adapted_basis

Basis adaptation model

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		truth_model_pointer	Pointer to specify a “truth” model, from which to construct a surrogate
Optional		truncation_tolerance	Convergence tolerance used to identify the (reduced) dimension of the rotation matrix

Description

A model that transforms the original model (given by truth_model_pointer) to one with a rotated set of variables. The current implementation does not support the reduction of the dimension for the new model and only transform the variable set in a rotated one in which the variables are arranged in a decreasing importance order.

An initial PCE representation is built with a sparse grid method and the PCE coefficients are then used to build a rotation matrix. This matrix is also reported as a result of the code execution. The new model can subsequently be used for an UQ workflow.

Expected Output A basis adaptation model will perform an initial sparse grid design to identify the rotation matrix.

Examples

Perform an initial sparse grid design (level 1) to evaluate the PCE expansion and evaluate the rotation matrix. Afterwards, 100 samples are generated for the model in the rotated space to obtain its statistics.

method,
  sampling
    model_pointer = 'SUBSPACE'
    samples = 100 seed = 1234567

model
  id_model = 'SUBSPACE'
  adapted_basis
    truth_model_pointer = 'FULLSPACE'
    sparse_grid_level = 1

Theory

The idea behind the Basis Adaptation method is to generate a PCE representation and rotate it such that the new set of rotated variables are organized in a decreasing importance ordering. Subsequently this rotation matrix can be truncated according to some criterion to only include the significant directions.

The first step of the Basis Adaptation is to compute a PCE expansion (we assume here \(\xi\) to have a standard multivariate Gaussian distribution)

\[Q(\xi) = \sum_{\alpha\in\mathcal{J}_{d,p}}Q_{\alpha}\psi_{\alpha}(\xi),\]

where \(\alpha = (\alpha_1,...,\alpha_d) \in \mathcal{J}_{d,p}:=(\mathbb{N}_0)^d\) with \(|\alpha| = \sum_{i=1}^{d} \alpha_i<= d\) is multi-index of dimension \(d\) and order up to \(p\) .

Afterwards, a multivariate Gaussian distribution \(\eta\) is sought such that

\[\eta = A \xi,\]

where \(A\) is an isometry such that \(A A^\mathrm{T} = I\) .

The basis adaptation model is obtained by expressing the original model with respect to the rotated set of variables \(\eta\) as

\[{Q}^{A}(\eta) = \sum_{\beta\in\mathcal{J}_{d,p}}Q_{\beta}^{A}\psi_{\beta}(\eta).\]

Since the basis for both the original and the adapted basis model span the same space, we know that \({Q}^{A}(\eta(\xi)) = Q(\xi)\) and therefore a relationships between the PCE coefficients exists

\[Q_{\alpha} = \sum_{\beta\in\mathcal{J}_{d,p}}Q_{\beta}^{A}\langle\psi_{\beta}^{A},\psi_{\alpha}\rangle, \ \alpha\in \mathcal{J}_{d,p}.\]

The linear adaptation strategy is used at this time in Dakota to obtain the rotation matrix \(A\) . The steps to obtain the matrix \(A\) are the following:

1. The first variable \(\eta_1\) is obtained by directly combining the first order terms, that is

\[\eta_1 = \sum_{ \alpha \in\mathcal{J}_{d,1}} Q_{ \alpha}\psi_{\alpha}(\xi) = \sum_{i=1}^{d}Q_{e_i} \xi_i,\]

where \(e_i\) is the \(d-\) dimensional multi-index with 1 at the \(i\) th location and zero elsewhere.

2. The remaining rows of the \(A\) matrix ( \(A \in \mathbb{R}^{d \times d}\) ) are obtained by using the largest coefficient, row after row, placed in the column that it occupies in the first row. All other terms are set to zero.

3. As last step, in order to obtain an isometry, the Gram-Schmidt procedure is followed on \(A.\)

See the theory manual for more details.