Regularization Method for Numerical Inversion of Laplace Transforms
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Regularization Method for Numerical Inversion of Laplace Transforms

Summary | Problem classification | Method's place and advantages | References

Summary:
           My last paper Numerical inversion of the Laplace transform: analysis via regularized analytic continuation concludes and summarizes my research. At the same time, it also generalizes the results of many research projects conducted in this field over the last 70 years. My presentation at the conference "Tikhonov and Contemporary Mathematics" (Moscow, June 2006) covers basic ideas of this paper. Download:



Second part of my presentation IPMS: The International Conference "Inverse Problems: Modeling and Simulation" (June, 2004. Turkey) contains a detailed explanation of the developed method and outlines possible applications.

Problem classification:
           It is only matters for a researcher what is possible to get from the information in hands. The choice of an appropriate method obviously depends on the input information available. Generally there are three cases encountered while solving a problem by means of Laplace transform:
  1. Laplace transform is obtained in analytically closed form. In simple cases the inverse transform can be found via analytical methods or with the help of tables.
  2. Laplace transform is computable in the complex half-plane of convergence. In this case the original function can be computed by evaluation of the complex integral of inverse transformation.
  3. Laplace transform is computable or measurable on the real and positive axis only. In this case the problem is extremely ill-posed. This case is much more complicated simply because of absence of the exact inversion formula.

In general, no one has needs in numerical inversion if the answer is known from the tables, unless it is too complicated. Similarly, if a LT is known in complex half plane, the numerical inversion based calculations of the Bromwich integral (whenever it is possible) should result in a better output. The third case is when the method I developed is applicable.

Method's place and advantages:
           The method under consideration is designed for inverting of a Laplace transforms given on the real and positive axis. Currently it is the best method designed for an invertion of real-valued Laplace transforms because of:
  1. Stability and applicability to any image function given on the real axis;
  2. Most accurate results for the same accuracy of input data;
  3. Predictable behavior and theoretical justification;
  4. Clearly stated method's restrictions and limitations.
  5. Simple rules for regularization parameter's selection. More details can be found in the paper [2] from the references list below.
From author's experience it can be stated that numerical inversion of a LT from the real and positive axis can be successfully applied in two cases:
  1. When a LT can be obtained only for real and positive values of its parameter. The problem of exponential analysis is an example. Other examples are specific heat-phonon spectrum inversion and inverse blackbody radiation problems.
  2. When it is simpler to calculate LT as a function of real parameter, and inversion from the real axis could lead to satisfactory results. If the output of a problem under consideration is expected to be not highly oscillating function, then one can expect to obtain acceptable results. For example, one can try to use this method in conjunction with another numerical method for solving PDE of the parabolic type. (See conference presentation above for a simple example)

References:
           The references are given in a logical and time order from the most general results covered in later papers to the first and earliest ones. Each paper reflects a step in the method development and generalization.
  1. Kryzhniy V.V. Numerical inversion of the Laplace transform: analysis via regularized analytic continuation.
    Inverse Problems, 22, 2006, pp. 579-597
    This paper generalizes the results of many research projects conducted in this field over the last 70 years. It summarizes my research and provides a comprehensive analysis of the problem, including problem's classification, criteria for comparison of real methods, and accuracy limitations of even the most effective real method.

  2. Kryzhniy V.V. High-resolution exponential analysis via regularized numerical inversion of Laplace transforms.
    J. Comp. Phys., Vol.199/2, 2004, pp. 618- 630
    Analytical background of the method of inverse Laplace transformation made it possible to overcome the known limitations for the problem of exponential analysis. This is due to the fact that zeroes of the inverse Laplace transform of a sum of exponential decays are as informative as its maxima.

  3. Kryzhniy V.V. On regularization of numerical inversion of Laplace transforms.
    J. of Inverse Ill-Posed problems, 2004, Vol.12, No.3, pp.279-296
    This paper deals with derivation of the most general formulas. Multiple regularizing operators are built. Arbitrary parameter is substituted with arbitrary function. Error analysis and recommendations for parameter selection are given for the most interesting choice of arbitrary function. Numerical results demonstrate advantages over other known methods.

  4. Kryzhniy V.V. Regularized inversion of integral transformations of Mellin convolution type.
    Inverse Problems, 19, 2003, pp. 1227-1240
    The results of the previous paper are generalized in two ways. First, the integer-valued parameter is substituted with real-valued one. This required creation of a new approach for building of regularizing operators. Sets of regularizing operators for inverting integral transformations of Mellin convolution type. The principal difference between Fourier-Hankel type transformations and Laplace and Meijer transformations is discussed. Numerical examples for inverting of Fourier sine and Laplace transformations are given.

  5. Kryzhniy V.V. Regularizing Operators of Real-valued Inverse Laplace transformation.
    Inverse Problems in Engineering, 11, 2003, pp.561- 574
    Paper provides first generalization of the method by introducing an integer-valued parameter. A set of regularizing operators is built. Short discussion on automated choice of regularization parameter might be of interest.

  6. Kryzhniy V.V. Direct regularization of Inversion of Real-Valued Laplace Transforms.
    Inverse Problems, 19, 2003, pp.573-583.
    This first paper deals with a construction and analysis of a regularizing operator for inverse Laplace transformation. The constructed operator is applicable to a certain subset of Laplace transforms. The initial idea of this paper was to find a stabilizing factor in Tikhonov-Arsenin sense that allows proceeding analytical development as far as possible. The obtained formulas were generalized later. Some details of the error analysis and examples might be of interest.
Sincerely,                              
Vladimir Kryzhniy


Last modified on August 17, 2006
 

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