interface to internal representation of OMD data More...

#include <DataStore.hpp>

Inheritance diagram for OMD::DataStore:

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 left snaps More...

const Block	LTY () const
	read-access to projected right snaps More...

	DataStore (const Eigen::Ref< const Eigen::MatrixXd > S, Eigen::Index r)
	standard contructor from snapshots 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 M = (L'YX'L) inv(L'XX'L) More...

const double	getResidual () const
	squared Frobenius residual: \|\|Y\|\| - \|\|(L'YX'L) inv(L'XX'L) (L'XY'L)\|\| More...

const Eigen::MatrixXd &	getProjector () const
	projector onto orthog. space of X'L More...

const Eigen::MatrixXd &	getYstar () const
	part of Y'L that is orthogonal to X'L as R from QR 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)

Protected Attributes
const Eigen::Ref< const Eigen::MatrixXd >	S
	reference to snapshots (no data is stored)

Eigen::MatrixXd	LTS
	product of current mode and snapshots [r x m]

const double	normY
	Frobenius norm of left matrices.

Eigen::MatrixXd	W
	projector such that I - Z pinv(Z) = W W' with Z = X'L

Eigen::MatrixXd	XTL_qr_rep
	QR representation of X'L as [r x r] (upper triangular)

Eigen::MatrixXd	YTL_qr_rep_top
	QR representation of Y'L as [r x r] (top part)

Eigen::MatrixXd	YTL_qr_rep_bottom
	QR representation of Y'L as [r x r] (upper triangular)

Detailed Description

interface to internal representation of OMD data

Constructor & Destructor Documentation

◆ DataStore()

OMD::DataStore::DataStore	(	const Eigen::Ref< const Eigen::MatrixXd >	S,
		Eigen::Index	r
	)

standard contructor from snapshots

The internal data member are initialized to correct dimensions.

Parameters

S	snapshots
r	dimension of reduced system

Member Function Documentation

◆ getProjector()

const Eigen::MatrixXd& OMD::DataStore::getProjector ( ) const

inline

projector onto orthog. space of X'L

The projector is represented by which is defined as

$W\,W^\top = \big( I - X^\top L (L^\top X \, X^\top L)^{-1} L^\top X \big)$

This projection is applied to $Y^\top L$ , such that the internal representation renders the result to

$W\,W^\top\, Y^\top L = Q \begin{bmatrix} 0 \\ I \end{bmatrix} Y^\ast = \Pi \, Y^\ast$

where this method gives access to $\Pi \in \mathbb{R}^{(m-1)\times r}$ , see DataStore::updateSystemMatrices() for the definition of the QR decomposition.

Returns: ref. to matrix [(m-1) x r]

◆ getResidual()

const double OMD::DataStore::getResidual ( ) const

squared Frobenius residual: ||Y|| - ||(L'YX'L) inv(L'XX'L) (L'XY'L)||

The cost function of OMD is simplified to

$\Vert Y -LML^\top X \Vert_F^2 = \Vert Y \Vert_F^2 - \Vert L^\top Y \Vert_F^2 + \Vert Y^\ast \Vert_F^2$

This form may be further simplified to unite the second and third term. Though, the chosen forms makes it reusable for OMDc

Returns: computed residual from current internal state

◆ getSystemMatrix()

const Eigen::MatrixXd OMD::DataStore::getSystemMatrix ( ) const

system matrix of linear system M = (L'YX'L) inv(L'XX'L)

the system matrix computes

$M = ( L^\top Y ) \, (L^\top X )^\dagger$

using SVD for the pseudo-inverse. It is internally simplified to

$M^\top = R_X^\dagger \, R_Y$

by the QR decomposition explained in DataStore::updateSystemMatrices()

Returns: system matrix from internal rep. [r x r]

◆ getYstar()

const Eigen::MatrixXd& OMD::DataStore::getYstar ( ) const

inline

part of Y'L that is orthogonal to X'L as R from QR

Note: only essential part of QR decomposition without Q, to be used with DataStore::getProjector()

Returns: ref. to intertal data [r x r]

◆ LTX()

const Block OMD::DataStore::LTX ( ) const

inline

read-access to projected left snaps

Returns: Eigen::Block: L'X

◆ LTY()

const Block OMD::DataStore::LTY ( ) const

inline

read-access to projected right snaps

Returns: Eigen::Block L'Y

◆ updateSystemMatrices()

void OMD::DataStore::updateSystemMatrices ( const Eigen::Ref< const Eigen::MatrixXd > L )

recomputes internal representations for given modes

The internal representation of the data reads

$\begin{bmatrix} X^\top L & Y^\top L \end{bmatrix} = Q \, \begin{bmatrix} R_X & R_Y \\ 0 & Y^\ast \end{bmatrix}$

simplifying the computation of OMD::getResidual() and OMD::getSystemMatrix() . The results of the decomposition are stored in protected members.

Parameters

L	projection matrix

◆ X()

const RefBlock OMD::DataStore::X ( ) const

inline

read-access to left snaps

Returns: Eigen::Block to left snaps

◆ Y()

const RefBlock OMD::DataStore::Y ( ) const

inline

read-access to right snaps

Returns: Eigen::Block to right snaps

The documentation for this class was generated from the following file:

/Users/studien/Documents/REPO/ModelReduction/omdc/libs/inc/OMD/DataStore.hpp

Public Types

Public Member Functions

Public Attributes

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ DataStore()

Member Function Documentation

◆ getProjector()

◆ getResidual()

◆ getSystemMatrix()

◆ getYstar()

◆ LTX()

◆ LTY()

◆ updateSystemMatrices()

◆ X()

◆ Y()