Search

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as$$r(x)=\sum _{i=0}^n a_i(x) f_i\big /\sum _{j=0}^m b_j(x)$$r(x)=∑i=0nai(x)fi/∑j=0mbj(x)in terms of data values$$f_i$$fiand some functions$$a_i$$aiand$$b_j$$bjfor$$i=0,\ldots ,n$$i=0,…,nand$$j=0,\dots ,m$$j=0,⋯,mwith a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows withn, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless ofn. We support our theoretical results with numerical experiments.

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D'Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185-1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix. © 2006 Elsevier B.V. All rights reserved.

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($$\lambda$$ λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the $$\lambda$$ λ -greedy algorithm enables us to define target-data-independent interpolation nodes.

In the numerical methods for the Laplace transform inversion the errors quantification in computational processes is a crucial issue. In this paper, we propose two inversion methods based on smoothing splines combined with a procedure for the derivation of error bounds. In particular, we numerically study the impact of the fitting error amplification through the analysis of several sources of error and their propagation.

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the considered dynamics, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo-Levi spaces. The performances of the method are then tested on functions which are common in the framework of the Laplace transform inversion. Comparisons with Support Vector Regression (SVR) are also carried out and show that the proposed method is effective particularly since it does not require to train complex architecture constructions.

An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and we also face up to instability issues. This produces an effective tool for real applications, as confirmed by comparisons with classical grid-based approaches.

Minimizing variational models by means of (un)constrained optimization algorithms is a well-known approach for dealing with the image denoising problem. In this paper, we propose a modification of the widely explored TV-ROF model named H-TV-ROF, in which a penalty term based on higher order derivatives is added. A Split Bregman iterative scheme is used to solve the proposed model and its convergence is proved. The performance of the new algorithm is analized and compared with TV-ROF on a set of numerical experiments.

Different methods for the numerical evaluations of the inverse Laplace and inverse of joint Laplace–Hankel integral transforms are applied to solve a wide range of initial-boundary value problems often arising in engineering and applied mathematics. The aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature. Most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half-space to concentrated impulse or those related to viscoelastic wave motion in layered media. These classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations, theoretical seismology, soil–structure-interaction in time domain and so on. Therefore, their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems. Some different methods are also investigated in detail and compared for the inversion of the joint Hankel–Laplace transforms, where more sophisticated integrand functions are encountered. It is shown that Durbin, Crump, D’Amore, Fixed-Talbot, Gaver–Whyn–Rho (GWR), and Direct Integration methods have excellent performance and produce good results when applied to the same problems. On the contrary, Gaver–Stehfest and Piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for “ simple ” transformed functions. Particularly the performance of GWR algorithm is very good even for transformed functions with infinite number of singularities, where the other methods fail. In addition, in case of double integral transforms, only the Fixed-Talbot, Durbin and Weeks methods are recommended.