Gauss-Kronrod Research Articles

Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification.

A now common goal in medical research is to investigate the inter‐relationships between a repeatedly measured biomarker, measured with error, and the time to an event of interest. This form of question can be tackled with a joint longitudinal‐survival model, with the most common approach combining a longitudinal mixed effects model with a proportional hazards survival model, where the models are linked through shared random effects. In this article, we look at incorporating delayed entry (left truncation), which has received relatively little attention. The extension to delayed entry requires a second set of numerical integration, beyond that required in a standard joint model. We therefore implement two sets of fully adaptive Gauss–Hermite quadrature with nested Gauss–Kronrod quadrature (to allow time‐dependent association structures), conducted simultaneously, to evaluate the likelihood. We evaluate fully adaptive quadrature compared with previously proposed non‐adaptive quadrature through a simulation study, showing substantial improvements, both in terms of minimising bias and reducing computation time. We further investigate, through simulation, the consequences of misspecifying the longitudinal trajectory and its impact on estimates of association. Our scenarios showed the current value association structure to be very robust, compared with the rate of change that we found to be highly sensitive showing that assuming a simpler trend when the truth is more complex can lead to substantial bias. With emphasis on flexible parametric approaches, we generalise previous models by proposing the use of polynomials or splines to capture the longitudinal trend and restricted cubic splines to model the baseline log hazard function. The methods are illustrated on a dataset of breast cancer patients, modelling mammographic density jointly with survival, where we show how to incorporate density measurements prior to the at‐risk period, to make use of all the available information. User‐friendly Stata software is provided. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Precision‐tuning and hybrid pricer for closed‐form solution‐based Heston calibration

SummaryCalibration methods are the heart of modeling any financial process. While for the Heston model (semi) closed‐form solutions exist for simple products, their evaluation involves complex functions and infinite integrals. So far, these integrals can only be solved with time‐consuming numerical methods. For that reason, calibration consumes a large portion of available compute power in the daily finance business.However, more and more theoretical and practical subtleties have been discovered over the years, and today, a large number of calibration methods are available. Currently, there is no clear indication which numerical method should be used for a specific calibration purpose under given speed and accuracy constraints. With this publication, we aim at closing this gap. We derive a novel methodology for systematically finding the best methods for a well‐defined accuracy target. For a practical setup, we study the available popular closed‐form solutions and integration algorithms. In total, we compare 14 numerical methods, including adaptive quadrature and Fourier methods. For a target accuracy of 10−3, we show that adaptive Gauss–Kronrod methods are best on CPUs for the unrestricted parameter set. Furthermore, we introduce hybrid pricer methods that combine quadrature and fast Fourier transform pricers, what gives us another 2.4× speedup. Copyright © 2015 John Wiley & Sons, Ltd.

A Gauss-Kronrod-Trapezoidal integration scheme for modeling biological tissues with continuous fiber distributions

Fibrous biological tissues may be modeled using a continuous fiber distribution (CFD) to capture tension–compression nonlinearity, anisotropic fiber distributions, and load-induced anisotropy. The CFD framework requires spherical integration of weighted individual fiber responses, with fibers contributing to the stress response only when they are in tension. The common method for performing this integration employs the discretization of the unit sphere into a polyhedron with nearly uniform triangular faces (finite element integration or FEI scheme). Although FEI has proven to be more accurate and efficient than integration using spherical coordinates, it presents three major drawbacks: First, the number of elements on the unit sphere needed to achieve satisfactory accuracy becomes a significant computational cost in a finite element (FE) analysis. Second, fibers may not be in tension in some regions on the unit sphere, where the integration becomes a waste. Third, if tensed fiber bundles span a small region compared to the area of the elements on the sphere, a significant discretization error arises. This study presents an integration scheme specialized to the CFD framework, which significantly mitigates the first drawback of the FEI scheme, while eliminating the second and third completely. Here, integration is performed only over the regions of the unit sphere where fibers are in tension. Gauss–Kronrod quadrature is used across latitudes and the trapezoidal scheme across longitudes. Over a wide range of strain states, fiber material properties, and fiber angular distributions, results demonstrate that this new scheme always outperforms FEI, sometimes by orders of magnitude in the number of computational steps and relative accuracy of the stress calculation.

Exact, approximate and numerical solutions for a variant of Stokes׳ first problem for a new class of non-linear fluids

Stress power-law fluids are a special sub-class of fluids defined through implicit constitutive relations, wherein the symmetric part of the velocity gradient depends on a power-law of the stress (see Eq. (2.2)), and were introduced recently to describe the non-Newtonian response of fluid bodies. Such fluids are counterparts to the classical power-law fluids wherein the stress is given in terms of a power-law for the symmetric part of the velocity gradient. Stress power-law fluids can describe phenomena that cannot be described by classical power-law fluids (see [1]). In this paper, first a new exact solution for a variant of Stokes׳ first problem for stress power-law fluids, when the exponent n=0 (Navier–Stokes fluid), is obtained. Such an exact solution for the stress is in terms of a convolution integral, for which we establish bounds. We then compute the convolution integral using Gauss–Kronrod quadrature by ensuring that its value always lies within the bounds. Using the validated quadrature, we can accurately evaluate the exact solution and we the exact solution it to validate the numerical scheme employed in solving the governing equations for stress-power law fluids with arbitrary exponent n. Finally, for stress power-law fluids wherein the exponent n<0 (stress-thickening fluids), we obtain an approximate solution for the stress that agrees well with the numerical solution.

Estimating the error of Gaussian quadratures with simple and multiple nodes by using their extensions with multiple nodes

The estimation of the error in a quadrature formula is an important problem. A simple and effective procedure for estimating the error of Gaussian quadrature formulas using their extensions with multiple nodes will be presented. Our method works for estimating the error of any interpolatory quadrature formula with simple or multiple nodes. We concentrate the most of our attention to the estimation of the error of standard Gauss quadratures, as the most known and popular ones. In that sense we offer an adequate alternative to Gauss–Kronrod quadratures, which have been intensively investigated in the last five decades, both from the theoretical and computational point of view. We found and a few cases of optimal, i.e., Kronrod extensions with multiple nodes for some Gauss quadrature formulas; we are not aware of any results of this kind in the mathematical literature. Numerical results are presented, in order to demonstrate the efficiency of the proposed method.

A single loop reliability-based design optimization using EPM and MPP-based PSO

A reliability-based design optimization (RBDO) incorporates a probabilistic analysis with an optimization technique to find a best design within a reliable design space. However, the computational cost of an RBDO task is often expensive compared to a deterministic optimization, which is mainly due to the reliability analysis performed inside the optimization loop. Theoretically, the reliability of a given design point can be obtained through a multidimensional integration. Integration with multiple variables over the safety domain is, unfortunately, formidable in most cases. Monte- Carlo simulation (MCS) is often used to solve this difficulty. However, the inherit statistic uncertainty associated with MCS sometimes causes an unstable RBDO solution. To avoid this unstable solution, this study transforms a multi-variable constraint into a single variable constraint using an exponential function with a polynomial coefficient (EPM). The adaptive Gauss-Kronrod quadrature is used to compute the constraint reliability. The calculated reliability and its derivative are incorporated with an optimizer such as sequential quadratic programming (SQP) or most probable point particle swarm optimization (MPP-based PSO) to conduct the RBDO task. To ensure the design accuracy, the stability of the RBDO algorithm with respect to the initial point is investigated through several numerical examples.

Imaging by a double negative metamaterial slab excited with an arbitrarily oriented dipole

The imaging capability of nonideal double negative metamaterial slabs excited by arbitrarily oriented electric or magnetic dipoles is investigated. A dipole oriented parallel to the interface of the slab excites transverse electric and transverse magnetic modes simultaneously, which are transmitted unequally by the slab if either the slab's relative permittivity (er) or permeability (μr) or both are perturbed from the ideal case, i.e., er=μr=−1 for a slab in vacuum. Approximations in the electrostatic limit, valid for modes with large wave numbers, fail to predict this phenomenon, and more accurate analysis of the field distribution in the vicinity of the image plane behind a nonideal slab becomes important. By expressing the fields in terms of a one‒dimensional Hankel transform in the spectral domain and using an adaptive Gauss‒Kronrod quadrature, a tool for evaluating the spatial field intensity in the vicinity of the image plane behind the slab is developed. The calculated field intensities reveal that a double negative slab does not focus in the three‒dimensional space and is only capable of two‒dimensional imaging.

Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function

We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss–Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G2, and G3 non-linear superspiral splines.

Analytical performance of orthogonal frequency division multiplexing systems impaired by a non-linear high-power amplifier with memory

The performance of modern communication transmitters utilising multi-carrier modulation techniques is highly sensitive to non-linear distortions arising mainly from the high-power amplifier (HPA). In addition, the wideband characteristics of multi-carrier signals result in frequency-dependent distortions, typically known as memory effects. This study aims to analytically evaluate the impact of the distortion induced by non-linear HPA with memory on a multi-carrier signal exemplified by orthogonal frequency division multiplexing (OFDM). The behavioural model of HPA considered for theoretical analysis is a memory polynomial model, which is a truncated form of the Volterra series. In the theoretical framework developed, this study shows that memory in HPA considerably degrades the performance of OFDM in terms of symbol error rate (SER), and the distortion itself can be canonically characterised by a complex attenuation component and a non-linear noise component. Closed-form expressions for SER in additive white Gaussian noise channel are derived and the SER for fading channel is approximated by the adaptive Gauss-Kronrod quadrature method. Simulation results are shown for a realistic HPA, based on the Wiener-Hammerstein model, and compared with the analytical results to validate the proposed analysis.

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.

One speed black sphere problem: Some benchmark results

Benchmark numerical results for the black sphere problem (one speed, isotropic scattering) are reported by extending some previous computations on the relevant integral transport equation [Loyalka, S.K., 1975. A numerical method for solving integral equations of neutron transport. Nuclear Science and Engineering 56, 317–319]. Spheres of radii (non-dimensionalized in terms of a mean free path) in the range 10−6–106 are considered, and a transformation of the range from (R,∞) to (0,1), is used together with successively high order quadratures, and high precision in all computations. Two different quadrature sets (Gauss–Legendre and Gauss–Kronrod) are explored and convergent and accurate numerical results for the extrapolation distance as well as the density profile are obtained.

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.

Modified anti-Gauss and degree optimal average formulas for Gegenbauer measure

For the practical estimation of the error of Gauss quadrature rules, Gauss–Kronrod formulas are widely used; but, for the Gegenbauer measure, d μ C = ( 1 − x 2 ) α − 1 / 2 d x , real positive Gauss–Kronrod formulas do not exist for α > 3 and n sufficiently large. Among the alternatives which are available in the literature, Gauss–Lobatto and anti-Gauss formulas are of particular interest. In this paper, using the modified anti-Gauss formulas introduced by Ehrich, we determine the degree optimal stratified extensions of Gauss–Gegenbauer formulas, and we investigate their properties.

Ultraspherical Stieltjes Polynomials and Gauss–Kronrod Quadrature Behave Nicely for $\lambda&lt; 0$

We show that the zeros of the Stieltjes polynomials associated with the ultraspherical weight function with parameter $\lambda<0$ are real and simple and, except for two of them, belong to $(-1,1)$. We also prove that they strictly interlace with the zeros of the corresponding ultraspherical polynomials. Consequently, a Gauss–Kronrod quadrature formula with two nodes outside the interval $[-1,1]$ is obtained. We prove that the coefficients of such quadrature rules are positive. Finally, an asymptotic representation of Stieltjes polynomials which converges uniformly on the whole interval $[-1,1]$ is provided.

A historical note on Gauss–Kronrod quadrature

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch.

Generalized Stieltjes Polynomials and Rational Gauss?Kronrod Quadrature

Generalized Stieltjes polynomials are introduced and their asymptotic properties outside the support of the measure are studied. As applications, we prove the convergence of sequences of interpolating rational functions, whose poles are partially fixed, to Markov functions and give an asymptotic estimate of the error of rational Gauss–Kronrod quadrature formulas when functions which are analytic on some neighborhood of the set of integration are considered.

Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions

Let w ?,beta (t)=(1?t)?(1+t)β, ?,β>?1, denote the Jacobi weight function. For 0≤?,β 5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions ${{w_\lambda(t) = \big(1-t^2\big)^{{\lambda-1/2}}}}$ for 0≤?≤2. For ?>3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently.

Asymptotics for Stieltjes polynomials, padé-type approximants, and Gauss-Kronrod quadrature

We study the asymptotic properties of Stieltjes polynomials outside the support of the measure as well as the asymptotic behaviour of their zeros. These properties are used to estimate the rate of convergence of sequences of rational functions, whose poles are partially fixed, which approximate Markovtype functions. An estimate for the speed of convergence of the Gauss-Kronrod quadrature formula in the case of analytic functions is also given.

Increasing reliability of Gauss–Kronrod quadrature by Eratosthenes' sieve method

The reliability of the local error estimates returned by the Gauss–Kronrod quadrature rules can be raised up to the theoretical 100% rate of success, under error estimate sharpening, provided a number of natural validating conditions are required. The self-validating scheme of the local error estimates, which is easy to implement and adds little supplementary computing effort, strengthens considerably the correctness of the decisions within the automatic adaptive quadrature.

Stopping functionals for Gaussian quadrature formulas

Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important Gauss–Kronrod and Patterson schemes, and methods which are based on Gaussian nodes, are presented and compared.