28,821 research outputs found
Application of Fredholm integral equations inverse theory to the radial basis function approximation problem
This paper reveals and examines the relationship between the solution and stability of Fredholm integral equations and radial basis function approximation or interpolation. The underlying system (kernel) matrices are shown to have a smoothing property which is dependent on the choice of kernel. Instead of using the condition number to describe the ill-conditioning, hence only looking at the largest and smallest singular values of the matrix, techniques from inverse theory, particularly the Picard condition, show that it is understanding the exponential decay of the singular values which is critical for interpreting and mitigating instability. Results on the spectra of certain classes of kernel matrices are reviewed, verifying the exponential decay of the singular values. Numerical results illustrating the application of integral equation inverse theory are also provided and demonstrate that interpolation weights may be regarded as samplings of a weighted solution of an integral equation. This is then relevant for mapping from one set of radial basis function centers to another set. Techniques for the solution of integral equations can be further exploited in future studies to find stable solutions and to reduce the impact of errors in the data
A Threshold Regularization Method for Inverse Problems
A number of regularization methods for discrete inverse problems consist in
considering weighted versions of the usual least square solution. However,
these so-called filter methods are generally restricted to monotonic
transformations, e.g. the Tikhonov regularization or the spectral cut-off. In
this paper, we point out that in several cases, non-monotonic sequences of
filters are more efficient. We study a regularization method that naturally
extends the spectral cut-off procedure to non-monotonic sequences and provide
several oracle inequalities, showing the method to be nearly optimal under mild
assumptions. Then, we extend the method to inverse problems with noisy operator
and provide efficiency results in a newly introduced conditional framework
Stellar Content from high resolution galactic spectra via Maximum A Posteriori
This paper describes STECMAP (STEllar Content via Maximum A Posteriori), a
flexible, non-parametric inversion method for the interpretation of the
integrated light spectra of galaxies, based on synthetic spectra of single
stellar populations (SSPs). We focus on the recovery of a galaxy's star
formation history and stellar age-metallicity relation. We use the high
resolution SSPs produced by PEGASE-HR to quantify the informational content of
the wavelength range 4000 - 6800 Angstroms.
A detailed investigation of the properties of the corresponding simplified
linear problem is performed using singular value decomposition. It turns out to
be a powerful tool for explaining and predicting the behaviour of the
inversion. We provide means of quantifying the fundamental limitations of the
problem considering the intrinsic properties of the SSPs in the spectral range
of interest, as well as the noise in these models and in the data.
We performed a systematic simulation campaign and found that, when the time
elapsed between two bursts of star formation is larger than 0.8 dex, the
properties of each episode can be constrained with a precision of 0.04 dex in
age and 0.02 dex in metallicity from high quality data (R=10 000,
signal-to-noise ratio SNR=100 per pixel), not taking model errors into account.
The described methods and error estimates will be useful in the design and in
the analysis of extragalactic spectroscopic surveys.Comment: 31 pages, 23 figures, accepted for publication in MNRA
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