Piecewise Polynomial Representation Research Articles

Interpolation of Subdivision Features for Curved Geometry Modeling

We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. The surfaces correspond to initial linear triangulations that determine the sharp point and curve features. The method automatically suggests a subset of sharp features to smooth which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdivision of the initial mesh, with no need of an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. We show numerical evidence that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for nodal distributions with sub-optimal Lebesgue constant. We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.

A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.

Irbasis: Open-source database and software for intermediate-representation basis functions of imaginary-time Green’s function

The open-source library, irbasis, provides easy-to-use tools for two sets of orthogonal functions named intermediate representation (IR). The IR basis enables a compact representation of the Matsubara Green’s function and efficient calculations of quantum models. The IR basis functions are defined as the solution of an integral equation whose analytical solution is not available for this moment. The library consists of a database of pre-computed high-precision numerical solutions and computational code for evaluating the functions from the database. This paper describes technical details and demonstrates how to use the library. Program summaryProgram Title: irbasisProgram Files doi:http://dx.doi.org/10.17632/2fh88ynxm6.1Licensing provisions: MIT licenseProgramming language:C++, Python.External routines/libraries: numpy, scipy and h5py for Python library, HDF5 C library for C++ library.Nature of problem: Numerical orthogonal systems for Green’s functionSolution method: Galerkin method, piece-wise polynomial representation

Piecewise Polynomial Model Reduction Method for Nonlinear Systems in Time Domain

AbstractModel order reduction (MOR) of nonlinear systems draws great attention in the past several decades. This paper presents a new MOR method in time domain for nonlinear dynamical systems. The new algorithm is based on the combination of the Taylor series expansion for the state variable x(t) and the trajectory piecewise polynomial technique. Firstly, the nonlinear system is approximated by a piecewise polynomial representation. Then, based on the Taylor series coefficients of x(t), we formulate the projection matrix V for the piecewise polynomial system and the compact model of the piecewise polynomial system is obtained in the following. Besides, error estimation and stability analysis are also presented in this paper. Finally, two nonlinear systems are tested to verify the effectiveness of the algorithm.

Multiple-correction hybrid k-exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids

A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.

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.

General linear formulations of stochastic dominance criteria

We develop and implement linear formulations of general Nth order stochastic dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean–variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.

Piecewise polynomial representations of genomic tracks.

Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/.

A second order Cartesian finite volume method for elliptic interface and embedded Dirichlet problems

We present a finite volume method to solve elliptic equations with immersed interface conditions. This method allows discontinuities on the solution and its normal derivatives on an interface inside the domain on a Cartesian grid. The main idea is to use a piecewise polynomial representation of the solution on a dual grid that avoid distinctions between the different interface configurations. The method achieves second order accuracy with a compact nine-point stencil. Moreover, we show that this method applies to solve embedded Dirichlet and Neumann problems.

General Linear Formulations of Stochastic Dominance Criteria: With an Analysis of Stock Market Portfolio Efficiency

We develop and implement linear formulations of general N-th order Stochastic Dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean-variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.

Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws

This paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A finite volume scheme is used for the deterministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with a Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in the context of problems involving a large number of stochastic parameters.

Estimation of continuously differentiable signal with allowance for constraints

Consideration was given to the estimation of a piecewise-polynomially representable signal subject to continuity and smoothness and with allowance for the constraints on the value of signal and its derivative. Such representation may be used also for perturbation generating the estimated signal. The measurement noise was assumed to have stochastic description. Given was an analytical definition of the domain of values of the coefficients of the piecewise-polynomial representation of the signal under which it satisfies the stipulated requirements. A two-step computation-efficient solution of the problem was proposed, and an example of its efficient use was given.

Online Segmentation of Time Series Based on Polynomial Least-Squares Approximations

The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs.

PPR based similarity search for gas monitoring data

A Piecewise Polynomial Representation (PPR) based similarity search approach for time series data in coal well gas monitoring was proposed. For experiments, sample data were real gas monitoring data obtained from Yuhua Coal Mine, and evaluation criterions were information loss and mean search efficiency. Experimental results compared with the performance of Discrete Fourier Transform (DFT) based approach and Discrete Wavelet Transform (DWT) based approach show: in the condition of the same compress rate, values of information loss of PPR, DFT and DWT based approaches are very close to each other; Whereas, mean search efficiency of PPR based approach is 32% higher than that of DFT based approach, and 34% higher than that of DWT based approach respectively. Therefore, PPR based approach is more suitable for similarity search of gas monitoring data.

Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

We introduce a family of box splines for efficient, accurate and smooth reconstruction of volumetric data sampled on the Body Centered Cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C(0) reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representation of the C(0) and C(2) box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space---generated by BCC-lattice shifts of these box splines---is {twice} as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order, and with the same sampling density). Practical evidence is provided demonstrating that not only the BCC lattice is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - IV. Anisotropy

SUMMARY We present a new numerical method to solve the heterogeneous elastic anisotropic wave equation with arbitrary high-order accuracy in space and time on unstructured tetrahedral meshes. Using the most general Hooke's tensor we derive the velocity-stress formulation leading to a linear hyperbolic system which accounts for the variation of the material properties depending on direction. This approach allows for the accurate modelling even of the most general crystalline symmetry class, the triclinic anisotropy, as no interpolation of material properties to particular mesh vertices is necessary. The proposed method combines the Discontinuous Galerkin method with the arbitrary high-order derivatives (ADER) time integration approach using arbitrary high-order derivatives of the piecewise polynomial representation of the unknown solution. The discontinuities of this piecewise polynomial approximation at element interfaces permit the application of the well-established theory of finite volumes and numerical fluxes across element interfaces obtained by the solution of derivative Riemann problems. Due to the novel ADER time integration technique the scheme provides the same approximation order in space and time automatically. A numerical convergence study confirms that the new scheme achieves the desired arbitrary high-order accuracy even for anisotropic material on unstructured tetrahedral meshes. Furthermore, it shows that higher accuracy can be reached with higher-order schemes while reducing computational cost and storage space. To this end, we also present a new Godunov-type numerical flux for anisotropic material and compare its accuracy with a computationally simpler Rusanov flux. As a further extension, we include the coupling of anisotropy and viscoelastic attenuation based on the Generalized Maxwell Body rheology and the mean and deviatoric stress concepts. Finally, we validate the new scheme by comparing the results of our simulations to an analytic solution as well as to spectral element computations.

Analyzing a generalized Loop subdivision scheme

In this paper a class of subdivision schemes generalizing the algorithm of Loop is presented. The stencils have the same support as those from the algorithm of Loop, but allow a variety of weights. By varying the weights a class of C 1 regular subdivision schemes is obtained. This class includes the algorithm of Loop and the midpoint schemes of order one and two for triangular nets. The proof of C 1 regularity of the limit surface for arbitrary triangular nets is provided for any choice of feasible weights. The purpose of this generalization of the subdivision algorithm of Loop is to demonstrate the capabilities of the applied analysis technique. Since this class includes schemes that do not generalize box spline subdivision, the analysis of the characteristic map is done with a technique that does not need an explicit piecewise polynomial representation. This technique is computationally simple and can be used to analyze classes of subdivision schemes. It extends previously presented techniques based on geometric criteria.

Low-temperature Data for Carbon Dioxide

We investigate the empirical data for the vapor pressure (154$ \leq$$T$$\leq$196 K) and heat capacity (15.52$ \leq$$T$$\leq$189.78 K) of the solid carbon dioxide. The approach is both theoretical and numerical, using a computer algebra system (CAS). From the latter point of view, we have adopted a cubic piecewise polynomial representation for the heat capacity and reached an excellent agreement between the available empirical data and the evaluated one. Furthermore, we have obtained values for the vapor pressure and heat of sublimation at temperatures below 195 right down to 0 K. The key prerequisites are the: 1) Determination of the heat of sublimation of 26250 J$\cdot$mol\textsuperscript{-1} at vanishing temperature and 2) Elaboration of a `linearized' vapor pressure equation that includes all the relevant properties of the gaseous and solid phases. It is shown that: 1) The empirical vapor pressure equation derived by Giauque & Egan remains valid below the assumed lower limit of 154 K (similar argument holds for Antoine's equation), 2) The heat of sublimation reaches its maximum value of 27211 J$\cdot$mol\textsuperscript{-1} at 58.829 K and 3) The vapor behaves as a (polyatomic) ideal gas for temperatures below 150 K.

Genetic design of feature spaces for pattern classifiers

Functional piecewise approximation seeks data representation that is compact, highly simplified and meaningful. This study presents a genetic algorithm (GA)-based approach for computing a piecewise polynomial representation of functions, with the focus being on piecewise linear approximation in an application of biomedical spectral data. The area of piecewise linear approximation has been researched in the past four decades approximately, and the method presented here is compared with another well-known approach. The expansion of this method to piecewise polynomial representation is shown to be straightforward. Finally, the application of this method as a feature extraction method for classification of a dataset of feature vectors, specifically biomedical spectra, is demonstrated.

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement.