interface to internal representation of OMDc data More...
#include <DataStore.hpp>
Public Types | |
| typedef Eigen::Block< const Eigen::MatrixXd, Eigen::Dynamic, Eigen::Dynamic, true > | Block |
| short-hand for matrix block expression | |
| typedef Eigen::Block< const Eigen::Ref< const Eigen::MatrixXd >, Eigen::Dynamic, Eigen::Dynamic, true > | RefBlock |
| short-hand for referenced matrix block | |
Public Member Functions | |
| const RefBlock | X () const |
| read-access to 'left' snaps More... | |
| const RefBlock | Y () const |
| read-access to 'right' snaps More... | |
| const Block | LTX () const |
| read-access to projected input 'left' snaps More... | |
| const Block | LTY () const |
| read-access to projected output 'right' snaps More... | |
| DataStore (const Eigen::Ref< const Eigen::MatrixXd > S, const Eigen::Ref< const Eigen::MatrixXd > U, Eigen::Index r) | |
| standard contructor from snapshots and inputs More... | |
| void | updateSystemMatrices (const Eigen::Ref< const Eigen::MatrixXd > L) |
| recomputes internal representations for given modes More... | |
| const Eigen::MatrixXd | getSystemMatrix () const |
| system matrix of linear system from internal representation More... | |
| const Eigen::MatrixXd & | getProjector () const |
| get state projector More... | |
| const Eigen::MatrixXd | getInputMatrix () const |
| input matrix of linear system from internal representation More... | |
| const double | getResidual () const |
| residual of squared Fobenius norm || Y - LML'X - LPU || More... | |
| const Eigen::VectorXd | getInitialState () const |
| computes intial state from identified system More... | |
Public Attributes | |
| const Eigen::Index | m |
| number of snapshots | |
| const Eigen::Index | r |
| rank of reduced space | |
| const Eigen::Index | n |
| number of spatial locations (n > m) | |
| const Eigen::Index | p |
| number of inputs | |
Detailed Description
interface to internal representation of OMDc data
The original snapshot data is held by reference. This class actually stores only three projection matrices:
- the snapshots projected onto the lifting matrix
- the orthogonal projector of the inputs
- an additional orthogonal projector for the cost function
The projectors are obtained from QR decompositions, and thus, ensure numerical accuracy by avoiding summation errors.
- Note
- The matrices of the identified system are only computed on demand by respective methods.
Constructor & Destructor Documentation
◆ DataStore()
| OMDc::DataStore::DataStore | ( | const Eigen::Ref< const Eigen::MatrixXd > | S, |
| const Eigen::Ref< const Eigen::MatrixXd > | U, | ||
| Eigen::Index | r | ||
| ) |
standard contructor from snapshots and inputs
- Parameters
-
S snapshots [n x m] U inputs [p x (m-1)] r dimension of reduced system
Member Function Documentation
◆ getInitialState()
| const Eigen::VectorXd OMDc::DataStore::getInitialState | ( | ) | const |
computes intial state from identified system
The initial reduced state 
is computed as the solution to
![\[ \min_{a_0} \sum_{k=0}^{m-1} \Vert L^\top s_k - a_k \Vert_2^2 \quad \mathrm{s.t.} \,\, a_{k+1} = M a_k + P u_k \]](form_27.png)
The initial state is found such that the identified model evolves as closely as possible to the data set. Observe that the OMDc problem states the residual differently: it defines a stepwise error.
- Note
- The optimization problem can be solved by substituting the identified dynamics into the cost function, and then stacking block-wise powers of the system and input matrix. Then, the resulting problem in the initial state may be solved by the pseudo-inverse of the (large) stacked matrices. However, this implementation does not use stacked matrices but operates with small memory foot print directly on the state vectors.
- Returns
- initial reduced state as
Eigen::Vector
◆ getInputMatrix()
| const Eigen::MatrixXd OMDc::DataStore::getInputMatrix | ( | ) | const |
input matrix of linear system from internal representation
The input matrix is computed by
![\[ P = ( L^\top Y - M L^\top X ) \, U^\dagger \]](form_24.png)
using SVD for the pseudo-inverse
- Returns
- input matrix [r x p]
◆ getProjector()
| const Eigen::MatrixXd& OMDc::DataStore::getProjector | ( | ) | const |
get state projector
The projector is represented by 
which is defined as
![\[ ZW\,W^\top Z^\top = Z \big( I - Z^\top X^\top L (L^\top X Z \, Z^\top X^\top L)^{-1} L^\top X Z \big) Z^\top \qquad Z\,Z^\top = \big(I - U^\top (U U^\top)^{-1} U \big) \]](form_23.png)
- Returns
- read-only reference to internal projection matrix
◆ getResidual()
| const double OMDc::DataStore::getResidual | ( | ) | const |
residual of squared Fobenius norm || Y - LML'X - LPU ||
The cost function of OMDc is simplified to
![\[ \Vert Y -LML^\top X-LPU\Vert_F^2 = \Vert Y \Vert_F^2 - \Vert L^\top Y \Vert_F^2 + \Vert L^\top Y \,Z W \Vert_F^2 \]](form_25.png)
using the projector stored internally.
- Returns
- computed residual from current internal state
◆ getSystemMatrix()
| const Eigen::MatrixXd OMDc::DataStore::getSystemMatrix | ( | ) | const |
system matrix of linear system from internal representation
The system matrix is computed by
![\[ M = ( L^\top Y Z ) \, (L^\top X Z )^\dagger \]](form_21.png)
using SVD for the pseudo-inverse
- Returns
- system matrix [r x r]
◆ LTX()
|
inline |
read-access to projected input 'left' snaps
- Returns
Eigen::BlockL'X
◆ LTY()
|
inline |
read-access to projected output 'right' snaps
- Returns
Eigen::BlockL'Y
◆ updateSystemMatrices()
| void OMDc::DataStore::updateSystemMatrices | ( | const Eigen::Ref< const Eigen::MatrixXd > | L | ) |
recomputes internal representations for given modes
- Parameters
-
L projection matrix [n x r]
◆ X()
|
inline |
read-access to 'left' snaps
- Returns
Eigen::Blockto 'left' snaps
◆ Y()
|
inline |
read-access to 'right' snaps
- Returns
Eigen::Blockto 'right' snaps
The documentation for this class was generated from the following file:
- /home/lre/Documents/REPO/omdc-pub-repo/libs/OMDc/DataStore.hpp
