The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various ...The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.展开更多
Sedimentary cyclothems at different scales show formations’ fractal structure which can be reflected on logs. The slope of the power spectrum of log is related to the fractal dimension of formations. The fractal dime...Sedimentary cyclothems at different scales show formations’ fractal structure which can be reflected on logs. The slope of the power spectrum of log is related to the fractal dimension of formations. The fractal dimensions from two logs with similar vertical resolutions are the same. Using fractal interpolating algorithm density log can be reconstructed. The reconstructed log can be compared with core density in washout intervals.展开更多
Iterated function system （IFS） models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 （R is the set of real numbers）. In this paper, the piece-wi...Iterated function system （IFS） models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 （R is the set of real numbers）. In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn （n is an integer greater than 3）, which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a ＂mapping partial derivative＂, and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.展开更多
文摘The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.
文摘Sedimentary cyclothems at different scales show formations’ fractal structure which can be reflected on logs. The slope of the power spectrum of log is related to the fractal dimension of formations. The fractal dimensions from two logs with similar vertical resolutions are the same. Using fractal interpolating algorithm density log can be reconstructed. The reconstructed log can be compared with core density in washout intervals.
文摘Iterated function system （IFS） models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 （R is the set of real numbers）. In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn （n is an integer greater than 3）, which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a ＂mapping partial derivative＂, and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.
