Parametric Polynomial Research Articles

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.

Interpolation of circular arcs by parametric polynomials of maximal geometric smoothness

The aim of this paper is a construction of parametric polynomial interpolants of a circular arc possessing maximal geometric smoothness. Two boundary points of a circular arc are interpolated together with higher order geometric data. Construction of interpolants is done via a complex factorization of the implicit unit circle equation. The problem is reduced to solving only one nonlinear equation determined by a monotone function and the existence of the solution is proven for any degree of the interpolating polynomial. Precise starting points for the Newton–Raphson type iteration methods are provided and the best solutions are then given in a closed form. Interpolation by parametric polynomials of degree up to six is discussed in detail and numerical examples confirming theoretical results are included.

Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation

ABSTRACTIn this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view.

PRM77 - An Open-Source Toolkit for Developing Flexible Evidence-Based Decision and Simulation Models for Value Assessment in Oncology with R

Cost-effectiveness models have historically been built using spreadsheet software, but recently researchers have begun developing models using programming languages, such as R, to overcome its limitations. However, current software does not have the flexibility to develop cost-effectiveness models in oncology representative of appropriate evidence synthesis of the available data. Our objective was to create open-source software for developing flexible evidence synthesis based decision models to help assess value in oncology. We developed an R package that provides a general framework for decision modeling using survival analysis, partitioned survival analysis, or multi-state modeling. Cost-effectiveness analysis can either be performed for competing single or multi-line sequential treatment strategies. Relative treatment effects used in the model can either be based on a single study or an evidence synthesis of multiple studies. Parametric, spline, and fractional polynomial based analyses are supported. Parametric distributions include those available in the R package flexsurv and corresponding reparameterizations appropriate for network meta-analysis. The core code is written in C++ for speed in order to facilitate probabilistic sensitivity analysis, structural uncertainty analysis, individual patient simulation, and integration with web applications. The R package is freely available for download on GitHub. The software and methodology is illustrated with a cost-effectiveness analysis in oncology. A new toolkit for developing decision and simulation models for value assessment of medical interventions in oncology has been developed. The R package facilitates the proper integration of parameter estimates obtained with advanced meta-analysis techniques to ensure disease progression and relative treatment effects reflect the available data. Moreover, the large number of available probability distributions reduces barriers to exploring the implications of different structural assumptions.

Evaluating accuracy of diagnostic tests without conditional independence assumption.

Evaluating the accuracy (ie, estimating the sensitivity and specificity) of new diagnostic tests without the presence of a gold standard is of practical meaning and has been the subject of intensive study for several decades. Existing methods use 2 or more diagnostic tests under several basic assumptions and then estimate the accuracy parameters via the maximum likelihood estimation. One of the basic assumptions is the conditional independence of the tests given the disease status. This assumption is impractical in many real applications in veterinary research. Several methods have been proposed with various dependence models to relax this assumption. However, these methods impose subjective dependence structures, which may not be practical and may introduce additional nuisance parameters. In this article, we propose a simple method for addressing this problem without the conditional independence assumption, using an empirical conditioning approach. The proposed method reduces to the popular Hui-Walter model in the case of conditional independence. Also, our likelihood function is of order-2 polynomial in parameters, while that of Hui-Walter is of order-3. The reduced model complexity increases the stability in estimation. Simulation studies are conducted to evaluate the performance of the proposed method, which shows overall smaller biases in estimation and is more stable than the existing method, especially when tests are conditionally dependent. Two real data examples are used to illustrate the proposed method.

Computing zeta functions of generic projective hypersurfaces in larger characteristic

We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the space complexity to $\tilde{O}(\log(p))$ while remaining polynomial in the other input parameters.

Multivariate Splines-Based Asymmetric Aerodynamic Modeling of Morphing Aircraft

The aerodynamic parameters of morphing aircraft can vary significantly in the morphing process; the aerodynamic data model is complex and high-order nonlinear. This paper presents a new method of aerodynamic modeling of morphing aircraft by using multivariate splines. Firstly, the aerodynamic coefficient model of morphing aircraft based on the multivariate splines model structure is presented. This model can describe symmetric and asymmetric aerodynamic forces. Then, the multivariate simple splines estimation process is introduced in detail and the concrete formulation is deduced. Finally, the final model structure is determined by error analysis, and the aerodynamic parameter polynomial model in the global coordinate system is obtained by coordinate transformation. The results show that this method can accurately describe the aerodynamic model of morphing aircraft during arbitrary symmetric and asymmetric morphing process without predicting the specific model structure of the aerodynamic parameters related to varying process.

Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 1: micro- and nanostructure of collagen fibers

Soft tissues are characterized by a nonlinear mechanical response, highly affected by the multiscale structure of collagen fibers. The effectiveness and the calibration of constitutive models play a major role on the reliability and the applicability of computational models in biomechanics. This paper presents a procedure for the identification of the relationship between collagen-related structural features in soft tissues with model parameters of classical polynomial- and exponential-based constitutive models. Histological features at microscale, as well as biochemical and biophysical properties at nanoscale, are addressed by employing a multiscale structural description of soft tissue mechanics as benchmark data set. Both the direct (from structure to parameters) and the inverse (from parameters to structure) problem are addressed. Suitable optimization problems are introduced for accurate numerical and approximated analytical direct relationships. The inverse identification has been addressed by providing also a measure of the reliability of the computed estimates. Results show the effectiveness of the proposed strategies and allow to discuss the fitting capabilities of classical constitutive approaches in terms of parameters identification.

A convex polynomial model for planar sliding mechanics: theory, application, and experimental validation

We propose a polynomial model for planar sliding mechanics. For the force–motion mapping, we treat the set of generalized friction loads as the 1-sublevel set of a polynomial whose gradient directions correspond to generalized velocities. The polynomial is confined to be convex even-degree homogeneous in order to obey the maximum work inequality, symmetry, shape invariance in scale, and fast invertibility. We present a simple and statistically efficient model identification procedure using a sum-of-squares convex relaxation. We then derive the kinematic contact model that resolves the contact modes and instantaneous object motion given a position controlled manipulator action. The inherently stochastic object-to-surface friction distributions are modeled by sampling polynomial parameters from distributions that preserve sum-of-squares convexity. Thanks to the model smoothness, the mechanics of patch contact is captured while being computationally efficient without mode selection at support points. Simulation and robotic experiments on pushing and grasping validate the accuracy and efficiency of our approach.

A generalized framework for reduced‐order modeling of a wind turbine wake

AbstractA reduced‐order model for a wind turbine wake is sought from large eddy simulation data. Fluctuating velocity fields are combined in the correlation tensor to form the kernel of the proper orthogonal decomposition (POD). Proper orthogonal decomposition modes resulting from the decomposition represent the spatially coherent turbulence structures in the wind turbine wake; eigenvalues delineate the relative amount of turbulent kinetic energy associated with each mode. Back‐projecting the POD modes onto the velocity snapshots produces dynamic coefficients that express the amplitude of each mode in time. A reduced‐order model of the wind turbine wake (wakeROM) is defined through a series of polynomial parameters that quantify mode interaction and the evolution of each POD mode coefficients. The resulting system of ordinary differential equations models the wind turbine wake composed only of the large‐scale turbulent dynamics identified by the POD. Tikhonov regularization is used to recalibrate the dynamical system by adding additional constraints to the minimization seeking polynomial parameters, reducing error in the modeled mode coefficients. The wakeROM is periodically reinitialized with new initial conditions found by relating the incoming turbulent velocity to the POD mode coefficients through a series of open‐loop transfer functions. The wakeROM reproduces mode coefficients to within 25.2%, quantified through the normalized root‐mean‐square error. A high‐level view of the modeling approach is provided as a platform to discuss promising research directions, alternate processes that could benefit stability and efficiency, and desired extensions of the wakeROM.

Circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples.

PPolyNets: Achieving High Prediction Accuracy and Efficiency With Parametric Polynomial Activations

Recently, to implement cloud-based machine learning approaches while maintaining data privacy, a scheme named CryptoNets is proposed to perform prediction on encrypted data using neural networks. By applying the leveled homomorphic encryption scheme, CryptoNets enables the cloud server to securely run the computation process without participations of other parties. Since the encryption scheme only supports polynomial operations, the authors simply use square activation as a substitution of the conventional activations, obtaining a relatively low prediction accuracy on the MNIST dataset (98.95%). Later work try to improve the accuracy by using polynomial approximations of the ReLU activation with large neural networks, which introduce heavy computation cost, making the prediction process impractical. In this paper, to achieve better prediction performance, we propose new parametric polynomial (PPoly) activations, which can adaptively learn the parameters during the training phase. Using our PPoly activations, we achieve higher accuracy (99.33%, 99.64%, and 99.70%) with shallow and narrow networks, guaranteeing the efficiency of prediction process. We conduct extensive experiments to show the expressiveness of our PPoly activations and discuss the tradeoff between accuracy and efficiency for the prediction on encrypted data.

Exact and Closed-Form CRLBS for High-Order Kinematic Parameters Estimation Using LFM Coherent Pulse Train

In radar signal processing, the detection and parameter estimation of high-speed maneuvering targets, which often utilize a coherent pulse train signal with linear frequency modulation, have been receiving increasing attention. Fundamentally and significantly, the Cramer–Rao lower bound (CRLB), as a cornerstone for evaluating the estimation performance of high-order kinematic parameters, has been derived and investigated here. In this paper, a 2-D echo signal model expressed in the fast-frequency and slow-time domains is adopted. The scaled orthogonal Legendre polynomials are deliberately introduced to solve the inverse problem of the Fisher information matrix, and then, a linear mapping relationship between different polynomial parameters can be used to obtain the analytical CRLB expressions. The main contributions included are: 1) the CRLBs, which are exact and closed form, have been extended to arbitrary motion model orders and reference time instants; 2) the influences of the motion model order, the reference time instant, as well as the radar parameters on the CRLBs are exploited comprehensively; and 3) some specific cases, including four low-order motion models and two preferred reference times, are also presented to better demonstrate the CRLB performance relationships. It highlights the fact that the reference time instant corresponding to the middle of the pulse train is a reasonable and compromised choice for parameter estimation, although it is not necessarily optimal for the kinematic parameters of all models and orders. The above research results are illustrated with numerical simulations and further verified using the maximum likelihood estimation method combined with Monte Carlo experiments.

Eigenvectors of Orthogonally Decomposable Functions

The eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely orthogonal tensor decompositions, independent component analysis (ICA), topic models, spectral clustering, and Gaussian mixture learning---we generalize the eigendecomposition from quadratic forms to a broad class of “orthogonally decomposable” functions. We identify a key role of convexity in our extension, and we generalize two traditional characterizations of eigenvectors: First, the eigenvectors of a quadratic form arise from the optima structure of the quadratic form on the sphere. Second, the eigenvectors are the fixed points of the power iteration. In our setting, we consider a simple first order generalization of the power method which we call gradient iteration. It leads to efficient and easily implementable methods for basis recovery. It includes influential machine learning methods such as cumulant-based FastICA and the tensor power iteration for orthogonally decomposable tensors as special cases. We provide a complete theoretical analysis of gradient iteration using the structure theory of discrete dynamical systems to show almost sure convergence and fast (superlinear) convergence rates. The analysis also extends to the case when the observed function is only approximately orthogonally decomposable, with bounds that are polynomial in dimension and other relevant parameters, such as perturbation size. Our perturbation results can be considered as a nonlinear version of the classical Davis--Kahan theorem for perturbations of eigenvectors of symmetric matrices.

Decoupled Parameter Estimation Methods for Hammerstein Systems by Using Filtering Technique

The implementation of parameter estimation of Hammerstein systems is much difficult due to the existing parameter products from the nonlinear block and the linear block. This paper directly decomposes the parameter coupling between the nonlinear part and the linear part in a Hammerstein system by using the estimated parameter polynomial of the coupled linear part to filter the Hammerstein system, transforms the Hammerstein system into two forms, and investigates two decoupled parameter estimation methods: the one-step decoupled least squares estimation method and the two-step decoupled least squares estimation method corresponding to the two forms. Furthermore, the computational complexity is compared between the proposed two estimation algorithms. The simulation results show the effectiveness of the proposed two estimation methods with a similar estimation accuracy.

Orbits of polynomial dynamical systems modulo primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C \mathbb {C} . Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates.

In-Situ Systematic Error Correction for Digital Volume Correlation Using a Reference Sample

The self-heating effect of a laboratory X-ray computed tomography (CT) scanner causes slight change in its imaging geometry, which induces translation and dilatation (i.e., artificial displacement and strain) in reconstructed volume images recorded at different times. To realize high-accuracy internal full-field deformation measurements using digital volume correlation (DVC), these artificial displacements and strains associated with unstable CT imaging must be eliminated. In this work, an effective and easily implemented reference sample compensation (RSC) method is proposed for in-situ systematic error correction in DVC. The proposed method utilizes a stationary reference sample, which is placed beside the test sample to record the artificial displacement fields caused by the self-heating effect of CT scanners. The detected displacement fields are then fitted by a parametric polynomial model, which is used to remove the unwanted artificial deformations in the test sample. Rescan tests of a stationary sample and real uniaxial compression tests performed on copper foam specimens demonstrate the accuracy, efficacy, and practicality of the presented RSC method.

T1 mapping with the variable flip angle technique: A simple correction for insufficient spoiling of transverse magnetization.

The variable flip angle method derives T1 maps from radiofrequency-spoiled gradient-echo data sets, acquired with different flip angles α. Because the method assumes validity of the Ernst equation, insufficient spoiling of transverse magnetization yields errors in T1 estimation, depending on the chosen radiofrequency-spoiling phase increment (Δϕ). This paper presents a versatile correction method that uses modified flip angles α' to restore the validity of the Ernst equation. Spoiled gradient-echo signals were simulated for three commonly used phase increments Δϕ (50°/117°/150°), different values of α, repetition time (TR), T1 , and a T2 of 85 ms. For each parameter combination, α' (for which the Ernst equation yielded the same signal) and a correction factor CΔϕ (α, TR, T1 ) = α'/α were determined. CΔϕ was found to be independent of T1 and fitted as polynomial CΔϕ (α, TR), allowing to calculate α' for any protocol using this Δϕ. The accuracy of the correction method for T2 values deviating from 85 ms was also determined. The method was tested in vitro and in vivo for variable flip angle scans with different acquisition parameters. The technique considerably improved the accuracy of variable flip angle-based T1 maps in vitro and in vivo. The proposed method allows for a simple correction of insufficient spoiling in gradient-echo data. The required polynomial parameters are supplied for three common Δϕ. Magn Reson Med 79:3082-3092, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

ESTIMATING EXTERIOR ORIENTATION PARAMETERS OF HYPERSPECTRAL BANDS BASED ON POLYNOMIAL MODELS

Abstract. Hyperspectral camera operating in sequential acquisition mode produces spectral bands that are not recorded at the same instant, thus having different exterior orientation parameters (EOPs) for each band. The study presents experiments on bundle adjustment with time-dependent polynomial models for band orientation of hyperspectral cubes sequentially collected. The technique was applied to a Rikola camera model. The purpose was to investigate the behaviour of the estimated polynomial parameters and the feasibility of using a minimum of bands to estimate EOPs. Simulated and real data were produced for the analysis of parameters and accuracy in ground points. The tests considered conventional bundle adjustment and the polynomial models. The results showed that both techniques were comparable, indicating that the time-dependent polynomial model can be used to estimate the EOPs of all spectral bands, without requiring a bundle adjustment of each band. The accuracy of the block adjustment was analysed based on the discrepancy obtained from checkpoints. The root mean square error (RMSE) indicated an accuracy of 1 GSD in planimetry and 1.5 GSD in altimetry, when using a minimum of four bands per cube.