7,711 research outputs found
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
Systematic calculation of molecular vibrational spectra through a complete Morse expansion
We propose an accurate and efficient method to compute vibrational spectra of
molecules, based on exact diagonalization of an algebraically calculated matrix
based on powers of Morse coordinate. The present work focuses on the 1D
potential of diatomic molecules: as typical examples, we apply this method to
the standard Lennard-Jones oscillator, and to the ab initio potential of the H2
molecule. Global cm-1 accuracy is exhibited through the H2 spectrum, obtained
through the diagonalization of a 30 x 30 matrix. This theory is at the root of
a new method to obtain globally accurate vibrational spectral data in the
context of the multi-dimensional potential of polyatomic molecules, at an
affordable computational cost.Comment: 30 pages including 6 figure
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov -widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the -widths can be controlled by that of the error achieved by best
-term approximations using polynomials in the parametric variable. However,
we prove that in certain relevant instances where the diffusion coefficients
are piecewise constant over a partition of the physical domain, the -widths
exhibit significantly faster decay. This, in turn, yields a theoretical
justification of the fast convergence of reduced basis or POD methods when
treating such parametric PDEs. Our results are confirmed by numerical
experiments, which also reveal the influence of the partition geometry on the
decay of the -widths.Comment: 27 pages, 6 figure
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