energy

Truncate the subspace based on eigenvalue energy

Specification

  • Alias: None

  • Arguments: None

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Optional

truncation_tolerance

Specify the maximum percentage (as a decimal) of the eigenvalue energy not captured by the active subspace representation.

Description

Uses a criterion based on the derivative matrix eigenvalue energy.

Usage Tips

This subspace truncation method may work best when working with non-normally distributed uncertain variables. If this automated diagnostic does not yield desirable results, consider using the explicit dimension truncation option or one of the other truncation methods.

Theory

Using the eigenvalue energy truncation metric, the subspace size is determined using the following equation:

\[n = \inf \left\lbrace d \in \mathbf{Z} \quad\middle|\quad 1 \le d \le N \quad \wedge\quad 1 - \frac{\sum_{i = 1}^{d} \lambda_i}{\sum_{i = 1}^{N} \lambda_i} \,<\, \epsilon \right\rbrace\]

where \(\epsilon\) is the truncation_tolerance, \(n\) is the estimated subspace size, \(N\) is the size of the full space, and \(\lambda_i\) are the eigenvalues of the derivative matrix.