Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory.

Area of Science:

• Data analysis
• Applied mathematics
• Scientific modeling

Background:

• Empirical orthogonal functions (EOFs) are derived from observed data distributions.
• Constructing EOFs requires a substantial library of observations, which is often unavailable.
• Mathematical functions can represent distributions when direct observations are scarce.

Purpose of the Study:

• To develop an inversion method using pseudo-empirical orthogonal functions (pseudo-EOFs) when observational data is insufficient.
• To leverage known mathematical forms of distributions to create a functional library.
• To enable function reconstruction from a limited set of mathematical functions.

Main Methods:

• Constructing a library of distributions from known mathematical functions.
• Developing pseudo-EOFs from this mathematical library.
• Employing a linear sum of pseudo-EOFs to represent any distribution.
• Implementing an inversion technique with smoothing and positivity constraints.

Main Results:

• Demonstrated the feasibility of constructing pseudo-EOFs from mathematical libraries.
• Successfully applied the pseudo-EOF method for function inversion without extensive observed data.
• Validated the effectiveness of incorporating smoothing and positivity constraints.

Conclusions:

• Pseudo-EOFs provide a viable alternative to empirical EOFs when observational data is limited.
• The developed inversion method is effective for reconstructing distributions using mathematical models.
• The technique offers a robust approach for data analysis in data-scarce scientific domains.