763,307 research outputs found
A Short Tale of Long Tail Integration
Integration of the form , where is either
or , is widely
encountered in many engineering and scientific applications, such as those
involving Fourier or Laplace transforms. Often such integrals are approximated
by a numerical integration over a finite domain , leaving a truncation
error equal to the tail integration in addition
to the discretization error. This paper describes a very simple, perhaps the
simplest, end-point correction to approximate the tail integration, which
significantly reduces the truncation error and thus increases the overall
accuracy of the numerical integration, with virtually no extra computational
effort. Higher order correction terms and error estimates for the end-point
correction formula are also derived. The effectiveness of this one-point
correction formula is demonstrated through several examples
Accuracy analysis of the box-counting algorithm
Accuracy of the box-counting algorithm for numerical computation of the
fractal exponents is investigated. To this end several sample mathematical
fractal sets are analyzed. It is shown that the standard deviation obtained for
the fit of the fractal scaling in the log-log plot strongly underestimates the
actual error. The real computational error was found to have power scaling with
respect to the number of data points in the sample (). For fractals
embedded in two-dimensional space the error is larger than for those embedded
in one-dimensional space. For fractal functions the error is even larger.
Obtained formula can give more realistic estimates for the computed generalized
fractal exponents' accuracy.Comment: 3 figure
A Posteriori Error Control for the Binary Mumford-Shah Model
The binary Mumford-Shah model is a widespread tool for image segmentation and
can be considered as a basic model in shape optimization with a broad range of
applications in computer vision, ranging from basic segmentation and labeling
to object reconstruction. This paper presents robust a posteriori error
estimates for a natural error quantity, namely the area of the non properly
segmented region. To this end, a suitable strictly convex and non-constrained
relaxation of the originally non-convex functional is investigated and Repin's
functional approach for a posteriori error estimation is used to control the
numerical error for the relaxed problem in the -norm. In combination with
a suitable cut out argument, a fully practical estimate for the area mismatch
is derived. This estimate is incorporated in an adaptive meshing strategy. Two
different adaptive primal-dual finite element schemes, and the most frequently
used finite difference discretization are investigated and compared. Numerical
experiments show qualitative and quantitative properties of the estimates and
demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl
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