Topic
Simple rational approximation
About: Simple rational approximation is a research topic. Over the lifetime, 6 publications have been published within this topic receiving 124 citations.
Papers
TL;DR: A main contribution of the present work is the observation that all approaches can, in fact, be interpreted as realizing different rational approximations of a univariate function over the spectrum of the original (non-fractional) diffusion operator.
Abstract: In recent years, a number of numerical methods for the solution of fractional Laplace and, more generally, fractional diffusion problems have been proposed. The approaches are quite diverse and include, among others, the use of best uniform rational approximations, quadrature for Dunford–Taylor-like integrals, finite element approaches for a localized elliptic extension into a space of increased dimensions, and time stepping methods for a parabolic reformulation of the fractional differential equation. A systematic comparison, both theoretical and experimental, of these approaches has thus far been lacking. A main contribution of the present work is the observation that all approaches mentioned above can, in fact, be interpreted as realizing different rational approximations of a univariate function over the spectrum of the original (non-fractional) diffusion operator. While this is obvious for some of the methods, it is a new result in particular for extension-based and time stepping approaches. This observation allows us to cast all described methods into a unified theoretical and computational framework, which has a number of benefits. Theoretically, it enables us to develop new convergence proofs for several of the studied methods, clarifies similarities and differences between the approaches, suggests how to design new and improved methods, and allows a direct comparison of the relative performance of the various methods. Practically, it provides a single, simple to implement, efficient and fully parallel algorithm for the realization of all studied methods; for instance, this immediately yields a fast and memory-efficient way of realizing all tensor product extension methods and lets us parallelize the otherwise inherently sequential time stepping approach. Finally, we present a detailed numerical study comparing all investigated methods for various fractional exponents and draw conclusions from the results. The comparison is made fair by the central insight that the computational effort of all these methods depends only on a single parameter, the degree of the underlying rational approximation. As a point of comparison, we also test a simple rational approximation method based on a black-box direct rational approximation algorithm which performs very well in practice.
76 citations
TL;DR: The problem of C 2 Hermite interpolation by Pythagorean Hodograph (PH) space curves is solved and a four-dimensional family of PH interpolants of degree 9 are constructed and a geometrically invariant parameterization of this family is introduced.
Abstract: We solve the problem of C 2 Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of C 2 space boundary data (two points with associated first and second derivatives) we construct a four-dimensional family of PH interpolants of degree 9 and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the "reversed" set of boundary data is simply the "reversed" original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree 9 which is globally C 2 , and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.
52 citations
5 Sep 2005
TL;DR: A geometrically invariant parameterization of the family of interpolants is introduced to identify a particular solution, which has the following properties: preserves planarity, i.e., the interpolant to planar data is a planar PH curve and has the best possible approximation order.
Abstract: As observed by Farouki et al.[9], any set of C1 space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. In general there exists a two dimensional family of interpolants.
In this paper we study the properties of this family in more detail. We introduce a geometrically invariant parameterization of the family of interpolants. This parameterization is used to identify a particular solution, which has the following properties. Firstly, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Secondly, it has the best possible approximation order (4). Thirdly, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. These observations lead to a fast and precise algorithm for converting any (possibly piecewise) analytical curve into a piecewise PH curve of degree 5 which is globally C1.
Finally we exploit the rational frames associated with any space PH curve (the Euler-Rodrigues frame) in order to obtain a simple rational approximation of pipe surfaces with a piecewise analytical spine curve and we analyze its approximation order.
40 citations
TL;DR: A simple rational approximation to the solution of the rough Heston Riccati equation valid in a region of its domain relevant to option valuation is presented and pricing using this approximation is both fast and very accurate.
Abstract: We present a simple rational approximation to the solution of the rough Heston Riccati equation valid in a region of its domain relevant to option valuation. Pricing under rough Heston using this approximation is both fast and very accurate.
32 citations
TL;DR: In this paper, the authors present a rational approximation that is more accurate, has a wider range of applicability, and is even simpler in mathematical form than the polynomial approximation, but its primary advantage is that it yields heat capacity estimates with error no greater than 0.03% for a wide range of measurement configurations.
Abstract: A number of mathematical expressions are available for calculating soil volumetric heat capacity from data obtained with the dual-probe heat-pulse (DPHP) method. One of the more attractive options is a polynomial approximation that is simple to evaluate and yields estimates of heat capacity with error no greater than about 2% for typical DPHP measurement configurations (i.e., configurations with a probe spacing of approximately 6 mm and a heating duration in the range of 8–15 s). Unfortunately, the polynomial approximation is less accurate for measurement configurations with relatively small probe spacing or a relatively long heating duration. In this note we present a rational approximation that is more accurate, has a wider range of applicability, and is even simpler in mathematical form than the polynomial approximation. The rational approximation has error no greater than 0.015% for typical DPHP measurement configurations, but its primary advantage is that it yields heat capacity estimates with error no greater than 0.03% for a wide range of measurement configurations.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 1 |
| 2015 | 1 |
| 2007 | 1 |
| 2005 | 1 |
| 1983 | 1 |