active_subspace

Active (variable) subspace model

Specification

• Alias: subspace

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		truth_model_pointer	Pointer to specify a full-space model, from which to construct a lower dimensional surrogate
Optional		initial_samples	Initial number of samples for sampling-based methods
Optional		sample_type	Selection of sampling strategy
Optional		truncation_method	Metric that estimates active subspace size
Optional		dimension	Explicitly specify the desired subspace size
Optional		bootstrap_samples	Number of bootstrap replicates used in truncation metrics
Optional		build_surrogate	Construct moving least squares surrogate over active subspace
Optional		normalization	Normalize gradient samples

Description

A model that transforms the original model (given by truth_model_pointer) to one with a reduced set of variables. This reduced model is identified by iteratively sampling the gradient of the original model and performing a singular value decomposition of the gradient matrix.

Expected Output

A subspace model will perform an initial sampling design to identify an active subspace using one of the truncation methods.

Usage Tips

If the desired subspace size is not identified, consider using the explicit dimension truncation option or one of the other truncation methods.

Examples

Perform an initial 100 gradient samples and use the bing_li truncation method to identify an active subspace. The truncation method uses 150 bootstrap samples to compute the Bing Li truncation metric.

model
  subspace
    id_model = 'SUBSPACE'
    truth_model_pointer = 'FULLSPACE'
    initial_samples  100
    truncation_method bing_li
    bootstrap_samples 150

Theory

The idea behind active subspaces is to find directions in the input variable space in which the quantity of interest is nearly constant. After rotation of the input variables, this method can allow significant dimension reduction. Below is a brief summary of the process.

1. Compute the gradient of the quantity of interest, \(q = f(\mathbf{x})\) ,

   at several locations sampled from the full input space,

\[\nabla_{\mathbf{x}} f_i = \nabla f(\mathbf{x}_i).\]

2. Compute the eigendecomposition of the matrix \(\hat{\mathbf{C}}\) ,

\[\hat{\mathbf{C}} = \frac{1}{M}\sum_{i=1}^{M}\nabla_{\mathbf{x}} f_i\nabla_{\mathbf{x}} f_i^T = \hat{\mathbf{W}}\hat{\mathbf{\Lambda}}\hat{\mathbf{W}}^T,\]

where \(\hat{\mathbf{W}}\) has eigenvectors as columns,

\(\hat{\mathbf{\Lambda}} = \text{diag}(\hat{\lambda}_1,\:\ldots\:,\hat{\lambda}_N)\) contains eigenvalues, and \(N\) is the total number of parameters.

3. Using a truncation_method or specifying a

   dimension to estimate the active subspace size, split the eigenvectors into active and inactive directions,

\[\hat{\mathbf{W}} = \left[\hat{\mathbf{W}}_1\quad\hat{\mathbf{W}}_2\right].\]

These eigenvectors are used to rotate the input variables.

4. Next the input variables, \(\mathbf{x}\) , are expanded in terms of active and

   inactive variables,

\[\mathbf{x} = \hat{\mathbf{W}}_1\mathbf{y} + \hat{\mathbf{W}}_2\mathbf{z}.\]

5. A surrogate is then built as a function of the active variables,

\[g(\mathbf{y}) \approx f(\mathbf{x})\]

For additional information, see:

1. Constantine, Paul G. “Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies”. Vol. 2. SIAM, 2015.

2. Constantine, Paul G., Eric Dow, and Qiqi Wang. “Active subspace methods in theory and practice: Applications to kriging surfaces.” SIAM Journal on Scientific Computing 36.4 (2014): A1500-A1524.

3. Constantine, Paul, and David Gleich. “Computing Active Subspaces.” arXiv preprint arXiv:1408.0545 (2014).