1,721,097 research outputs found
Penalized hyperbolic-polynomial splines
No full textWith the aim of generalizing P-splines, we here define a special type of penalized splines, called HP-splines, where polynomial splines are replaced by the richer class of hyperbolic-polynomial splines and a suitably tailored discrete penalty term is used. Hyperbolic-polynomial splines, important in several applications, are a natural generalization of polynomial splines consisting of piecewise-defined functions with segments spanned by ‘atoms'of type xreαx where r=0,...,l and α∈R. HP-splines, that reduce to P-splines for α=0, are more suitable to data with an exponential trend which is frequent in applications
Data-Driven Extrapolation Via Feature Augmentation Based on Variably Scaled Thin Plate Splines
Full text linkThe 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 target functions, 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 showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions
Feature Augmentation for Numerical Inversion of Multi-exponential Decay Curves
No full textMulti-exponential decay curves describe the nuclear magnetic resonance response to radiofrequency exposure. This problem is strongly related to finding a non-negative function given a finite number of noisy values of its Laplace Transform. The solution of this inverse problem takes advantage from a functional modelling of the data. We propose a fitting procedure based on the definition of a meshfree interpolant computed via a variably scaled kernel strategy. A parametric weighted sum of a finite number of exponential terms is introduced to describe the data behaviour; a data-driven procedure estimates its free parameters to scale the kernel, acting as a feature augmentation strategy. The performances of this procedure are investigated on real data sets from nuclear magnetic resonance acquisitions in the context of food science
Quality assurance of Gaver’s formula for multi-precision Laplace transform inversion in real case
No full textWe are concerned with Gaver’s formula, which is at the heart of a numerical algorithm, widely used in scientific and engineering applications, for computing approximations of inverse Laplace transform in multi-precision arithmetic systems. We demonstrate that, once parameters n (i.e. the number of terms of Gaver’s formula) and δ (i.e. an upper bound on noise on data) are given, then the number of correct significant digits of computed values of the inverse function is bounded above by −┌log10 (δ)┐ + 1. In case of noise free data this number is arbitrarily large, as it is bounded below by n. We establish the requirement of the multi-precision system ensuring that the quality of numerical results is fulfilled. Experiments and comparisons validate the effectiveness of such approach
An analytical model for carrier-facilitated solute transport in weakly heterogeneous porous media
No full textCarrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer's heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase
On the numerical stability of linear barycentric rational interpolation
No full textThe 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) = Sigma(n)(i=0) a(i) (x) f(i) / Sigma(m)(j=0) b(j) (x) in terms of data values f(i) and some functions a(i) and b(j) for i = 0, ..., n and j = 0, ..., m with 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 with n, 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 of n. We support our theoretical results with numerical experiments.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) = Sigma(n)(i=0) a(i) (x) f(i) / Sigma(m)(j=0) b(j) (x) in terms of data values f(i) and some functions a(i) and b(j) for i = 0, ..., n and j = 0, ..., m with 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 with n, 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 of n. We support our theoretical results with numerical experiments
Publisher Correction to: On the numerical stability of linear barycentric rational interpolation (September, 10.1007/s00211-022-01316-w, 2022)
No full textUsing local PHS+poly approximations for Laplace transform inversion by Gaver-Stehfest algorithm
No full textThe Laplace transform inversion is a well-known ill-conditioned problem and many numerical schemes in literature have investigated how to solve it. In this paper, we revise the Gaver-Stehfest method by using polyharmonic splines augmented with polynomials to approximate the Laplace transform into the numerical inversion formula. Theoretical accuracy bounds for the fitting model are given. Discussions on the effectiveness of the inversion algorithm are produced and confirmed by numerical experiments about approximation errors and inversion results. Comparisons with an existing model are also presented
An efficient algorithm for regularization of Laplace transform inversion in real case
No full textAbstractWe 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
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