate band-limited
representations for arbitrarily-long time signals, and 2) A novel Fourier
transform approach which, in a time-parallel manner and without causing
spurious periodicity effects, delivers numerically dispersionless
spectrally-accurate solutions. A similar hybrid technique can be obtained on
the basis of Laplace transforms instead of Fourier transforms, but we do not
consider the Laplace-based method in the present contribution. The algorithm
can handle dispersive media, it can tackle complex physical structures, it
enables parallelization in time in a straightforward manner, and it allows for
time leaping---that is, solution sampling at any given time at
-bounded sampling cost, for arbitrarily large values of ,
and without requirement of evaluation of the solution at intermediate times.
The proposed frequency-time hybridization strategy, which generalizes to any
linear partial differential equation in the time domain for which
frequency-domain solutions can be obtained (including e.g. the time-domain
Maxwell equations), and which is applicable in a wide range of scientific and
engineering contexts, provides significant advantages over other available
alternatives such as volumetric discretization, time-domain integral equations,
and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now
including direct comparisons to existing CQ and TDIE solver implementations)
(Part I of II
A Flexible Implementation of a Matrix Laurent Series-Based 16-Point Fast Fourier and Hartley Transforms
This paper describes a flexible architecture for implementing a new fast
computation of the discrete Fourier and Hartley transforms, which is based on a
matrix Laurent series. The device calculates the transforms based on a single
bit selection operator. The hardware structure and synthesis are presented,
which handled a 16-point fast transform in 65 nsec, with a Xilinx SPARTAN 3E
device.Comment: 4 pages, 4 figures. IEEE VI Southern Programmable Logic Conference
201
Fast Computation of Fourier Integral Operators
We introduce a general purpose algorithm for rapidly computing certain types
of oscillatory integrals which frequently arise in problems connected to wave
propagation and general hyperbolic equations. The problem is to evaluate
numerically a so-called Fourier integral operator (FIO) of the form at points given on
a Cartesian grid. Here, is a frequency variable, is the
Fourier transform of the input , is an amplitude and
is a phase function, which is typically as large as ;
hence the integral is highly oscillatory at high frequencies. Because an FIO is
a dense matrix, a naive matrix vector product with an input given on a
Cartesian grid of size by would require operations.
This paper develops a new numerical algorithm which requires operations, and as low as in storage space. It operates by
localizing the integral over polar wedges with small angular aperture in the
frequency plane. On each wedge, the algorithm factorizes the kernel into two components: 1) a diffeomorphism which is
handled by means of a nonuniform FFT and 2) a residual factor which is handled
by numerical separation of the spatial and frequency variables. The key to the
complexity and accuracy estimates is that the separation rank of the residual
kernel is \emph{provably independent of the problem size}. Several numerical
examples demonstrate the efficiency and accuracy of the proposed methodology.
We also discuss the potential of our ideas for various applications such as
reflection seismology.Comment: 31 pages, 3 figure
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