Abstract
The Weeks method for the numerical inversion of the Laplace transform utilizes a Möbius transformation which is parameterized by two real quantities, σ and b. Proper selection of these parameters depends highly on the Laplace space function F(s) and is generally a nontrivial task. In this paper, a convolutional neural network is trained to determine optimal values for these parameters for the specific case of the matrix exponential. The matrix exponential e^{A} is estimated by numerically inverting the corresponding resolvent matrix (Formula presented.) via the Weeks method at (Formula presented.) pairs provided by the network. For illustration, classes of square real matrices of size three to six are studied. For these small matrices, the CayleyHamilton theorem and rational approximations can be utilized to obtain values to compare with the results from the network derived estimates. The network learned by minimizing the error of the matrix exponentials from the Weeks method over a large data set spanning (Formula presented.) pairs. Network training using the Jacobi identity as a metric was found to yield a selfcontained approach that does not require a truth matrix exponential for comparison.
Original language  English (US) 

Journal  Journal of Algorithms and Computational Technology 
Volume  15 
DOIs  
State  Published  2021 
Keywords
 Numerical Laplace transform inversion
 Weeks’ method
 machine learning
 matrix exponential
ASJC Scopus subject areas
 Numerical Analysis
 Computational Mathematics
 Applied Mathematics
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sjzip1act10.1177_1748302621999621  Supplemental material for Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning
Kano, P. O. (Creator), Brio, M. (Contributor) & Bailey, J. (Creator), SAGE Journals, 2021
DOI: 10.25384/sage.14337082.v1, https://sage.figshare.com/articles/dataset/sjzip1act10_1177_1748302621999621__Supplemental_material_for_Optimal_parameter_selection_in_Weeks_method_for_numerical_Laplace_transform_inversion_based_on_machine_learning/14337082/1
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Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning
Kano, P. O. (Creator), Brio, M. (Contributor) & Bailey, J. (Creator), SAGE Journals, 2021
DOI: 10.25384/sage.c.5357908.v1, https://sage.figshare.com/collections/Optimal_parameter_selection_in_Weeks_method_for_numerical_Laplace_transform_inversion_based_on_machine_learning/5357908/1
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