Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ε-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.     

On the equivalence between stable inversion for nonminimum phase systems and reciprocal transfer functions defined by the two-sided Laplace transform

In this paper, the two-sided Laplace transform, a classical but not very common mathematical tool, is revived to express the stable inversion for linear nonminimum phase systems that was recently proposed from the viewpoint of state-space representations. It is demonstrated that those two different expressions for the stable inversion are mathematically equivalent. Simple examples are presented to illustrate the two-sided Laplace transform as a direct and intuitive approach to stable inversion. The two-sided Laplace transform approach is also applied to the development of an iterative learning control for nonminimum phase systems that needs neither a precise inversion model nor Fourier-Transform computations, but instead requires only measuring the system response with time reversals.     

Laplace变换的代数数值反演方法

１．引 言 在过去的几十年中,以 Ｌａｐｌａｃｅ变换理论为基础的运算法［１］在自动控制领域中成功地简化并解决了许多实际问题．运算法中最困难的就是从变像（即 Ｌａｎｌａｃｅ变换）找出原函数即反演）．虽然有许多变换表中由变像可直接查出相应的反演,但变换表远远不能包括所有实际中遇到的问题,而且在一般情况下,反演的表达式是一个复积分［２］： 这个积分在很多情况下是十分复杂以致难以算出,这时我们只能通过近似的方法求出反演的近似值,即求数值反演． Ｌａｐｌａｃｅ变换的数值反演是个病态问题．尽管在九十年代以前许多学…     

On the numerical approximation of the Laplace transform

A method is developed for the numerical approximation of the Laplace transform of a given function of time. This method is an extension of the method proposed in [2] An example is included which compares the accuracy of the original method and the extension proposed.     

A direct method to solve integral and integro-differential equations of convolution type by using improved operational matrix

In this article, we use improved operational matrix of block pulse functions on interval [0,?1) to solve Volterra integral and integro-differential equations of convolution type without solving any system and projection method. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by improved operational matrix of integration. Numerical examples show that the approximate solutions have a good degree of accuracy.     

An efficient numerical method to solve a special cass of integro-differential equations relating to the Levy models

The paper presents a new efficient numerical method for solving a special class of integro-differential equations with numerical derivatives. These problems arise in applications used to calculate special functionals of the Levy processes. The method is based on the Wiener-Hopf approximated factorization and numerical inversion of the Laplace transform.     

A recursive method for determining the Laplace transform of exp (At)

A method for recursively determining the Laplace transform of the state transition matrix for a linear time-invariant system is presented. The method does not require matrix inversion, nor does the complexity of the operations involved increase with the dimensionality of the system. It is equally applicable to systems with distinct or multiple roots.     

Laplace transform analysis of the carbon cycle

A Laplace transform representation is used to describe the changes in atmospheric CO₂ in response to emissions. The formalism gives an explicit representation of generic relations that are less clear when model results are presented as numerical integrations with particular parameter values. In particular, the Laplace transform formalism clarifies some issues involved in inversion of ice-core data and analysis of geosequestration. The airborne fraction is expressed as the emission growth rate multiplied by the Laplace transform of the atmospheric response function, evaluated at the growth rate. This representation emphasises that historical data only capture carbon cycle dynamics over a limited range of time-scales. The Laplace transform formalism provides a basis for expressing uncertainties in the response function in terms of the Padé–Laplace transformation used for fitting sums of exponentials.     

Thermally induced vibrations of beam structures

A general numerical method is developed for determining the dynamic response of beam structures to rapidly applied thermal loads. The method consists of formulating and solving the dynamic problem in the Laplace transform domain with the aid of dynamic stiffness influence coefficients defined for a beam element in that domain and of obtaining the response by a numerical inversion of the transformed solution. Thus, the solution of the associated heat conduction problem, usually obtained by Laplace transform and needed for the computation of the thermal load, can be used in its transformed form. The effects of damping and of axial compressive forces on the structural response are also studied. Three examples are presented in detail to illustrate the proposed method and demonstrate its advantages.     

The accuracy of numerical inversion of the Laplace transform applied to the time-dependent reflection function from a semi-infinite atmosphere

In order to check the accuracy of the numerical inversion of the Laplace transform when it is applied to evaluate the time-dependent diffuse reflection function from a semi-infinite atmosphere, numerical experiments are done in virtue of comparing the numerical value with the exact one. It becomes clear that the numerical inversion method is sufficiently applicable if the time dependence of incident radiation is given by the Dirac δ function. A technique which reduces the error of the numerical inversion is proposed. It gives a remarkable improvement.     

Dynamic response of plate systems by combining finite differences, finite elements and laplace transform

A numerical method for the determination of the dynamic response of large rectangular plates or plate systems to lateral loads is proposed. The method is a combination of the finite difference method, the finite element method and the Laplace transform with respect to time. The plate system is considered as an assemblage of a small number of big rectangular superelements whose stiffness matrices are derived with the aid of the finite difference method in the Laplace transform domain. These superelements are then used to formulate and solve the problem by the finite element method in the transformed domain. The dynamic response is finally obtained by a numerical inversion of the transformed solution. External viscous or internal viscoelastic damping as well as the elastic foundation interaction effect can easily be taken into account. The method is illustrated and its merits demonstrated by means of numerical examples.     

Solution of plane transient elastodynamic problems by finite elements and laplace transform

The general transient linear elastodynamic problem under conditions of plane stress or plane strain is numerically solved by a special finite element method combined with numerical Laplace transform. A rectangular finite element with eight degrees of freedom is constructed on the basis of the governing equations of motion in the Laplace transformed domain. Thus the problem is formulated and numerically solved in the transformed domain and the time domain response is obtained by a numerical inversion of the transformed solution. Viscoelastic material behavior is easily taken into account by invoking the correspondence principle. The method appears to have certain advantages over conventional finite element techniques.     

Estimation of parameters in fractional subdiffusion equations by the time integral characteristics method

This paper presents an extension to the time integral characteristics method for estimation of parameters in fractional subdiffusion equations containing Riemann-Liouville and Caputo fractional time derivatives. The explicit representations of the fractional diffusion coefficient and order of fractional differentiation via a Laplace transform of the concentration field are obtained. A technique of optimal Laplace parameter determination by minimization of relative errors bounds is described. The effectivity of the proposed approach is illustrated by numerical example.     

Numerical inversion of integral transforms

The difficulties associated with the numerical inversion of Fourier and Laplace transforms are identified and a method, based on the modified Fourier transform, developed to overcome them. The method will deal with problems which are non-linear in the sense that the parameters are frequency dependent. Computed results compare favourably with test results on electrical power systems.     

Inverting fractional order transfer functions through Laguerre approximation

This paper concerns the approximate inversion of fractional order transfer functions. The approximating solution is of the form of a weighted sum of Laguerre functions. The weights of the sum are found from a simple generating expression. As case study, the inversion of the Laplace transform of the Abel's equation is considered.     

基于结合扩展精度技术的基本解方法的非线性功能梯度材料热传导问题求解

采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量,将时域问题转化为频域问题求解;采用基本解方法计算得到高精度的频域数值解,再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例,分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点,能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.     

Use of dynamic influence coefficients in forced vibration problems with the aid of Laplace transform

The use and importance of dynamic stiffness influence coefficients in flexural forced vibrations of structures composed of beams are described. The dynamic forces can be either harmonic or general transient forces. The dynamic influence coefficients are defined in the Laplace transform plane, are computed there and are given in tables for uniform beams under various end conditions. The dynamic response is obtained, in general, by a matrix inversion in the Laplace transform plane and a numerical inversion, based on interpolation concepts, of the transformed solution. Structural examples of forced vibrations of a simple beam and a rigid frame illustrate the use of dynamic coefficients and demonstrate their advantages over other known methods in accuracy, simplicity of formulation and speed of computation.     

A Flexible Inverse Laplace Transform Algorithm and its Application

A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotient-difference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotient-difference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm.