Discovering an active subspace in a single‐diode solar cell model

Abstract

Predictions from science and engineering models depend on the values of the model's input parameters. As the number of parameters increases, algorithmic parameter studies like optimization or uncertainty quantification require many more model evaluations. One way to combat this curse of dimensionality is to seek an alternative parameterization with fewer variables that produces comparable predictions. The active subspace is a low-dimensional linear subspace defined by important directions in the model's input space; input perturbations along these directions change the model's prediction more, on average, than perturbations orthogonal to the important directions. We describe a method for checking if a model admits an exploitable active subspace, and we apply this method to a single-diode solar cell model with five input parameters. We find that the maximum power of the solar cell has a dominant one-dimensional active subspace, which enables us to perform thorough parameter studies in one dimension instead of five

Global sensitivity metrics from active subspaces

“…In contrast to the total sensitivity index τ i from (8) and the derivative-based measures ν i from (9), the β i 's are signed, so they indicate if f will increase or decrease given a positive change in x i . We can compute reference values for β i by noting the relationship between the m-dimensional Legendre series truncated to degree 1 (i.e., only a constant and m linear terms) and the least-squares linear approximation (12); the Legendre series is often referred to as the generalized polynomial chaos associated with the uniform density [21,29]. In particular, the degree-1 Legendre approximation is the best linear approximation in the L 2 norm.…”

“…Its coefficients admit a simple integral representation by virtue of the Legendre polynomials' orthogonality. Thus, we can estimate the coefficients with high order Gauss quadrature [30] and scale them by a quadrature-based estimate of the variance to get the coefficients of a linear monomial approximation as in (12).…”

Time‐dependent global sensitivity analysis with active subspaces for a lithium ion battery model

“…A component with a relatively large magnitude indicates that the corresponding parameter is important in defining the important subspace. Often in practice, the eigenvector components with large magnitudes correspond to parameters with relatively large standard sensitivity metrics, for example, Sobol' indices [5,7]. Moreover, if the functional relationship in the summary plot is monotonic-that is, f appears to be an increasing or decreasing function of w T x as assessed by the summary plot-then the sign of each eigenvector component reveals the direction in which f changes in response to changes in the corresponding parameter, on average.…”

Dimension reduction in magnetohydrodynamics power generation models: Dimensional analysis and active subspaces

“…wherex denotes the gradient with respect tox, andg is the gradient ofg with respect to its argument, A Tx . Plugging Equation 19 into Equation 8,…”