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In this paper, we extendthe structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples. [ABSTRACT FROM AUTHOR]

We investigate the piecewise linear nonconforming Crouzeix–Raviart and the lowest order Raviart–Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix–Raviart and the Raviart–Thomas finite-element approximate problems. We next present the equivalence between the Raviart–Thomas finite-element method and the enriched Crouzeix–Raviart finite-element method. We emphasize that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained. [ABSTRACT FROM AUTHOR]

• A mental stress state diagnostic for timely intervention from clinicians. • Deep neural network with hierarchical learning capabilities using IoT based biomarkers. • Learns using both wrist-based and chest-based sensor bio-signal features. • Model-level fusion of high-level representations for stress state classification. Affective state recognition at an early stage can help in mood stabilization, stress and depression management for mental well-being. Pro-active and remote mental healthcare warrants the use of various biomarkers to detect the affective mental state of the individual by evaluating the daily activities. With the easy accessibility of IoT-based sensors for healthcare, observable and quantifiable characteristics of our body, physiological changes in the body can be measured and tracked using various wearable devices. This work puts forward a model for mental stress state detection using sensor-based bio-signals. A multi-level deep neural network with hierarchical learning capabilities of convolution neural network is proposed. Multivariate time-series data consisting of both wrist-based and chest-based sensor bio-signals is trained using a hierarchy of networks to generate high-level features for each bio-signal feature. A model-level fusion strategy is proposed to combine the high-level features into one unified representation and classify the stress states into three categories as baseline, stress and amusement. The model is evaluated on the WESAD benchmark dataset for mental health and compares favourably to state-of-the-art approaches giving a superlative performance accuracy of 87.7%. [ABSTRACT FROM AUTHOR]

It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results. [ABSTRACT FROM AUTHOR]

In this article we present a new adaptive algorithm for solving 2D interpolation problems of large scattered data sets through the radial basis function partition of unity method. Unlike other time-consuming schemes this adaptive method is able to efficiently deal with scattered data points with highly varying density in the domain. This target is obtained by decomposing the underlying domain in subdomains of variable size so as to guarantee a suitable number of points within each of them. The localization of such points is done by means of an efficient search procedure that depends on a partition of the domain in square cells. For each subdomain the adaptive process identifies a predefined neighborhood consisting of one or more levels of neighboring cells, which allows us to quickly find all the subdomain points. The algorithm is further devised for an optimal selection of the local shape parameters associated with radial basis function interpolants via leave-one-out cross validation and maximum likelihood estimation techniques. Numerical experiments show good performance of this adaptive algorithm on some test examples with different data distributions. The efficacy of our interpolation scheme is also pointed out by solving real world applications. [ABSTRACT FROM AUTHOR]

We propose a fast algorithm for the variable-order (VO) space-fractional advection-diffusion equations with nonlinear source terms on a finite domain. Due to the impact of the space-dependent the VO, the resulting coefficient matrices arising from the finite difference discretization of the fractional advection-diffusion equation are dense without Toeplitz-like structure. By the properties of the elements of coefficient matrices, we show that the off-diagonal blocks can be approximated by low-rank matrices. Then we present a fast algorithm based on the polynomial interpolation to approximate the coefficient matrices. The approximation can be constructed in O (k N) operations and requires O (k N) storage with N and k being the number of unknowns and the approximants, respectively. Moreover, the matrix-vector multiplication can be implemented in O (k N log N) complexity, which leads to a fast iterative solver for the resulting linear systems. The stability and convergence of the new scheme are also studied. Numerical tests are carried out to exemplify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We introduce a new local meshfree method for the approximation of the Laplace–Beltrami operator on a smooth surface in R 3 . It is a direct method that uses radial basis functions augmented with multivariate polynomials. A key element of this method is that it does not need an explicit expression of the surface, which can be simply defined by a set of scattered nodes. Likewise, it does not require expressions for the surface normal vectors or for the curvature of the surface, which are approximated using explicit formulas derived in the paper. An additional advantage is that it is a local method and, hence, the matrix that approximates the Laplace–Beltrami operator is sparse, which translates into good scalability properties. The convergence, accuracy and other computational characteristics of the proposed method are studied numerically. Its performance is shown by solving two reaction–diffusion partial differential equations on surfaces; the Turing model for pattern formation, and the Schaeffer's model for electrical cardiac tissue behavior. [ABSTRACT FROM AUTHOR]

In this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms. [ABSTRACT FROM AUTHOR]

We offer the algorithm for choosing tension parameters of the generalized splines for convexity preserving interpolation. The resulting spline minimally differs from the classical cubic spline and coincides with it if sufficient convexity conditions are satisfied for the last one. We consider specific algorithms for different generalized cubic splines such as rational, exponential, variable power, hyperbolic splines, and splines with additional knots. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of finding an optimal value of the shape parameter in radial basis function interpolation. In particular, we propose the use of a leave-one-out cross validation (LOOCV) technique combined with univariate global optimization methods, which involve strategies of global optimization with pessimistic improvement (GOPI) and global optimization with optimistic improvement (GOOI). This choice is carried out to overcome serious issues of commonly used optimization routines that sometimes result in shape parameter values are not globally optimal. New locally-biased versions of geometric and information Lipschitz global optimization algorithms are presented. Numerical experiments and applications to real-world problems show a promising performance and efficacy of the new algorithms, called LOOCV-GOPI and LOOCV-GOOI, in comparison with their direct competitors. [ABSTRACT FROM AUTHOR]

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition. [ABSTRACT FROM AUTHOR]

This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments. FERMILAB-CONF-19-468-E-T, LHCHXSWG-2019-005

In 2012 the ATLAS and CMS collaborations discovered at the LHC the Higgs boson decaying to vector bosons. This discovery has provided a strong indication that the mechanism of Electroweak Symmetry Breaking (EWSB) is similar to the one predicted by Brout-Englert-Higgs (BEH) nearly 50 years before. Since then, one of the priorities of the LHC program, as well as of the majority of the future collider proposals, is to measure directly the parameters of the EWSB potential. The goal is to identify if it has indeed the straightforward quartic shape predicted by BEH or it is more complex, as the result of an unexplored physics nature. The answer to this major scientific question will have a considerable impact on our understanding of vacuum properties and the history of the universe through the EWSB during the Big Bang. The only direct way to probe these couplings is through the measure of the production of multiple Higgs bosons, two being the simplest case. In this paper, we present a comprehensive review of the current searches and the state of the art insights on the topic. In particular, we explain why this ambitious project is even more challenging than the discovery of the Higgs boson itself. Finally, we sketch the plans of the HEP community for how to access the parameters of the BEH mechanism. This review is adapted to a curious reader familiar with particle physics in general or a scientist who wants to have a landscape overview of the topic.

Imaging methods based on the absorption or scattering of atmospheric muons, collectively named under the neologism “muography”, exploit the abundant natural flux of muons produced from cosmic-ray interactions in the atmosphere. Recent years have seen a steep rise in the development of muography methods in a variety of innovative multidisciplinary approaches to study the interior of natural or human-made structures, establishing synergies between usually disconnected academic disciplines such as particle physics, geology, and archaeology. Muography also bears promise of immediate societal impact through geotechnical investigations, nuclear waste surveys, homeland security, and natural hazard monitoring. Our aim is to provide an introduction to this vibrant research area, starting from the physical principles at the basis of the methods and describing the main detector technologies and imaging tools, including their combination with conventional techniques from other disciplines, where appropriate. Then, we discuss critically some outstanding issues that affect a broad variety of applications, and the current state of the art in addressing them. Finally, we review several recent developments in the application of muography methods to specific use cases, without any pretence of exhaustiveness.

Demonstration of the light radiation pressure on a microscopic level by A. Ashkin led to the invention of optical tweezers (OT). Applied in the studies of living systems, OT have become a preferable instrument for the noninvasive study of microobjects, allowing manipulation and measurement of the mechanical properties of molecules, organelles, and cells. In the present paper, we overview OT applications in hemorheological research, placing emphasis on red blood cells but also discussing OT applications for the investigation of the biomechanics of leukocytes and platelets. Blood properties have always served as a primary parameter in medical diagnostics due to the interconnection with the physiological state of an organism. Despite blood testing being a well-established procedure of conventional medicine, there are still many complex processes that must be unraveled to improve our understanding and contribute to future medicine. OT are advancing single-cell research, promising new insights into individual cell characteristics compared to the traditional approaches. We review the fundamental and practical findings revealed in blood research through the optical manipulation, stretching, guiding, immobilization, and inter-/intracellular force measurements of single blood cells.

The complexity of collider data analyses has dramatically increased from early colliders to the CERN LHC. Reconstruction of the collision products in the particle detectors has reached a point that requires dedicated publications documenting the techniques, and periodic retuning of the algorithms themselves. Analysis methods evolved to account for the increased complexity of the combination of particles required in each collision event (final states) and for the need of squeezing every last bit of sensitivity from the data; physicists often seek to fully reconstruct the final state, a process that is mostly relatively easy at lepton colliders but sometimes exceedingly difficult at hadron colliders to the point of requiring sometimes using advanced statistical techniques such as machine learning. The need for keeping the publications documenting results to a reasonable size implies a greater level of compression or even omission of information with respect to publications from twenty years ago. The need for compression should however not prevent sharing a reasonable amount of information that is essential to understanding a given analysis. Infrastructures like Rivet or HepData have been developed to host additional material, but physicists in the experimental Collaborations often still send an insufficient amount of material to these databases. In this manuscript I advocate for an increase in the information shared by the Collaborations, and try to define a minimum standard for acceptable level of information when reporting the results of statistical procedures in High Energy Physics publications.

This article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

Abstract A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319–333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor–corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately $$4-\alpha $$ 4-α , where $$\alpha $$ α is the order of time derivative.

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of H -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using H -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using H -matrices is of almost linear complexity with respect to the number of centers. [ABSTRACT FROM AUTHOR]

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential exp (A) v has to be approximated by p m (s - 1 A) s v = exp (A + Δ A) v . We propose here a rigorous bound for the relative backward error Δ A 2 / A 2 , which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrödinger operators. The numerical results confirm the superiority of the new approach with respect to methods based on the classical power series expansion of the backward error for the matrices of our interest, both in terms of computational cost and achieved accuracy. [ABSTRACT FROM AUTHOR]

In this paper a special class of one-dimensional L-splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L-splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions permits a simple and direct numerical implementation. Applications are provided to financial data of the index S&P500, for the fast calculation of statistically interesting quantities, as cross validation (scores), generalized cross validation (scores) for finding the best smoothing parameter α , and the residual sum of squares. [ABSTRACT FROM AUTHOR]

• Enhance face verification performance for demographics effected by decision bias. • Operates on comparison score-level to enable using existing face recognition models. • Based on the notation of individual fairness to treat similar face groups similarly. • Effectiveness proved on three databases and two face embeddings. • Unsupervised enhancement of the overall and intra-class recognition performance. Current face recognition systems achieve high progress on several benchmark tests. Despite this progress, recent works showed that these systems are strongly biased against demographic sub-groups. Consequently, an easily integrable solution is needed to reduce the discriminatory effect of these biased systems. Previous work mainly focused on learning less biased face representations, which comes at the cost of a strongly degraded overall recognition performance. In this work, we propose a novel unsupervised fair score normalization approach that is specifically designed to reduce the effect of bias in face recognition and subsequently lead to a significant overall performance boost. Our hypothesis is built on the notation of individual fairness by designing a normalization approach that leads to treating "similar" individuals "similarly". Experiments were conducted on three publicly available datasets captured under controlled and in-the-wild circumstances. Results demonstrate that our solution reduces demographic biases, e.g. by up to 82.7% in the case when gender is considered. Moreover, it mitigates the bias more consistently than existing works. In contrast to previous works, our fair normalization approach enhances the overall performance by up to 53.2% at false match rate of 10 − 3 and up to 82.9% at a false match rate of 10 − 5. Additionally, it is easily integrable into existing recognition systems and not limited to face biometrics. [ABSTRACT FROM AUTHOR]

In this paper we aim to compare a popular numerical method with a new, recently proposed meshless approach for Heston PDE resolution. In finance, most famous models can be reformulated as PDEs, which are solved by finite difference and Monte Carlo methods. In particular, we focus on Heston model PDE and we solve it via radial basis functions (RBF) methods and alternating direction implicit. RBFs have become quite popular in engineering as meshless methods: they are less computationally heavy than finite differences and can be applied for high-order problems. [ABSTRACT FROM AUTHOR]

Polynomial interpolation or regression models are an important tool in Derivative-free Optimization, acting as surrogates of the real function. In this work, we propose the use of these models in the multiobjective framework of directional direct search, namely the one of Direct Multisearch. Previously evaluated points are used to build quadratic polynomial models, which are minimized in an attempt of generating nondominated points of the true function, defining a search step for the algorithm. Numerical results state the competitiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The Floater–Hormann family of the barycentric rational interpolants has recently gained popularity because of its excellent stability properties and highly order of convergence. The purpose of this paper is to design highly accurate and stable schemes based on this family of interpolants for the numerical solution of stiff Volterra integral equations of the second kind. [ABSTRACT FROM AUTHOR]

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. [ABSTRACT FROM AUTHOR]

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach. [ABSTRACT FROM AUTHOR]

In this paper, a proficient numerical technique for the time-fractional telegraph equation (TFTE) is proposed. The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme. This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space. A stability analysis of the scheme is presented to confirm that the errors do not amplify. A convergence analysis is also presented. Computational experiments are carried out in addition to verify the theoretical analysis. Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling. [ABSTRACT FROM AUTHOR]

In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell' Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

As an extension of Lagrange interpolation, we introduce a class of interpolation formulas and study their existence and uniqueness. In the sequel, we consider some particular cases and construct the corresponding weighted quadrature rules. Numerical examples are finally given and compared. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing. [ABSTRACT FROM AUTHOR]

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields n points at one sitting for a given integer n. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the P-greedy algorithm, which is known to provide nearly optimal points. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images. [ABSTRACT FROM AUTHOR]

We discuss the problem of fitting a smooth regular curve γ : [ 0 , T ] → E n based on reduced data Q m = { q i } i = 0 m in arbitrary Euclidean space E n . The respective interpolation knots T = { t i } i = 0 m satisfying q i = γ (t i) are assumed to be unknown. In our setting the substitutes T λ = { t ^ i } i = 0 m of T are selected according to the so-called exponential parameterization governed by Q m and λ ∈ [ 0 , 1 ] . A modified Hermite interpolant γ ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit (T ^ λ , Q m) . The case of λ = 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating γ ∈ C 4 by γ ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4) the remaining λ ∈ [ 0 , 1) render a linear convergence order in γ ^ H ≈ γ for any Q m sampled more-or-less uniformly. The related analysis relies on comparing the difference γ - γ ^ H ∘ ϕ H in which ϕ H forms a special mapping between [0, T] and [ 0 , T ^ ] with T ^ = t ^ m . In this paper: (a) several sufficient conditions enforcing ϕ H to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of γ - γ ^ H ∘ ϕ H [from Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica. [ABSTRACT FROM AUTHOR]

In this work, we use a kind of C 1 rational interpolation splines in one and two dimensions to generate curves and surfaces with region control. Simple data-dependent sufficient constraints are derived on the local control parameters to generate C 1 interpolation curves lying strictly between two given piecewise linear curves and C 1 interpolation surfaces lying strictly between two given piecewise bi-cubic blending linear interpolation surfaces. [ABSTRACT FROM AUTHOR]

We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges to a polynomial method. In an additional one-dimensional extension, we allow also the points to be selected optimally and show that in this case convergence is to the unique Gaussian quadrature–type method that achieves the maximal polynomial degree of exactness. The proofs are based on an explicit characterisation of the reproducing kernel Hilbert space of the Gaussian kernel in terms of exponentially damped polynomials. [ABSTRACT FROM AUTHOR]