Piecewise Polynomial Functions Research Articles

Identification of Wiener state–space models utilizing Gaussian sum smoothing

In this paper, we address the problem of system identification for Wiener state–space models. Our approach is based on the Maximum Likelihood method and the Expectation–Maximization algorithm. In the problem of interest, we model the output nonlinearity as a piecewise polynomial function and we jointly estimate the parameters of the linear system with the coefficients of each polynomial section. In our proposal, the computation of the cost function in the Expectation–Maximization algorithm requires the computation of the joint distribution of the state and the output of the linear system given the output of the nonlinear block. These quantities are obtained from an approximation that leads to a novel Gaussian sum smoothing algorithm. Additionally, we show that our method also addresses the identification of state–space systems in which the output is produced by a known quantizer. We present numerical examples to illustrate the benefits of the proposed identification technique.

Biclustering analysis on tree-shaped time-series single cell gene expression data of Caenorhabditis elegans

BackgroundIn recent years, gene clustering analysis has become a widely used tool for studying gene functions, efficiently categorizing genes with similar expression patterns to aid in identifying gene functions. Caenorhabditis elegans is commonly used in embryonic research due to its consistent cell lineage from fertilized egg to adulthood. Biologists use 4D confocal imaging to observe gene expression dynamics at the single-cell level. However, on one hand, the observed tree-shaped time-series datasets have characteristics such as non-pairwise data points between different individuals. On the other hand, the influence of cell type heterogeneity should also be considered during clustering, aiming to obtain more biologically significant clustering results.ResultsA biclustering model is proposed for tree-shaped single-cell gene expression data of Caenorhabditis elegans. Detailedly, a tree-shaped piecewise polynomial function is first employed to fit non-pairwise gene expression time series data. Then, four factors are considered in the objective function, including Pearson correlation coefficients capturing gene correlations, p-values from the Kolmogorov-Smirnov test measuring the similarity between cells, as well as gene expression size and bicluster overlapping size. After that, Genetic Algorithm is utilized to optimize the function.ConclusionThe results on the small-scale dataset analysis validate the feasibility and effectiveness of our model and are superior to existing classical biclustering models. Besides, gene enrichment analysis is employed to assess the results on the complete real dataset analysis, confirming that the discovered biclustering results hold significant biological relevance.

Minimax Regret Robust Screening with Moment Information

Problem definition: We study a robust screening problem where a seller attempts to sell a product to a buyer knowing only the moment and support information of the buyer’s valuation distribution. The objective is to maximize the competitive ratio relative to an optimal hindsight policy equipped with full valuation information. Methodology/results: We formulate the robust screening problem as a linear programming problem, which can be solved efficiently if the support of the buyer’s valuation is finite. When the support of the buyer’s valuation is continuous and the seller knows the mean and the upper and lower bounds of the support for the buyer’s valuation, we show that the optimal payment is a piecewise polynomial function of the valuation with a degree of at most two. Moreover, we derive the closed-form competitive ratio corresponding to the optimal mechanism. The optimal mechanism can be implemented by a randomized pricing mechanism whose price density function is a piecewise inverse function adjusted by a constant. When the mean and variance are known to the seller, we propose a feasible piecewise polynomial approximation of the optimal payment function with a degree of at most three. We also demonstrate that the optimal competitive ratio exhibits a logarithmic decay with respect to the coefficient of variation of the buyer’s valuation distribution. Managerial implications: Our general framework provides an approach to investigating the value of moment information in the robust screening problem. We establish that even a loose upper bound of support or a large variance can guarantee a good competitive ratio. Funding: The research of S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72394395]. The research of S. Liu is partly supported by the National Natural Science Foundation of China [Grant NSFC-72072117]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0072 .

Theorems on direct and inverse approximation by algebraic polynomials and piecewise polynomials in the spaces H<sup>m</sup>(a, b) and B<sup>s</sup><sub>2,q</sub>(a, b)

The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.

Neural network-based method for determining vessel position by seabed relief

A seabed relief-based vessel position fixing model on the basis of a four-layered feedforward neural network is proposed. Hidden neurons have hyperbolic tangent activation functions. The model is constructed for 1-D case that can be considered as vessel motion throw a narrow channel or alongside fairway axis. A sequence of spot soundings is given for the network input. The linear coordinate registered for the last sounding forms the network output. The training set is formed by means of the intentional pseudorandom alteration of input samples in accordance with suspected limits of sea level variations and the constant error of its measurements. The validation set is not used. The Adamax algorithm is implemented for the neural network training. The maximum of absolute value of the prediction error is used as a performance criterion of the net. Modeling has been conducted with the Python programming language. The Tensorflow library is used for the creation, training and testing of the neural network. The depth is modelled as a piecewise polynomial function of the coordinate. The results of neural network testing with the use of noised input samples let to state that the neural net can determine a ship position by means of soundings with acceptable accuracy. Different combinations of the sea level error and the number of hidden neurons have been considered. For each of such combinations the network accuracy indicators have been calculated. The best results are obtained for the network with 100 hidden neurons per each layer.

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

A lower bound for the dimension of tetrahedral splines in large degree

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.

Unsolvability of Some Problems about Piecewise-Polynomial Functions

Unsolvability of Some Problems about Piecewise-Polynomial Functions

Построение С1-гладких кусочно-квадратичных функций при решении краевых задач уравнений 4-го порядка на треугольной сетке

In this paper, we present one approach to constructing continuously differentiable piecewise-quadratic functions on a triangular grid, based on smoothing a piecewise-linear function in the vicinity of edges and triangulation nodes. In the works [8], [7], the issues of approximation of the functional (1) in triangular grids and the convergence of the variational method for solving the boundary value problem of the equation (2) were studied. However, in the numerical solution there are difficulties associated with the construction of continuously differentiable piecewise polynomial functions on triangulations. In particular, their construction requires solving large systems of equations at each step of the variational method. When trying to get by with only continuous piecewise polynomial functions, we got a negative result (divergence of approximate solutions was found) [9]. In this paper, we circumvent the difficulties that arise — we indicate a method for constructing piecewise polynomial functions that have continuous partial derivatives. With the help of this class of functions, we have obtained formulas for approximating the functional (1) and tested the method on the example of a biharmonic equation.

Logarithmic moduli of roots of line bundles on curves

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho–Pandharipande–Schmitt and Holmes–Schwarz).

Tracking Nonradial Outflows in Extreme Ultraviolet and White Light Solar Images

Understanding the solar corona requires knowledge of its dynamics through its various layers and subsequent connectivity to the heliosphere. This requires understanding the nature of the outflows and the physical transitions through the middle corona (∼1.5–6.0 R s ). While this region is still inaccessible to in situ measurements, remote sensing observations are available, but their interpretation can be controversial due to line-of-sight effects and the nonradial motion of outflowing structures close to the Sun (≤3.0 R s ). In this work, we describe a method to mitigate these challenges by generating nonradial height–time profiles of outflows by using advanced image processing techniques. The northern and southern boundaries of a large equatorial streamer during the 2008 solar minimum were identified in STEREO/SECCHI solar images, using two different methodologies based on thresholds of brightness and piecewise polynomial function fitting. To address line-of-sight issues, we used tomographic reconstruction of the 3D distribution of the coronal electron density based on SECCHI/COR2 images. Spectral analysis of the time series of the position angle of the streamer boundary revealed its oscillatory nature at some heights at 36–48 and 10.5–14.6 hr. Dividing the distance between the northern and southern streamer boundaries in equal parts at each height, we obtained nonradial height–time paths from which we generated nonradial profiles of corona/solar wind plasma outflow. We tracked outflows as they moved uninterruptedly from the Sun in EUVI, through COR1 and into COR2. Finally, we discuss the preliminary results of nonradial plane-of-sky velocities for a coronal mass ejection and two small-scale features.

An algebraic characterization of B-splines

B-splines of order k can be viewed as a mapping N taking a (k+1)-tuple of increasing real numbers a0<⋯<ak and giving as a result a certain piecewise polynomial function. Looking at this mapping N as a whole, basic properties of B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) has local support contained in the interval [a0,ak]; (2) N(a0,…,ak) allows refinement, i.e. for every a∈∪j=0k−1(aj,aj+1) we have that if (α0,…,αk+1) is the increasing rearrangement of the points {a0,…,ak,a}, the ‘old’ function N(a0,…,ak) is a linear combination of the ‘new’ functions N(α0,…,αk) and N(α1,…,αk+1); (3) N is translation and dilation invariant. It is easy to see that derivatives of N(a0,…,ak) satisfy properties (1)–(3) as well.Let F be a mapping taking (k+1)-tuples of increasing real numbers to some generalized function. In this paper we show that under some additional mild condition on the size of the supports of F(a0,…,ak) relative to the interval [a0,ak], properties (1)–(3) are already sufficient to deduce that F(a0,…,ak) is a non-zero multiple of (some derivative of) a B-spline function. However, and somewhat surprisingly, we explicitly give examples of choices of F satisfying (1)–(3) but are not of this form.

Characters of classical groups, Schur-type functions and discrete splines

We study a spectral problem related to finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$ and $SO(2N)$, which form the classical series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$, respectively. Irreducible characters of these three series are given by $N$-variate symmetric polynomials. The spectral problem in question consists in the decomposition of characters after their restriction to subgroups of the same type but smaller rank $K&lt;N$. The main result of the paper is the derivation of explicit determinantal formulae for the coefficients of the expansion. In fact, first we compute these coefficients in greater generality - for the multivariate symmetric Jacobi polynomials depending on two continuous parameters. Next, we show that the formulae are drastically simplified in the three special cases of Jacobi polynomials corresponding to characters of the series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$. In particular, we show that then these coefficients are given by piecewise polynomial functions. This is where a link with discrete splines arises. For characters of the series $\mathcal{A}$ (that is, of the unitary groups $U(N)$) similar results were obtained previously by Borodin and this author [5], and then reproved by Petrov [39] by another method. The case of symplectic and orthogonal characters is more intricate. Bibliography: 58 titles.

SPLINE MODEL OF GRIDDED DATA OPERATION AS A PRINCIPLE OF ELECTRONIC MAPPING SEABED TOPOGRAPHY

The influence of technologies of automated processing of hydrographic survey results on the potential of the human factor in the transformation of polygraphic cartographic products into digital ones is analyzed. When configuring the digital bathymetric model, the concept of the navigation surface is used as a perspective principle of automated mapping. The approach to the problem of generating bathymetric contours from measurement results in the navigation surface paradigm is construed on continuous spline interpolation of geospatial data for reasonable cartographic generalization when creating electronic cartographic products. From the theoretical positions of the spline approach, the subjectivity of the method of artificial displacement of generalized isobaths to deep-water areas is excluded when creating a safe digital model of the bottom relief, interpreted in a mathematical sense as a navigational isosurface. The principle of electronic mapping based on the use of a spline in tension as an effective approach for the process of generalizing isobaths in order to obtain a wide range of morphometric characteristics of underwater topography has been developed. The generalized line of the active depth contour is estimated in the form of algorithmic reproduction on electronic charts of the safe convexity of the isobate towards the deep-sea area due to the practical implementation of the B-spline “snake model” by analogy with the serpentine configuration of the bathymetric isoline in the form of a piecewise polynomial function. When using the spline approach, an innovative principle of electronic mapping of the underwater landscape based on operating with a set of gridded data is implemented. The latter are interpreted as the results of depth measurements with the formalization of a two-dimensional frame of fixed values of bathymetric measurements for their representation as a navigational isosurface in three-dimensional Euclidean space. The actual synthesis of the seabed topography is implemented on the basis of a proven hybrid spline model for a specific indicative test case based on the processing of experimental gridded data. Hypothetically, the possibility of intellectual assistance to the watch officer in the strategy of instant orientation in conditions of a minimum depth reserve under the keel is organized when using computer three-dimensional visualization of the topography of the underwater relief in an unaffiliated graphic environment with foreign software.

A Review of Collocation Approximations to Solutions of Differential Equations

This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results.

Extension operators for trimmed spline spaces

We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree p with k continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.

A Novel Robotic Exoskeleton for Finger Rehabilitation: Kinematics Analysis.

A novel robotic exoskeleton for fingers rehabilitation is developed, which is driven by linear motors through Bowden cables. For each finger, in addition to three links acting as phalanxes, two more links acting as knuckles are also implemented. Links are connected through passive joints, by which translational and rotary movements can be realized simultaneously. Either flexion or extension motion is accomplished by one cable of adequate stiffness. This exoskeleton possesses good adaptability to finger length of different subjects and length variations during movement. The exoskeleton's kinematics model is built by the statistics method, and piecewise polynomial functions (PPF) are chosen to describe the relationship between motor displacement and joint variables. Finally, the relationship between motor displacement and the finger's total bending angle is obtained, which can be used for rehabilitation trajectory planning. Experimental results show that this exoskeleton achieves nearly the maximum finger bending angle of a healthy adult person, with the maximum driving force of 68.6 N.

Logarithmic intersections of double ramification cycles

We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).

Numerical solutions of advection diffusion equations involving Atangana–Baleanu time fractional derivative via cubic B-spline approximations

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fractional advection diffusion equation (TFADE) involving the Atangana–Baleanu time fractional derivative (ABTFD). The finite difference scheme (FDS) is utilized to discretize the ABTFD with a non-singular kernel. The spatial derivative is discretized by using NCBS functions. The stability analysis of the proposed technique is studied. Convergence analysis of the current technique is also analyzed. The proposed technique is examined on a variety of problems, and the numerical outcomes are contrasted with the previously published technique’s results to ensure the correctness and accuracy of the current technique.

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.