A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Characterising ontogenetic niche shifts in Nile crocodile using stable isotope (δ13C, δ15N) analyses of scute keratin

Nile crocodiles undergo a three to five order of magnitude increase in body size during their lifespan. This shift coincides with a change in resource and habitat use which influences the strength, type and symmetry of interactions with other species. Identifying size-specific crocodile groups displaying similar traits is important for conservation planning. Here, we illustrate how stable carbon (δ¹³ C) and nitrogen (δ¹⁵ N) isotope analysis of scute keratin, together with breakpoint modelling analysis can be used to characterise ontogenetic niche shifts. Using a sample set of 238 crocodiles from the Okavango Delta, Botswana (35–463 cm total length), we found prominent size-related changes in the scute keratin δ¹³ C and δ¹⁵ N profiles close to 40 and 119 cm snout-vent length. The first shift corroborated the findings of a traditional stomach-content study conducted on the same population at the same time, and the second conformed to known crocodile ecology. This approach can be used as a first approximation to identify size-specific groups within crocodile populations, and these can then be investigated further using isotopic or other methods.     

Piecewise linear prewavelets over type-2 triangulations

In this article, we study the construction of piecewise linear prewavelets over type-2 triangulations. Different from a so-called semi-prewavelet approach, we investigate the orthogonal conditions directly and obtain parameterized prewavelets with a smaller support. The conditions for parameterized prewavelet basis on the parameters are also given.     

Variable Selection in Varying-Coefficient Models Using P-Splines

In this article, we consider nonparametric smoothing and variable selection in varying-coefficient models. Varying-coefficient models are commonly used for analyzing the time-dependent effects of covariates on responses measured repeatedly (such as longitudinal data). We present the P-spline estimator in this context and show its estimation consistency for a diverging number of knots (or B-spline basis functions). The combination of P-splines with nonnegative garrote (which is a variable selection method) leads to good estimation and variable selection. Moreover, we consider APSO (additive P-spline selection operator), which combines a P-spline penalty with a regularization penalty, and show its estimation and variable selection consistency. The methods are illustrated with a simulation study and real-data examples. The proofs of the theoretical results as well as one of the real-data examples are provided in the online supplementary materials.     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

Nikolskii type inequality for cardinal splines

The Nikolskii type inequality for cardinal splines

Collocation by singular splines

Splines determined by the kernel of the differential operator are known to be useful to solve the singular boundary value problems of the form . One of the most successful methods is the collocation method based on special Chebyshev splines. We investigate the construction of the associated B-splines based on knot-insertion algorithms for their evaluation, and their application in collocation at generalized Gaussian points. Specially, we show how to obtain these points as eigenvalues of a symmetric tridiagonal matrix of order k. This research was supported by Grant 037-1193086-2771, by the Ministry of science, education and sports of the Republic of Croatia.     

Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

Fast evaluation of polyharmonic splines in three dimensions

M. J. D. Powell ^{This paper concerns the fast evaluation of radial basis functions.It describes the mathematics of hierarchical and fast multipolemethods for fast evaluation of splines of the form where is a positive integer andp is a low-degree polynomial. Splines s of this form are polyharmonicsplines in 3 and have been found to be very useful for providingsolutions to scattered data interpolation problems in 3. Asit is now well known, hierarchical methods reduce the incrementalcost of a single extra evaluation from O(N) to O(log N) operationsand reduce the cost of a matrix–vector product (evaluationof s at all the centres) from O(N2) to O(N log N) operations.We give appropriate far- and near-field expansions, togetherwith error estimates, uniqueness theorems and translation formulae.A hierarchical code based on these formulae is detailed andsome numerical results are given.}