73,625 research outputs found
Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis
The maximum entropy principle is a powerful tool for solving underdetermined
inverse problems. This paper considers the problem of discretizing a continuous
distribution, which arises in various applied fields. We obtain the
approximating distribution by minimizing the Kullback-Leibler information
(relative entropy) of the unknown discrete distribution relative to an initial
discretization based on a quadrature formula subject to some moment
constraints. We study the theoretical error bound and the convergence of this
approximation method as the number of discrete points increases. We prove that
(i) the theoretical error bound of the approximate expectation of any bounded
continuous function has at most the same order as the quadrature formula we
start with, and (ii) the approximate discrete distribution weakly converges to
the given continuous distribution. Moreover, we present some numerical examples
that show the advantage of the method and apply to numerically solving an
optimal portfolio problem.Comment: 20 pages, 14 figure
On certain non-unique solutions of the Stieltjes moment problem
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form (2rn)! and [(rn)!]2. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for r > 1 both forms give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems
Histogram Tomography
In many tomographic imaging problems the data consist of integrals along
lines or curves. Increasingly we encounter "rich tomography" problems where the
quantity imaged is higher dimensional than a scalar per voxel, including
vectors tensors and functions. The data can also be higher dimensional and in
many cases consists of a one or two dimensional spectrum for each ray. In many
such cases the data contain not just integrals along rays but the distribution
of values along the ray. If this is discretized into bins we can think of this
as a histogram. In this paper we introduce the concept of "histogram
tomography". For scalar problems with histogram data this holds the possibility
of reconstruction with fewer rays. In vector and tensor problems it holds the
promise of reconstruction of images that are in the null space of related
integral transforms. For scalar histogram tomography problems we show how bins
in the histogram correspond to reconstructing level sets of function, while
moments of the distribution are the x-ray transform of powers of the unknown
function. In the vector case we give a reconstruction procedure for potential
components of the field. We demonstrate how the histogram longitudinal ray
transform data can be extracted from Bragg edge neutron spectral data and
hence, using moments, a non-linear system of partial differential equations
derived for the strain tensor. In x-ray diffraction tomography of strain the
transverse ray transform can be deduced from the diffraction pattern the full
histogram transverse ray transform cannot. We give an explicit example of
distributions of strain along a line that produce the same diffraction pattern,
and characterize the null space of the relevant transform.Comment: Small corrections from last versio
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