Spline Kernel Research Articles

A new kernel function for SPH with applications to free surface flows

Smoothed particle hydrodynamics (SPH) is a popular meshfree Lagrangian particle method, which uses a kernel function for numerical approximations. The kernel function is closely related to the computational accuracy and stability of the SPH method. In this paper, a new kernel function is proposed, which consists of two cosine functions and is referred to as double cosine kernel function. The newly proposed double cosine kernel function is sufficiently smooth, and is associated with an adjustable support domain. It also has smaller second order momentum, and therefore it can have better accuracy in terms of kernel approximation. SPH method with this double cosine kernel function is applied to simulate a dam-break flow and water entry of a horizontal circular cylinder. The obtained SPH results agree very well with the experimental results. The double cosine kernel function is also comparatively studied with two frequently used SPH kernel functions, Gaussian and cubic spline kernel functions.

On the correction of the boundary deficiency in SPH for the frictional contact simulation

Smoothed particle hydrodynamics (SPH) is a mesh-free method which is powerful for large deformation computation of soils. However, the algorithm for the simulation of frictional contact which is common in geotechnical engineering is still quite immature due to the boundary deficiency. In this study, the cause of boundary deficiency in the SPH simulation for frictional contact is analysed. Then, based on mathematical derivation, the method to correct boundary deficiency related to frictional contact is discussed theoretically, where the frictional contact algorithm is established by dividing the computational domain into several subdomains according to the existing contact boundaries and by using contact forces as bridges of these subdomains to fulfil problem solving, and the value of correction coefficient is obtained by comparing the SPH outcome of the contact particles with that calculated through Newton’s second law of motion. At the same time, from the perspective of numerical computation, an optimized value for the correction coefficient is proposed, and a thorough investigation is performed on the cubic spline kernel function and quintic spline kernel function, whose correction coefficients are found to be 2.0 and [2.0, 2.16], respectively. Finally, numerical tests are carried out to verify the proposed method. The outcome of the study is helpful to providing theoretical support for the research of frictional contact simulation within the framework of SPH.

Kernel based quadrature on spheres and other homogeneous spaces

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate--optimally so in many cases--and stable under an increasing number of nodes and in the presence of noise, provided the set $$X$$ X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to $$\mathbb S ^n$$ S n , oblate spheroids for instance. The weights are obtained by solving a single linear system. For $$\mathbb S ^2$$ S 2 , and the restricted thin plate spline kernel $$r^2\log r$$ r 2 log r , these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.

Regularized spectrum estimation using stable spline kernels

This paper presents a new regularized kernel-based approach for the estimation of the second order moments of stationary stochastic processes. The proposed estimator is defined by a Tikhonov-type variational problem. It contains few unknown parameters which can be estimated by cross validation solving a sequence of problems whose computational complexity scales linearly with the number of noisy moments (derived from the samples of the process). The correlation functions are assumed to be summable and the hypothesis space is a reproducing kernel Hilbert space induced by the recently introduced stable spline kernel. In this way, information on the decay to zero of the functions to be reconstructed is incorporated in the estimation process. An application to the identification of transfer functions in the case of white noise as input is also presented. Numerical simulations show that the proposed method compares favorably with respect to standard nonparametric estimation algorithms that exploit an oracle-type tuning of the parameters.

Influence of Different Interpolation Techniques on the Determination of the Critical Conditions for the Onset of Dynamic Recrystallisation

Accurate modeling of dynamic recrystallization (DRX) is highly important for forming processes like hot rolling and forging. To correctly predict the overall level of dynamic recrystallization reached, it is vital to determine and model the critical conditions that mark the start of DRX. For the determination of the critical conditions, a criterion has been proposed by Poliak and Jonas. It states that the onset of DRX can be detected from an inflection point in the work hardening rate as a function of flow stress. The work hardening rate is the derivative of the flow stress with respect to strain. Flow curves are in general measured at a certain sampling rate, yielding tabular stress-strain data, which are per se not continuously differentiable. In addition, inevitable jitter occurs in measured flow curves. Hence, flow curves need to be interpolated and smoothed before the work hardening rate and further derivatives necessary for evaluating the criterion by Poliak and Jonas can be computed. In this paper, the polynomial interpolation originally proposed by Poliak and Jonas is compared to a new approach based on radial basis functions using a thin plate spline kernel, which combines surface interpolation of various flow curves and smoothing in a single step. It is shown for different steel grades that the interpolation method used has a crucial influence on the resulting critical conditions for DRX, and that a simultaneous evaluation by surface interpolation might yield consistent critical conditions over a range of testing temperatures.

Stable local volatility function calibration using spline kernel

We propose an optimization formulation using the l 1 norm to ensure accuracy and stability in calibrating a local volatility function for option pricing. Using a regularization parameter, the proposed objective function balances calibration accuracy with model complexity. Motivated by the support vector machine learning, the unknown local volatility function is represented by a spline kernel function and the model complexity is controlled by minimizing the 1-norm of the kernel coefficient vector. In the context of support vector regression for function estimation based on a finite set of observations, this corresponds to minimizing the number of support vectors for predictability. We illustrate the ability of the proposed approach to reconstruct the local volatility function in a synthetic market. In addition, based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is simple and resembles the observed implied volatility surface in shape. Stability is illustrated by calibrating local volatility functions using market option data from different dates.

Localized Bases for Kernel Spaces on the Unit Sphere

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of $N$ scattered sites, the standard basis for such a space utilizes $N$ \emph{globally} supported kernels; computing with it is prohibitively expensive for large $N$. Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on $\sphere$, with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on $\mathbb{S}^2$ involving over one hundred fifty thousand data sites.

On the density of polyharmonic splines

This article treats the question of fundamentality of the translates of a polyharmonic spline kernel (also known as a surface spline) in the space of continuous functions on a compact set Ω⊂Rd when the translates are restricted to Ω. Fundamentality is not hard to demonstrate when a low degree polynomial may be added or when translates are permitted to lie outside of Ω; the challenge of this problem stems from the presence of the boundary, for which all successful approximation schemes require an added polynomial.When Ω is the unit ball, we demonstrate that translates of polyharmonic splines are fundamental by considering two related problems: the fundamentality in the space of functions vanishing at the boundary and fundamentality of the restricted kernel in the space of continuous functions on the sphere. This gives rise to a new approximation scheme composed of two parts: one which approximates purely on ∂Ω, and a second part involving a shift invariant approximant of a function vanishing outside of a neighborhood Ω.

Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method

We have presented a multiphase incompressible smoothed particle hydrodynamics method with an improved interface treatment procedure. To demonstrate the effectiveness of the interface treatment which can handle multiphase flow problems with high density and viscosity ratios, we have modeled several challenging two phase flow problems; namely, single vortex flow, square droplet deformation, droplet deformation in shear flow, and finally the Newtonian bubble rising in viscous and viscoelastic liquids. The proposed interface treatment includes the usage of (i) different smoothing functions (in this work, cubic spline kernel function for discretizing equations associated with the calculation of the surface tension force while the quintic spline for the discretization of governing equations and the relevant boundary conditions), and (ii) a new discretization scheme for calculating the pressure gradient. It is shown that with the application of the improved interface treatment, it becomes possible to model multiphase flow problems with the density and viscosity ratios up to 1000 and 100 respectively while using standard projection method. The utilization of cubic spline for the continuum surface force model significantly improves the quality of the calculated interface, thereby eliminating the interphase particle penetrations, and in turn leading to the calculation of more accurate velocity and pressure fields.

Improving convergence in smoothed particle hydrodynamics simulations without pairing instability

The numerical convergence of smoothed particle hydrodynamics (SPH) can be severely restricted by random force errors induced by particle disorder, especially in shear flows, which are ubiquitous in astrophysics. The increase in the number NH of neighbours when switching to more extended smoothing kernels at fixed resolution (using an appropriate definition for the SPH resolution scale) is insufficient to combat these errors. Consequently, trading resolution for better convergence is necessary, but for traditional smoothing kernels this option is limited by the pairing (or clumping) instability. Therefore, we investigate the suitability of the Wendland functions as smoothing kernels and compare them with the traditional B-splines. Linear stability analysis in three dimensions and test simulations demonstrate that the Wendland kernels avoid the pairing instability for all NH, despite having vanishing derivative at the origin (disproving traditional ideas about the origin of this instability; instead, we uncover a relation with the kernel Fourier transform and give an explanation in terms of the SPH density estimator). The Wendland kernels are computationally more convenient than the higher-order B-splines, allowing large NH and hence better numerical convergence (note that computational costs rise sub-linear with NH). Our analysis also shows that at low NH the quartic spline kernel with NH ~= 60 obtains much better convergence then the standard cubic spline.

Efficient algorithms for large scale linear system identification using stable spline estimators

A new nonparametric approach for system identification has been recently proposed where, in place of postulating parametric classes of impulse responses, the estimation process starts from an infinite-dimensional space. In particular, the impulse response is seen as the realization of a zero-mean Gaussian process. Its covariance, the so called stable spline kernel, encodes information on system stability and depends on few hyperparameters estimated from data via marginal likelihood optimization. This approach has been proved to compare much favorably with classical parametric methods but, in data rich situations, a possible drawback may be represented by its computational complexity which scales with the cube of the number of available samples. In this work we design a new computational strategy which may reduce significantly the computational load required by the stable spline estimator, thus extending its practical applicability also to large-scale scenarios.

Spline-Kernelled Chirplet Transform for the Analysis of Signals With Time-Varying Frequency and Its Application

The conventional time-frequency analysis (TFA) methods, including continuous wavelet transform, short-time Fourier transform, and Wigner-Ville distribution, have played important roles in analyzing nonstationary signals. However, they often show less capability in dealing with nonstationary signals with time-varying frequency due to the bad energy concentration in the time-frequency plane. On the other hand, by introducing an extra transform kernel that matches the instantaneous frequency of the signal, parameterized TFA methods show powerful ability in characterizing time-frequency patterns of nonstationary signals with time-varying frequency. In this paper, a novel time-frequency transform, called spline-kernelled chirplet transform (SCT), is proposed. By introducing a frequency-rotate operator and a frequency-shift operator constructed with spline kernel function, the SCT is particularly powerful for the strongly nonlinear frequency-modulated signals. In addition, an effective algorithm is developed to estimate the parameters of transform kernel in the SCT. The capabilities of the SCT and parameter estimation algorithm are validated by their applications for numerical signals and a set of vibration signal collected from a rotor test rig.

Comparison of a new kernel method and a sampling volume method for estimating indoor particulate matter concentration with Lagrangian modeling

This study develops a new concentration estimation method for Lagrangian modeling of airborne particulate matter (PM) in indoor environment. This new method, based on the cubic spline kernel function with the smoothing length and solid building boundary treatment, is considered herein to study the accuracy and efficiency of indoor airborne PM concentration calculation. The commonly used sampling volume method is also adopted for numerical performance test. Reliable measured PM concentration profiles in indoor chambers are used to verify the accuracy and efficiency of the above methods. A comparison of the relative merits and shortcomings of these two methods is emphasized. The results show that the sampling volume method is simple and easy to formulate in indoor environment. However, it produces sensitive and scattered concentration profiles, especially in recirculation zones, if the released particle number is not large enough. This problem can only be solved by using a sufficient number of particles, which should be determined by the sensitivity analysis. The kernel method can provide accurate and smooth profiles even though the released particle number is one order less than the sampling volume method. Under the same PM concentration estimation accuracy, the kernel method is much more efficient compared to the sampling volume method.

Partial logistic relevance vector machines in survival analysis

The use of relevance vector machines to flexibly model hazard rate functions is explored. This technique is adapted to survival analysis problems through the partial logistic approach. The method exploits the Bayesian automatic relevance determination procedure to obtain sparse solutions and it incorporates the flexibility of kernel-based models. Example results are presented on literature data from a head-and-neck cancer survival study using Gaussian and spline kernels. Sensitivity analysis is conducted to assess the influence of hyperprior distribution parameters. The proposed method is then contrasted with other flexible hazard regression methods, in particular the HARE model proposed by Kooperberg et al. [16]. A simulation study is conducted to carry out the comparison. The model developed in this paper exhibited good performance in the prediction of hazard rate. The application of this sparse Bayesian technique to a real cancer data set demonstrated that the proposed method can potentially reveal characteristics of the hazards, associated with the dynamics of the studied diseases, which may be missed by existing modeling approaches based on different perspectives on the bias vs. variance balance.

Spline-backfitted kernel smoothing of partially linear additive model

A spline-backfitted kernel smoothing method is proposed for partially linear additive model. Under assumptions of stationarity and geometric mixing, the proposed function and parameter estimators are oracally efficient and fast to compute. Such superior properties are achieved by applying to the data spline smoothing and kernel smoothing consecutively. Simulation experiments with both moderate and large number of variables confirm the asymptotic results. Application to the Boston housing data serves as a practical illustration of the method.

A new kernel-based approach for linear system identification

This paper describes a new kernel-based approach for linear system identification of stable systems. We model the impulse response as the realization of a Gaussian process whose statistics, differently from previously adopted priors, include information not only on smoothness but also on BIBO-stability. The associated autocovariance defines what we call a stable spline kernel. The corresponding minimum variance estimate belongs to a reproducing kernel Hilbert space which is spectrally characterized. Compared to parametric identification techniques, the impulse response of the system is searched for within an infinite-dimensional space, dense in the space of continuous functions. Overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization. The proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two-step procedure that projects the nonparametric estimate onto the space of nominal models. The continuous-time derivation immediately extends to the discrete-time case. On several continuous- and discrete-time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques.

Penalised spline support vector classifiers: computational issues

We study computational issues for support vector classification with penalised spline kernels. We show that, compared with traditional kernels, computational times can be drastically reduced in large problems making such problems feasible for sample sizes as large as ~106. The optimisation technology known as interior point methods plays a central role. Penalised spline kernels are also shown to allow simple incorporation of low-dimensional structure such as additivity. This can aid both interpretability and performance.

Nonlinear Model Predictive Control Based on Support Vector Machine with Multi-kernel

Multi-kernel-based support vector machine (SVM) model structure of nonlinear systems and its specific identification method is proposed, which is composed of a SVM with linear kernel function followed in series by a SVM with spline kernel function. With the help of this model, nonlinear model predictive control can be transformed to linear model predictive control, and consequently a unified analytical solution of optimal input of multi-step-ahead predictive control is possible to derive. This algorithm does not require online iterative optimization in order to be suitable for real-time control with less calculation. The simulation results of pH neutralization process and CSTR reactor show the effectiveness and advantages of the presented algorithm.

Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.

1111 SPH法を用いた材料界面の力学解析(GS-1 計算力学)

Deformation and fracture simulations of material with an interface which joins two different species are performed by SPH (Smoothed Particle Hydrodynamics) method. First, as a check of the SPH program using a cubic spline kernel function, validity of elastic response in a metal simple substance is confirmed on 2-dimensional problem. Secondly, stress-strain diagram and stress concentration of bi-material are considered by introducing boundary layers with intermediate material properties. It is certain that stress is higher in boundary region. We also find that the smaller the length of boundary layer (interface region) is, the stronger the stress concentration is.