Numerical Steepest Descent Method Research Articles

On the Chebyshev spectral collocation method for the solution of highly oscillatory Volterra integral equations of the second kind

Abstract Based on Chebyshev spectral collocation and numerical techniques for handling highly oscillatory integrals, we propose a numerical method for a class of highly oscillatory Volterra integral equations frequently encountered in engineering applications. Specifically, we interpolate the unknown function at Chebyshev points, and substitute these points into the integral equation, resulting in a system of linear equations. The highly oscillatory integrals are treated using either the numerical steepest descent method or the Filon-Clenshaw-Curtis method. Additionally, we derive an error estimation formula for this method using error analysis techniques and validate the convergence and effectiveness of the proposed approach through numerical examples.

On the convergence rate of collocation methods for Volterra integral equations with weakly singular oscillatory trigonometric kernels

This paper presents efficient collocation methods for linear Volterra integral equations with weakly singular highly oscillatory kernels. The numerical steepest descent method and generalized Gauss–Laguerre rule are utilized to calculate the weakly singular oscillatory integrals, which is the main challenge of the problem. Moreover, we derive the corresponding error estimation formula in terms of the frequency and the step length. This formula reflects, to some extent, the global convergence of the method. However, numerical examples show that the method is very effective and verify the correctness of the theoretical results.

Numerical methods for Cauchy principal value integrals of oscillatory Bessel functions

In this paper, we first propose different combination methods to compute the Cauchy principal value integrals of oscillatory Bessel functions. By special transformations, the considered integrals are converted to finite integrals and infinite integrals. Then, the finite integrals can be calculated through the Filon-type method, the Clenshaw–Curtis–Filon method and the Clenshaw–Curtis–Filon-type method, respectively. We compute the infinite integral through the numerical steepest descent method. Moreover, the error analysis with respect to frequency ω is given through theoretical analysis. Eventually, we present several numerical experiments which are in accord with our analysis. Particularly, the accuracy can be improved by either using more nodes or adding more derivatives interpolation at endpoints. The accuracy will increase drastically with the growth of frequency ω if both the number of nodes and interpolated multiplicity are fixed.

Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by $J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]$ , where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on $[0,+\infty )$ . Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

Optimal control and stochastic modelling of acid pre-treatment of bagasse

Abstract Acid pre-treatment is a preferred pre-treatment process for ethanol production from bagasse, a lignocellulosic feedstock. Better control of the acid pre-treatment is desired to improve yields and thereby its economic feasibility. This work applies the theory of optimal control for developing control strategies for a batch acid pre-treatment reactor. The Pontryagin's maximum principle in combination with the numerical method of steepest descent of Hamiltonian is used to solve the optimal control problem. Three different types of optimal control problems, namely, maximum concentration problem (MCP), minimum time problem (MTP), and maximum productivity problem (MPP), have been studied. The MCP results showed the significant improvement in xylose yield (around 9.23%) for 300 minutes of batch time and MTP reported the significant reduction in batch time (around 28%) to achieve 0.2 gram xylose concentration. However, pre-treatment is influenced by various sources of uncertainties. Therefore, it is essential to optimize the acid pre-treatment process after considering these variabilities. As a step towards solving the stochastic optimal control problem, this work has developed the stochastic model for the acid pre-treatment. Variability in feedstock composition has been considered, and stochastic model has been developed to capture time dependent and independent uncertainty in feedstock composition.

A remedy for the failure of the numerical steepest descent method on a class of oscillatory integrals

In this paper we demonstrate that the numerical method of steepest descent fails when applied in a straight forward fashion to the most commonly occurring highly oscillatory integrals in scattering theory. Through a polar change of variables, however, the integral can be reformulated so that it can be solved efficiently using a combination of oscillatory integration techniques and classical quadrature. The approach is described in detail and demonstrated numerically with some oscillatory integral examples. The numerical examples demonstrate that our approach avoids the failure in some special cases, such as in an acoustic scattering model oscillatory integral with observation point located in the illuminated region. This paves the way for using the framework of numerical steepest descent methods on a wider class of problems, like the 3D high frequency scattering from convex obstacles, up to now only handled in a satisfactory way by methods due to Ganesh and Hawkins (J Comp Phys 230:104–125, 2011).

Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method

A numerical steepest descent method, based on the Laguerre quadrature rule, is developed for integration of one-dimensional highly oscillatory functions on [0,???) of a general class. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The method is extended to the case when there are a finite number of stationary points in [0,???). It can be further extended to the case when the integrand is only smooth to some degree (not necessarily analytic). We illustrate the theoretical results using some numerical experiences.

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

We propose a variant of the numerical method of steepest descent for oscillatory integrals by using a low-cost explicit polynomial approximation of the paths of steepest descent. A loss of asymptotic order is observed, but in the most relevant cases the overall asymptotic order remains higher than a truncated asymptotic expansion at similar computational effort. Theoretical results based on number theory underpinning the mechanisms behind this effect are presented.

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.

HIGH FREQUENCY SCATTERING BY AN IMPENETRABLE SPHERE

The high frequency scattering of a scalar plane wave from an impenetrable sphere with a diameter of several thousand wavelengths is treated by the Sommerfeld-Watson transformation, the saddle-point technique (SPT), and the numerical steepest descent method (NSDM). Both the near and far flelds for the sphere are computed within the observation angle range of 0 to 180 degree. First, with the aid of the Watson transformation, the fast-convergent residue series replacing the slow-convergent Mie series is derived. Second, a new algorithm for flnding the zeros of the Hankel functions is developed. Third, a novel NSDM, which is adaptive to frequency and is hence frequency independent, is proposed to overcome the breakdown of the traditional SPT in the transition region. Numerical results show that when the observation angle is very small, the Mie series solution of the near-fleld will not be accurate due to error accumulation. Furthermore, using the proposed methods, the CPU times for both the near-fleld and far-fleld calculations are frequency independent with controllable error. This work can be used to benchmark future works for high-frequency scattering.

Hydrodynamic Forces on a Ship Moving with Forward Speed in Waves

A new numerical evaluation scheme is developed of the Green function which is essential in the boundary-value problem of the motions of a ship with forward speed in waves. The single integral expression derived by Bessho' is used and its integration is performed along the path of steepest descent on which the integrand does not oscillate. Using this numerical steepest-descent method, we can evaluate the Green function with less computing time as well as enough accuracy, even if both of the field and source points are close to the free surface.With this efficient scheme incorporated in the panel method, radiation and diffraction problems are solved for a submerged spheroid. Convergence and accuracy of the solution are numerically checked by increasing the number of discretized panels on the body.In the radiation problem, contributions of the steady perturbation potential to the body boundary condition and to the unsteady pressure are consistently taken into account. These effects on the added-mass and damping coefficients are investigated by comparing the obtained results with another numerical results taking only the unform-flow contribution into consideration. In the diffraction problem, computations are performed for the wave exciting forces and the diffraction wave pattern on the free surface. These computed results are compared with corresponding experiments measured by the authors, and found to be in favorable agreement.