Chebyshev Interpolation Research Articles

Chebyshev interpolation for parametric option pricing

Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce run-times while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.

Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media

In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.

An Adaptive Partition of Unity Method for Chebyshev Polynomial Interpolation

For a function that is analytic on and around an interval, Chebyshev polynomial interpolation provides spectral convergence. However, if the function has a singularity close to the interval, the rate of convergence is near one. In these cases splitting the interval and using piecewise interpolation can accelerate convergence. Chebfun includes a splitting mode that finds an optimal splitting through recursive bisection, but the result has no global smoothness unless conditions are imposed explicitly at the breakpoints. An alternative is to split the domain into overlapping intervals and use an infinitely smooth partition of unity to blend the local Chebyshev interpolants. A simple divide-and-conquer algorithm similar to Chebfun's splitting mode can be used to find an overlapping splitting adapted to features of the function. The algorithm implicitly constructs the partition of unity over the subdomains. This technique is applied to explicitly given functions as well as to the solutions of singularly pertur...

Chebyshev Methods for Ultra-Efficient Risk Calculations

Financial institutions now face the important challenge of having to do multiple portfolio revaluations for their risk computation. The list is almost endless: from XVAs to FRTB, stress testing programs, etc. These computations require from several hundred up to a few million revaluations. The cost of implementing these calculations via a “brute-force” full revaluation is enormous. There is now a strong demand in the industry for algorithmic solutions to the challenge. In this paper we show a solution based on Chebyshev interpolation techniques. It is based on the demonstrated fact that those interpolants show exponential convergence for the vast majority of pricing functions that an institution has. In this paper we elaborate on the theory behind it and extend those techniques to any dimensionality. We then approach the problem from a practical standpoint, illustrating how it can be applied to many of the challenges the industry is currently facing. We show that the computational effort of many current risk calculations can be decreased orders of magnitude with the proposed techniques, without compromising accuracy. Illustrative examples include XVAs and IMM on exotics, XVA sensitivities, Initial Margin Simulations, IMA-FRTB and AAD.

Numerical investigation of the inverse nodal problem by Chebisyhev interpolation method

In this study, we deal with the inverse nodal problem for Sturm-Liouville equation with eigenparameter-dependent and jump conditions. Firstly, we obtain reconstruction formulas for potential function, q, under a condition and boundary data, ?, as a limit by using nodal points to apply the Chebyshev interpolation method. Then, we prove the stability of this problem. Finally, we calculate approximate solutions of the inverse nodal problem by considering the Chebyshev interpolation method. We then present some numerical examples using Matlab software program to compare the results obtained by the classical approach and by Chebyshev polynomials for the solutions of the problem.

Chebyshev Spectral Element Analysis for Pore-Pressure of ERDs during Construction Period

Chebyshev spectral elements are applied to dissipation analysis of pore-pressure of roller compaction earth-rockfilled dams (ERD) during their construction. Nevertheless, the conventional finite element, for its excellent adaptability to complex geometrical configuration, is the most common way of spatial discretization for the pore-pressure solution of ERDs now [1]. The spectral element method, by means of the spectral isoparametric transformation, surmounts the disadvantages of disposing with complex geometry. According to the illustration of numerical examples, one can conclude that the spectral element methods have the following obvious advantages: 1) large spectral elements can be used in spectral element methods for the domains of homogeneous material; 2) in the application of large spectral elements to spatial discretization, only a few leading terms of Chebyshev interpolation polynomial are taken to arrive at the solutions of better accuracy; 3) spectral element methods have excellent convergence as well-known. Spectral method also is used to integrate the evolution equation in time to avoid the limitation of conditional stability of time-history integration

Efficient characterization of phase space mapping in axially symmetric optical systems

Phase space mapping, typically between an object and image plane, characterizes an optical system within a geometrical optics framework. We propose a novel conceptual frame to characterize the phase mapping in axially symmetric optical systems for arbitrary object locations, not restricted to a specific object plane. The idea is based on decomposing the phase mapping into a set of bivariate equations corresponding to different values of the radial coordinate on a specific object surface (most likely the entrance pupil). These equations are then approximated through bivariate Chebyshev interpolation at Chebyshev nodes, which guarantees uniform convergence. Additionally, we propose the use of a new concept (effective object phase space), defined as the set of points of the phase space at the first optical element (typically the entrance pupil) that are effectively mapped onto the image surface. The effective object phase space provides, by means of an inclusion test, a way to avoid tracing rays that do not reach the image surface.

Normalized Bernstein polynomials in solving space-time fractional diffusion equation

In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, we offer an algebraic map to make the rational normalized Bernstein functions. This study uses Galerkin and collocation methods. The integrals in the Galerkin method are established with Chebyshev interpolation. We implemented the proposed methods for some examples that are presented to demonstrate the theoretical results. To confirm the accuracy, error analysis is carried out.

Solving inverse Sturm–Liouville problem with separated boundary conditions by using two different input data

ABSTRACTIn this study, we consider Sturm–Liouville (SL) equation under the separated boundary conditions on a finite interval. We get the approximate solutions of inverse SL problem by using different input data as eigenvalues and nodes (zeros of eigenfunctions), separately, and calculate the computed errors related to the obtained approximate solutions. To calculate the approximate solutions, we use Chebyshev interpolation technique by applying first kind of Chebyshev polynomials. Eventually, the numerical results are presented by providing some examples.

An analysis of a butterfly algorithm

Butterfly algorithms are an effective multilevel technique to compress discretizations of integral operators with highly oscillatory kernel functions. The particular version of the butterfly algorithm presented in Candès, et al. (2009) realizes the transfer between levels by Chebyshev interpolation. We present a refinement of the analysis given in Demanet, et al. (2012) for this particular algorithm.

Multidimensional Chebyshev interpolation for warm and hot dense matter.

We propose a scheme based on a multidimensional Chebyshev interpolation to approximate smooth functions that depend on more than one variable. The present method generalizes the one dimensional Chebyshev approximation. The multidimensional approach can be used for generating databases like equation of state in the warm and hot dense matter. It is well suited to the present advance of massively parallel supercomputers.

Convergent Power Series for Boundary Value Problems and Eigenproblems with Application to Atmospheric and Oceanic Tides

It is usually impossible to apply power series to solve boundary value and eigenvalue problems because the radius of convergence does not extend to the boundaries. Hough functions, which solve a second order eigenproblem, are always entire functions or can be computed wholly in terms of entire functions. The eigenvalues ϵ are approximated by the zeros of the highest computed power series coefficient, which is always a polynomial in ϵ. The rate of convergence with series truncation N is proportional to exp(-μN) for some positive constant A brief table contains the entire Maple code for the algorithm. The scheme can also be extended to eigenproblems on an unbounded domain as here illustrated by the quantum harmonic oscillator and quartic oscillator. Computer algebra is very useful; the power series method can also be implemented in a stable, purely numerical fashion by applying trigonometric interpolation on a disk in the complex x-plane and Chebyshev interpolation on a real interval in e. Power series are briefly compared with Chebyshev, Fourier, and spherical harmonic spectral methods, which are mostly superior. The success of power series raises an interesting history-of-science issue: Why was there so little progress in solving Laplace's tidal equations in the nineteenth century?

A numerical approach for a class of astrophysics equations using piecewise spectral-variational iteration method

PurposeThe main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.Design/methodology/approachThis method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.FindingsSome well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.Originality/valueIn this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis, and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten $p$-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

Experimental study on f-ω regulation model under abnormal methane emission

In view of the difficulty of automatic adjustment, the recovery lag and the major accident potential of the mine ventilation system, an experimental model of the pipe net was established according to the typical one mine and one working face ventilation system of Daliuta coal mine. Using the best uniform approximation method of Chebyshev interpolation to fit the fan performance curve, we experimentally determined fan characteristics with different frequencies and establish the data base for the curves. Based on ventilation network monitoring theory, we designed a monitoring system for ventilation network parameter monitoring and fan operating frequency automatic control. Using the absolute methane emission quantity to predict the air quantity requirement of branch and fan frequency, we established a f-ω regulation model based on fan frequency and absolute methane emission quantity. After analysing methane emission and distribution characteristics, using CO2 to simulate the methane emission characteristics from a working face, we verified the correctness and rationality of the f-ω regulation model. The fan operation frequency is adjusted by the method of air adjustment change with methane emission quantity and the curve searching method after determining air quantity requirements. The results show that the air quantity in a branch strictly changes according to the f-ω regulation model, in the air-increasing dilution by fan frequency regulation, the CO2 concentration is limited to the set threshold value. The paper verifies the practicability of a frequency regulation system and the feasibility of the frequency adjustment scheme and provides guidance for the construction of automatic frequency conversion control system in coal mine ventilation networks.

An Efficient Pseudo-Spectral Method for Nonsmooth Dynamical Systems

In this paper, we present a new approach for solving nonsmooth dynamical systems. We first convert the nonsmooth system to the smooth form, using Chebyshev interpolation. Then, we solve the smooth system using Chebyshev pseudo-spectral method. The efficiency of our approach is shown for two practical nonsmooth mechanical systems.

Resolvent sampling based Rayleigh–Ritz method for large-scale nonlinear eigenvalue problems

A new algorithm, denoted by RSRR, is presented for solving large-scale nonlinear eigenvalue problems (NEPs) with a focus on improving the robustness and reliability of the solution, which is a challenging task in computational science and engineering. The proposed algorithm utilizes the Rayleigh–Ritz procedure to compute all eigenvalues and the corresponding eigenvectors lying within a given contour in the complex plane. The main novelties are the following. First and foremost, the approximate eigenspace is constructed by using the values of the resolvent at a series of sampling points on the contour, which effectively circumvents the unreliability of previous schemes that using high-order contour moments of the resolvent. Secondly, an improved Sakurai–Sugiura algorithm is proposed to solve the projected NEPs with enhancements on reliability and accuracy. The user-defined probing matrix in the original algorithm is avoided and the number of eigenvalues is determined automatically by the provided strategies. Finally, by approximating the projected matrices with the Chebyshev interpolation technique, RSRR is further extended to solve NEPs in the boundary element method, which is typically difficult due to the densely populated matrices and high computational costs. The good performance of RSRR is demonstrated by a variety of benchmark examples and large-scale practical applications, with the degrees of freedom ranging from several hundred up to around one million. The algorithm is suitable for parallelization and easy to implement in conjunction with other programs and software.

Fast method of approximate particular solutions using Chebyshev interpolation

The fast method of approximate particular solutions (FMAPS) is based on the global version of the method of approximate particular solutions (MAPS). In this method, given partial differential equations are discretized by the usual MAPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions.

Numerical Pricing of European Options with Arbitrary Payoffs

In this paper we introduce the CHEB method, a quadrature-based methodology for the fast and accurate pricing of European options with arbitrary payoffs. The method comes as a natural application of Chebfun, a numerical computing software package built on the approximation properties of Chebyshev series and Chebyshev interpolants. For the methodology to be useful for practical purposes, we address two considerations: the recovery of the underlying’s density from the characteristic function, and the estimation of the truncation error. We highlight an important connection between the proposed technique and the very efficient COS method, in which the payoff and density function are approximated by cosine series.