Inverse Laplace Transform of Multidimensional Relaxation Data Without Non-Negativity Constraint

Area of Science:

• Applied Mathematics
• Computational Science
• Data Analysis

Background:

• Multidimensional data analysis often requires spectrum conditioning.
• Inverse Laplace transforms are crucial for analyzing such data.
• Existing methods may impose strict non-negativity constraints, limiting applicability.

Purpose of the Study:

• To develop an algorithm for inverse Laplace transform of multidimensional data.
• To perform spectrum conditioning without a non-negativity constraint.
• To introduce novel regularization techniques for improved spectral accuracy.

Main Methods:

• Utilized Tikhonov regularization in generalized form.
• Implemented Uniform Penalty (UP) regularization to relax non-negativity requirements.
• Introduced a zero-crossing (ZC) penalty weighted by spectral slope.

Main Results:

• The algorithm effectively performs inverse Laplace transform without a non-negativity constraint.
• Uniform Penalty and zero-crossing penalties improve spectrum conditioning.
• The method reduces nonphysical undershooting near narrow peaks.

Conclusions:

• The developed algorithm offers a robust approach for inverse Laplace transforms in multidimensional data.
• The novel regularization strategy enhances spectral quality and reduces artifacts.
• This method provides a valuable tool for spectrum conditioning in various scientific applications.