A novel mollifier inversion scheme for the Laplace transform

In this article we present a novel inversion method for the Laplace transform for non‐equidistant scanning points applying the approximate inverse to this transform. The approximate inverse is a regularization technique for inverse problems based on evaluations of scalar products of the given data with so called reconstruction kernels. Each kernel solves a system of linear equations defined by the adjoint of the Laplace transform and dilatation invariant mollifiers, which are designed articularly for this operator. The paper includes numerical results.     

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

一种基于小波尺度函数变换的Laplace反演方法

该文基于Ｄａｕｂｅｃｈｉｅｓ小波尺度函数变换建立了关于Ｌａｐｌａｃｅ变换的一种反演数值方法．通过对小波尺度函数的低带通谱特性的定性与定量讨论,给出了这一反演方法所得原像函数的适用域．结果发现：其区域大小随着小波尺度函数的分辨指标（ｒｅｓｏｌｕｔｉｏｎｌｅｖｅｌ）选取的升高而增大．最后,以颤振曲线、具有指数增长的复函数、和一维振动弦的初边值问题等为例,定量给出了其反演方法的数值结果．通过与相应的原像精确结果对比发现：在反演的有效区域内,其数值反演的原像几乎与精确的原像图象重合．这表明这一Ｌａｐｌａｃｅ反演数值方法是有效和可靠的．     

非饱和土层一维固结问题的解析解

对一有限厚度,处于一维受荷状态,表面为透水透气面,底面为不透水不透气面的非饱和土层,依据Fredlund的非饱和土一维固结理论,由液相及气相的控制方程、Darcy定律及Fick定律,经Laplace变换及Cayley-Hamilton数学方法构造了顶面状态向量与任意深度处状态向量间的传递关系;通过引入初始及边界条件,得到了Laplace变换域内的超孔隙水压力、超孔隙气压力以及土层沉降的解;实现Laplace逆变换,得到了时间域内的解析解;用一典型算例,与差分法结果进行对比,验证了其正确性．     

ReLaTIve. An Ansi C90 software package for the Real Laplace Transform Inversion

A software package for numerical inversion of Laplace transforms computable everywhere on the real axis is described. Besides the function to invert the user has only to provide the numerical value (even if it is an approximate value) of the abscissa of convergence and the accuracy required for the inverse function. The software provides a controlled accuracy, i.e. it dynamically computes the so-called maximum attainable accuracy such that numerical results are provided within the greatest value between the user’s required accuracy and the maximum attainable accuracy. This is done because the intrinsic ill posedness of the real inversion problem sometime may prevent to reach the desired accuracy. The method implemented is based on a Laguerre polynomial series expansion of the inverse function and belongs to the class of polynomial-type methods of inversion of the Laplace transform, formally characterized as Collocation methods (C-methods).     

有势场逆问题的边界元法

本文给出了位势方程逆问题的一种最小二乘边界元解法。控制方程为Ｌａｐｌａｃｅ方程，但一部分边界上未给出任何边值，而只在某些内点上给出了势函值。这一问题在数学上属不适定问题，但在一定条件下存在唯一解。本文同时给出了一种估计解的可靠性的方法。数值试验表明，这类逆问题采用边界元法是非常有效的。     

Application of high order numerical quadratures to numerical inversion of the Laplace transform

In this paper, new algorithms are proposed for Fredholm integral equations of the first kind corresponding to the inverse Laplace transform. We apply high order numerical quadratures to the truncated integral equation and apply regularization to the discretized linear systems. The resulted regularized least square problems are then solved by the reduced QR factorization method. Several examples taken from the literature are tested. Numerical results show that the approximate inverse Laplace transform obtained by our approach can be very accurate.     

A method for the numerical inversion of Laplace transforms

In this paper a numerical inversion method for Laplace transforms, based on a Fourier series expansion developed by Durbin [5], is presented. The disadvantage of the inversion methods of that type, the encountered dependence of discretization and truncation error on the free parameters, is removed by the simultaneous application of a procedure for the reduction of the discretization error, a method for accelerating the convergence of the Fourier series and a procedure that computes approximately the ‘best’ choice of the free parameters. Suitable for a given problem, the inversion method allows the adequate application of these procedures. Therefore, in a big range of applications a high accuracy can be achieved with only a few function evaluations of the Laplace transform. The inversion method is implemented as a FORTRAN subroutine.     

Transformation of surface waves in a liquid with local changes in depth

The propagation of a pulse on the surface of a liquid of finite depth is studied when the depth decreases over a finite interval between liquids with constant depths to the left and right. The decrease in depth is specified by a parabolic function and the pulse, which increases sharply in time and then decays, is turned on at the initial time some distance to the right of the section with a variable depth. A Laplace transform method is used to solve the corresponding initial value-boundary value problem and this makes it possible to obtain a solution in hypergeometric functions in the transform space. In the limiting case of a linear variation in the depth, a numerical inversion of the Laplace transform is used to construct solutions which are analyzed for various geometric parameters and at different times. Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 131–142, 1999.     

Axisymmetrical vibrations of shells of revolution under abrupt loading

A frequency method is proposed for solving the problem of the vibrations of shells of revolution taking into account the energy dissipation under arbitrary force loading and on collision with a rigid obstacle. The Laplace transform is taken of the equation of the vibrations of a shell of revolution with non-zero initial conditions. For the inhomogeneous differential equation obtained, a variational method is used to solve the boundary-value problem, which consists of finding the Laplace-transformed boundary transverse and longitudinal forces and bending moments as functions of the boundary displacements. The equations of equilibrium of nodes, i.e. the corresponding equations of the finite-element method, are then compared, using results obtained earlier [1–4]. Amplitude-phase-frequency characteristics (APFCs) for the shell cross-sections selected are plotted. An inverse Laplace transformation is carried out using the clear relationship between the extreme points of the APFCs and the coefficients of the corresponding terms of the series in an expansion vibration modes [3]. In view of the fact that the proposed approach is approximate, numerical testing is used.     

A problem on heat conduction in an insulated wire solved by numerical inverse Laplace transform

A problem of transient heat conduction in an insulated wire is solved by use of Laplace transform and numerical inversion. The problem is solved for the radiation boundary condition and also for the boundary condition of no heat flux through the outer surface of the insulation. The results are presented both numerically with four significant figures and graphically. Asymptotic expansions are derived for small and large values of the time variable. The numerical inversion of the Laplace transform is checked by comparison with the asymptotic expansions and with the numerical results obtained by a numerical inversion formula utilizing one more abscissa than the previous one.     

An electromagnetic free convection flow of a micropolar fluid with relaxation time

In the present investigation, we study the influence of a transverse magnetic field on the one-dimensional motion of an electrically conducting micropolar fluid through a porous medium. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the microrotation and the velocity distributions as well as for the induced magnetic and electric fields are carried out and represented graphically.     

AIR Tools II: algebraic iterative reconstruction methods,improved implementation

This paper is devoted to designing a practical algorithm to invert the Laplace transform by assuming that the transform possesses the Puiseux expansion at infinity. First, the general asymptotic expansion of the inverse function at zero is derived, which can be used to approximate the inverse function when the variable is small. Second, an inversion algorithm is formulated by splitting the Bromwich integral into two parts. One is the main weakly oscillatory part, which is evaluated by a composite Gauss–Legendre rule and its Kronrod extension, and the other is the remaining strongly oscillatory part, which is integrated analytically using the Puiseux expansion of the transform at infinity. Finally, some typical tests show that the algorithm can be used to invert a wide range of Laplace transforms automatically with high accuracy and the output error estimator matches well with the true error.     

Best rational function approximation to Laplace transform inversion using a window function

The best rational function approximation for Laplace transform inversion due to Longman is modified by the introduction of an appropriate “window” function. This window function enables one to approximate the inverse transform f(t) by a linear combination g_n(t) of n exponential functions accurately in a given interval about a given point along the t-axis. It is proved that the sequence of approximants {g_n(t)}_{n = 1}^∞ converges to f(t) in the mean. The method is illustrated by some numerical examples.     

Least squares coefficients for a quadrature formula for Laplace transform inversion

A linear iterative method of least squares approximation of functions by exponentials due to Miller [9] is adapted to derive a set of least squares coefficients for an approximate Laplace transform inversion formula eq. (1). An earlier assumption made by Zakian [2] - that the approximation to the Laplace transform inverse will improve provided the approximation to the Dirac delta function is improved - is shown to be not substantiated for a number of test functions.     

An efficient algorithm for regularization of Laplace transform inversion in real case

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution.

In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.     

Inverse two-sided Laplace transform for probability density functions

We present a method for the numerical inversion of two-sided Laplace transform of a probability density function. The method assumes the knowledge of the first M derivatives at the origin of the function to be antitransformed. The approximate analytical form is obtained by resorting to maximum entropy principle. Both entropy and L₁-norm convergence are proved. Some numerical examples are illustrated.     

二维抛物型方程参数反演的遗传算法

本文研究了二维抛物型方程参数反演问题.利用遗传算法求解此反演问题的方法,把参数反演问题转化为优化问题,通过演化计算方法求解.它从多个初始点开始寻优,借助交叉和变异算子来获得参数的全局最优解.且数值模拟结果表明,具有精度高、编程简单、易于计算机实现等特点.     

A perturbation-boundary integral equation method for transient dynamics of nonlinear elastic materials

This study presents a formulation for solving the transient dynamics of nonlinear elastic materials. By using a perturbation expansion to linearize the basic equations and applying the Laplace transform to the subsequent perturbation equations, the boundary value problem of the transformed equations is further reduced to various boundary integral equations. After discretization of the integral equations, these are solved numerically, completing the solution in the Laplace transform space. Performing a numerical inversion of the Laplace transform yields the solution of the problem in the time domain.     

A Laplace transform certified reduced basis method; application to the heat equation and wave equation

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.