Laplace inversion methods

The effect of normalization errors on PSD prediction has been studied in the context of one Laplace inversion method (511. but the implications for the general problem of data weighting (Section II.B.2) have not been addressed. 

Hence, in case these pgfs are obtained for a given range of (complex) z-values, the CLD can be reconstructed via an appropriate Laplace inversion method. The z-based pgfs needed for this inversion are obtained by deriving the right-hand sides of... 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

Progress has been recently made in constructing an iterative inverse Laplace transform method which is not exponentially sensitive to noise. This Short Time Inverse Laplace Transform (STILT) method is based on rewriting the Bromwich inversion formula as ... 

In any case, the rate distribution function P(r), or the discrete pk, must first be determined from t,het(t) or g (t). Since this Laplace inversion is highly unstable, it is necessary to impose additional constraints, and various algorithms have been proposed in the literature [53,54,56,57,58,59]. Thorough discussions of these methods with respect to PCS data can be found in Ref. [60,61,62]. Here we will concentrate on the CONTIN program developed by Provencher [53,54,57]. 

Gregory, R.B. and Zhu, Y. (1990) Analysis of positron annihilation lifetime data by numerical laplace inversion with the program CONTIN , Nucl. Instrum. Methods Phys. Res., Sect. A 290,1172. 

Selection of an appropriate delay time range and a scheme for spacing them among correlator channels is necessary to ensure that intensity contributions are sampled from all of the particles in the sample [49,85]. Shaumeyer et al. [62] have explored the effect of the selected value of last delay time on particle size results based on the method of cumulants and single exponential methods. Stepanek [49] discusses the effect of incomplete data sets on Laplace inversion techniques. 

The objective of this study is to develop an analytical model for a soil column s response to a sinusoidally varying tracer loading function by applying the familiar Laplace transform method in which the convolution integral is used to obtain the inverse transformation. The solution methodology will use Laplace transforms and their inverses that are available in most introductory texts on Laplace transforms to develop both the quasi steady-state and unsteady-state solutions. Applications of the solutions will be listed and explained. 

The determination of the density of states for s classical oscillators by the method of Laplace transforms is of limited value because this can be obtained by other methods as well. Of much greater interest is the fact that the product of the quantum oscillators in Eq. (6.48) can be inverted by the Laplace transform method. However, it requires solving the inverse Laplace transform integral (Forst, 1971, 1973 Hoare and Ruijgrok, 1970) ... 

The amplitudes of the histogram of the distribution function are calculated by a non-negative least square method. This procedure is known as the exponential sampling method and is applicable to both monomodal and bimodal distributions. However, in view of the limitations of the Laplace inversion, it is difficult to resolve bimodal distributions with a ratio between the two particle species below 2. 

Practical Inversion Methods 363 Taking Laplace transforms of both sides gives... 

Criticisms of the inverse Laplace transform method to the integral for E[Pg.37]

Similar equations arise for the transverse relaxation rate l/7 2 . The problem now is to extract the pore distribution, W , from the observed RJ,t). From a mathematical point of view, this could be done by a Laplace inversion of Eqn (28.3). Examples of this method can be found in the review paper of W. P. Halperin et al. Other methods have used a prerequisite distribution that must be verified a posteriori. This has been carried out recently for the nuclear relaxation of and of methanol and nitromethane adsorbed on an organic polymeric resin crosslinked by paramagnetic divalent metal ions (Fig. 28.2). The results have been interpreted with a fractal distribution of categories of quasi-disconnected spherical pores, each being composed of N" spherical pores of radius R = RJ 1), with = log(/ o/ n.in)/log Introducing iht fractal dimension Df through the relation N a , leads to... 

Here 6 (r) is the distribution function to characterize the fall rate of structural reorganizations fluctuations. Generally, the Laplace inverse transformation (Equation 75) is required to deU rmine (7(1 ), but the function 7i(l) is suitable for this operation if only determined with a high accuracy, not achievable even by the up-to-date methods. 

Kipp (1985) transformed the above equations to the dimensionless form by dimensionless factors and variable parameters and solved the equations by the method of Laplace transformation, and finally got the solution through Laplace inverse transformation principle. 

The inversion of Eq. 9 from the complex s plane to the time plane can be made by a numerical Laplace transform inversion method. However, for high values of R, the dimensionless concentration in the reservoir is given by [3]... 

Thns, in either homodyne or heterodyne mode, the DLS autocorrelation function can yield D at any given q from an exponential fit of the form in Eqnation 5.63. For polydis-perse systems, DLS provides an average over the diffnsion coefficient distribution of the particles. A nnmber of approaches have been developed for analyzing polydisperse systems, including the robust cumulant method (see Section 8.2.2), histograms, and the inverse Laplace transform method (see Section 8.2.3). To turn the coefficient distribntion into the MWD requires a known relationship between D and M, often of the form D=AM y. 

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