1,677 research outputs found
Conditioning bounds for traveltime tomography in layered media
This paper revisits the problem of recovering a smooth, isotropic, layered
wave speed profile from surface traveltime information. While it is classic
knowledge that the diving (refracted) rays classically determine the wave speed
in a weakly well-posed fashion via the Abel transform, we show in this paper
that traveltimes of reflected rays do not contain enough information to recover
the medium in a well-posed manner, regardless of the discretization. The
counterpart of the Abel transform in the case of reflected rays is a Fredholm
kernel of the first kind which is shown to have singular values that decay at
least root-exponentially. Kinematically equivalent media are characterized in
terms of a sequence of matching moments. This severe conditioning issue comes
on top of the well-known rearrangement ambiguity due to low velocity zones.
Numerical experiments in an ideal scenario show that a waveform-based model
inversion code fits data accurately while converging to the wrong wave speed
profile
Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
Inspired by ideas taken from the machine learning literature, new
regularization techniques have been recently introduced in linear system
identification. In particular, all the adopted estimators solve a regularized
least squares problem, differing in the nature of the penalty term assigned to
the impulse response. Popular choices include atomic and nuclear norms (applied
to Hankel matrices) as well as norms induced by the so called stable spline
kernels. In this paper, a comparative study of estimators based on these
different types of regularizers is reported. Our findings reveal that stable
spline kernels outperform approaches based on atomic and nuclear norms since
they suitably embed information on impulse response stability and smoothness.
This point is illustrated using the Bayesian interpretation of regularization.
We also design a new class of regularizers defined by "integral" versions of
stable spline/TC kernels. Under quite realistic experimental conditions, the
new estimators outperform classical prediction error methods also when the
latter are equipped with an oracle for model order selection
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
Model order reduction of time-delay systems using a laguerre expansion technique
The demands for miniature sized circuits with higher operating speeds have increased the complexity of the circuit, while at high frequencies it is known that effects such as crosstalk, attenuation and delay can have adverse effects on signal integrity. To capture these high speed effects a very large number of system equations is normally required and hence model order reduction techniques are required to make the simulation of the circuits computationally feasible. This paper proposes a higher order Krylov subspace algorithm for model order reduction of time-delay systems based on a Laguerre expansion technique. The proposed technique consists of three sections i.e., first the delays are approximated using the recursive relation of Laguerre polynomials, then in the second part, the reduced order is estimated for the time-delay system using a delay truncation in the Laguerre domain and in the third part, a higher order Krylov technique using Laguerre expansion is computed for obtaining the reduced order time-delay system. The proposed technique is validated by means of real world numerical examples
Stable super-resolution limit and smallest singular value of restricted Fourier matrices
Super-resolution refers to the process of recovering the locations and
amplitudes of a collection of point sources, represented as a discrete measure,
given of its noisy low-frequency Fourier coefficients. The recovery
process is highly sensitive to noise whenever the distance between the
two closest point sources is less than . This paper studies the {\it
fundamental difficulty of super-resolution} and the {\it performance guarantees
of a subspace method called MUSIC} in the regime that .
The most important quantity in our theory is the minimum singular value of
the Vandermonde matrix whose nodes are specified by the source locations. Under
the assumption that the nodes are closely spaced within several well-separated
clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our
estimate is given as a weighted sum, where each term only depends on
the configuration of each individual clump. This implies that, as the noise
increases, the super-resolution capability of MUSIC degrades according to a
power law where the exponent depends on the cardinality of the largest clump.
Numerical experiments validate our theoretical bounds for the minimum singular
value and the resolution limit of MUSIC.
When there are point sources located on a grid with spacing , the
fundamental difficulty of super-resolution can be quantitatively characterized
by a min-max error, which is the reconstruction error incurred by the best
possible algorithm in the worst-case scenario. We show that the min-max error
is closely related to the minimum singular value of Vandermonde matrices, and
we provide a non-asymptotic and sharp estimate for the min-max error, where the
dominant term is .Comment: 47 pages, 8 figure
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