On the inverse Laplace transform for C0-semigroups in UMD-spaces

We consider a two dimensional free boundary problem for an inviscid, incompressible and irrotational fluid. Imbedded in the fluid is a gas bubble of prescribed shape. Given some initial configuration we investigate the long time evolution of the potential flow. We prove existence, uniqueness and regularity of a solution.     

Review of inverse Laplace transform algorithms for Laplace-space numerical approaches

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time.     

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.     

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on (0,∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.     

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Computation of the inverse Laplace transform based on a collocation method which uses only real values

We develop a numerical algorithm for inverting a Laplace transform (LT), based on Laguerre polynomial series expansion of the inverse function under the assumption that the LT is known on the real axis only. The method belongs to the class of Collocation methods (C-methods), and is applicable when the LT function is regular at infinity. Difficulties associated with these problems are due to their intrinsic ill-posedness. The main contribution of this paper is to provide computable estimates of truncation, discretization, conditioning and roundoff errors introduced by numerical computations. Moreover, we introduce the pseudoaccuracy which will be used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x   with respect to e^σx${\mathrm{e}}^{\sigma x}$. These estimates are then employed to dynamically truncate the series expansion. In other words, the number of the terms of the series acts like the regularization parameter which provides the trade-off between errors.     

Spectral regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation with one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

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.     

Motivating the Laplace transform

The Laplace transform clearly provides a rigorization of the operational calculus. This paper proceeds in the opposite direction. It relies heavily on work by Carson, but removes it from its explicitly electrical context.     

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.     

The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations

The differential transform method is one of the approximate methods which can be easily applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. In this paper, we present the definition and operation of the one-dimensional differential transform and investigate the particular exact solutions of system of ordinary differential equations that usually arise in mathematical biology by a one-dimensional differential transform method. The numerical results of the present method are presented and compared with the exact solutions that are calculated by the Laplace transform method.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.     

More on the weeks method for the numerical inversion of the Laplace transform

Summary Most of the numerical methods for the inversion of the Laplace Transform require the values of several incidental parameters. Generally, these parameters are related to the properties of the algorithm and to the analytical properties of the Laplace Transform functionF(s).One of the most promising inversion methods, the Weeks methods, computes the inverse functionf(t) as a series expansion of Laguerre functions involving two parameters, usually denoted by andb. In this paper we characterize the optimal choiceb _opt ofb, which maximizes the rate of convergence of the series, in terms of the location of the singularities ofF(s).     

多层介质中逆热传导问题的傅里叶正则化方法

逆热传导问题是数学物理反问题中的热点和前沿课题之一,在钢铁生产等领域中具有重要的应用背景.讨论一个多层介质中的逆热传导问题,它是一个极度不适定问题.通过傅里叶截断方法构造正则化近似解,并给出相应的稳定性估计.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Approximations for optimal Laplace transform inversion

A function (p) of the Laplace transform operatorp is approximated by a finite linear combination of functions (p+ _r ), where (p) is a specific function ofp having a known analytic inverse (t), and is chosen in accordance with various considerations. Then parameters _r ,r=1, 2,...,n, and then corresponding coefficientsA _r of the (p + _r ) are determined by a least-square procedure. Then, the corresponding approximation to the inversef(t) of (p) is given by analytic inversion of _r=1 ⁿ A _r (p+ _r ). The method represents a generalization of a method of best rational function approximation due to the author [which corresponds to the particular choice (t)1], but is capable of yielding considerably greater accuracy for givenn.The computations for this paper were carried out on the CDC-6600 computer at the Computation Center of Tel-Aviv University. The author is grateful to Dr. H. Jarosch of the Weizmann Institute of Science Computer Center for use of their Powell minimization subroutine (Ref. 1).     

A generalized Laplace transform of generalized functions

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A comparative approach to Laplace transform inversion

A complex Laplace transform function was inverted by three numerical methods and compared to the small time and large time approximation curves. This technique enabled the best choice of an inversion method to be made, since one method gave excellent results, at both small and large times and moved smoothly from one curve to the other.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.