DEIM: Tests regarding the DEIM method. More...

Detailed Description

DEIM: Tests regarding the DEIM method.

See Also: approx.DEIM

Author: Daniel Wirtz

Date: 2012-05-03

New in 0.6:: (Daniel Wirtz, 2012-05-03) Added this class.

This class is part of the framework

KerMor - Model Order Reduction using Kernels

Homepage http://www.morepas.org/software/index.html
Documentation http://www.morepas.org/software/kermor/index.html
License KerMor license conditions

Definition at line 18 of file DEIM.m.

Public Member Functions
	DEIM (sys, dim)

function fx =	evaluateCoreFun (unused1,double t)

Public Member Functions inherited from dscomponents.ACompEvalCoreFun
	ACompEvalCoreFun (sys)

function fx =	evaluateComponentSet (integer nr,colvec< double > x,double t)
	Computes the full or reduced component functions of the given point set. More...

function fx =	evaluateComponentSetMulti (integer nr,matrix< double > x,rowvec< double > t,matrix< double > mu)
	Computes the full component functions of the given point set. More...

function dfx =	evaluateComponentSetGradientsAt (integer nr,colvec< double > x,double t)
	Computes the full/reduced gradients of all component functions of the given point set. More...

function J =	evaluateJacobianSet (integer nr,colvec< double > x,double t)
	Returns the jacobian entries of the point set that have been specified using setPointSet's argument jpd. More...

function J =	evaluateJacobianSetMulti (integer nr,matrix< double > x,rowvec< double > t,colvec< double > mu)
	Returns the jacobian entries at multiple locations/times/parameters of the point set that have been specified using setPointSet's argument jpd. More...

function	setPointSet (nr, pts, jpd)
	Parameters: pts: A row vector with the desired points jpd: ("Jacobian Partial Derivatives") A cell array of size equal to the number of points. Each cell contains the indices for which the partial derivatives of the corresponding component function will be computed when calling evaluateJacobianSet. More...

function target =	project (V, W, target)

function copy =	clone (copy)
	The interface method with returns a copy of the current class instance. More...

function res =	test_ComponentEvalMatch (xsize)
	Tests if the local implementation of evaluateComponents matches the full evaluation. More...

Public Member Functions inherited from dscomponents.ACoreFun
	ACoreFun (sys)

function	setSystem (sys)

function target =	project (V, W, target)
	Sets the protected \(\vV,\vW\) matrices that can be utilized on core function evaluations after projection. More...

function fx =	evaluate (x, t)
	Evaluates the f-approximation. Depending on a possible projection and the CustomProjection-property the function either calls the inner evaluation directly which assumes \(f = f^r(z)\) or projects the reduced state variable z into the original space and evaluates the function there, so via \(f = V'f(Vz)\). More...

function fx =	evaluateMulti (colvec< double > x,double t,colvec< double > mu)
	Evaluates this function on multiple locations and maybe multiple times and parameters. More...

function	prepareSimulation (colvec< double > mu)
	A method that allows parameter-dependent computations to be performed before a simulation using this parameter starts. More...

function J =	getStateJacobian (x, t)
	Default implementation of jacobian matrix evaluation via finite differences. More...

function J =	getStateJacobianImpl (colvec< double > x,double t)
	Default implementation of state jacobians. uses finite differences. More...

function copy =	clone (copy)
	The interface method with returns a copy of the current class instance. More...

function [ matrix< double > J , dx ] =	getStateJacobianFD (x, t,rowvec< integer > partidx)
	Implementation of jacobian matrix evaluation via finite differences. More...

virtual function fx =	evaluateCoreFun (colvec< double > x,double t)
	Actual method used to evaluate the dynamical sytems' core function. More...

function res =	test_MultiArgEval (mudim)
	Convenience function that tests if a custom MultiArgumentEvaluation works as if called with single arguments. More...

function logical res =	test_Jacobian (matrix< double > xa,rowvec< double > ta,matrix< double > mua)
	Tests the custom provided jacobian matrix against the default finite difference computed one. More...

Public Member Functions inherited from KerMorObject
	KerMorObject ()
	Constructs a new KerMor object. More...

function	display ()
	disp(object2str(this)); More...

function bool =	eq (B)
	Checks equality of two KerMor objects. More...

function bool =	ne (B)
	Checks if two KerMorObjects are different. More...

function cn =	getClassName ()
	Returns the simple class name of this object without packages. More...

Public Member Functions inherited from DPCMObject
	DPCMObject ()
	Creates a new DPCM object. More...

	DPCMObject ()

Public Member Functions inherited from general.AProjectable
function handle target =	project (matrix< double > V,matrix< double > W,handle target)
	Returns a NEW INSTANCE of the projected object that does not rely on data of the old one via references (everything must be copied to ensure separability of reduced(=projected) versions and full versions, unless. More...

function copy =	clone (copy)
	The interface method with returns a copy of the current class instance. More...

Static Public Member Functions
static function res =	test_DEIMNoSpatialDependence ()

static function	analysis_DEIM_approx (m)

static function res =	computeDEIMErrors (general.DEIM deim,data.ApproxTrainData atd,rowvec< integer > orders,rowvec< integer > errorders)
	% DEIM approximation analysis over training data Computes the DEIM approximation error over the given training data for specified DEIM orders (M) and DEIM error orders (M'). More...

static function pm =	plotDEIMErrs (res, pm)

static function [ double t , nof , nor , o , pm ] =	jacobian_tests (m, pm)
	Tis function computes the jacobians of both the full and DEIM approximated system functions. More...

static function [ efull , ered , fxno ] =	getApproxErrorFullRed (r, xr,double t,colvec< double > mu, V)

static function [ etrue , EE , ED , pm ] =	getTrajApproxErrorDEIMEstimates (models.ReducedModel r,colvec< double > mu,integer inputidx)
	[etrue, EE, ED, pm] = getTrajApproxErrorDEIMEstimates(this, mu, inputidx) Computes the DEIM approximation errors for given mu and input. More...

static function pm =	getTrajApproxErrorDEIMEstimates_plots (r, etrue, EE, ED, pm)
	Visualizes the results of the ModelAnalyzer.getTrajApproxErrorDEIMEstimates method. More...

static function [ double t , values ] =	getMinReqErrorOrdersTable (errordata, relerrs, tsize, maxorder)
	Return values: t: A PrintTable containing the results. If not specified as a nargout argument, the table will be printed instead. More...

static function [ matrix < double > mui , fxi , afxi ] =	getDEIMErrorsAtXForParams (models.BaseFullModel m,colvec< double > x,integer numExtraSamples)
	% DEIM approximation analysis over parameters for specific state space location More...

static function pm =	getDEIMErrorsAtXForParams_plots (m,matrix< double > mui, fxi, afxi, pm)

static function [ errs , relerrs , times , rowvec < integer > deim_orders ] =	getDEIMReducedModelErrors (models.ReducedModel r,colvec< double > mu,integer inidx,rowvec< integer > deim_orders)
	% Model DEIM reduction quality assessment pics More...

static function pm =	getDEIMReducedModelErrors_plots (r, errs, relerrs, times, deim_orders, pm, tag)

static function [ e , aln , rowvec < integer > orders ] =	compareDEIM_Full_Jacobian (models.BaseFullModel m,data.ApproxTrainData atd,rowvec< integer > orders)
	% Matrix DEIM approximation analysis Compares the MDEIM approximation with the full jacobian More...

static function pm =	compareDEIM_Full_Jacobian_plots (m, e, aln, orders, pm, atdsubset)

static function pm =	effectivityAnalysis (models.ReducedModel r,colvec< double > mu,integer inputidx)
	% Effectivity analysis of error estimators Plots an effectivity graph for the current error estimator. More...

static function struct est =	getDEIMEstimators_MDEIM_ST (models.ReducedModel rmodel,struct est,rowvec< integer > jdorders,rowvec< integer > stsizes)
	% Error estimator struct compilation Computes a set of DEIMEstimators for given MDEIM orders and simtrans-sizes. More...

static function est =	getDEIMEstimators_ErrOrders (models.ReducedModel rmodel, est, errororders)

Protected Member Functions
function fx =	evaluateComponents (pts, ends, idx, self,colvec< double > x,double t)

function fx =	evaluateComponentsMulti (pts, ends, idx, self,colvec< double > x,double t,colvec< double > mu)

Protected Member Functions inherited from dscomponents.ACompEvalCoreFun
function matrix < double > dfx =	evaluateComponentGradientsAt (rowvec< integer > pts,rowvec< integer > ends,rowvec< integer > idx,rowvec< integer > self,colvec< double > x,double t)
	Default implementation of gradient computation via finite differences. More...

function dfx =	evaluateComponentPartialDerivatives (rowvec< integer > pts,rowvec< integer > ends,rowvec< integer > idx,rowvec< integer > deriv,rowvec< integer > self,colvec< double > x,double t, dfxsel)
	Computes specified partial derivatives of \(f\) of the components given by pts and the selected partial derivatives by dfxsel. More...

function dfx =	evaluateComponentPartialDerivativesMulti (pts, ends, idx, deriv, self,colvec< double > x,double t,colvec< double > mu, dfxsel)
	Multi-argument evaluation method for partial derivatives. Not used so far in KerMor, this is "legacy code" to keep around if needed at any stage as default finite difference-implementation. More...

function fx =	evaluateComponentsMulti (pts, ends, idx, self,colvec< double > x,double t,colvec< double > mu)

virtual function fx =	evaluateComponents (rowvec< integer > pts,rowvec< integer > ends,rowvec< integer > idx,rowvec< integer > self,matrix< double > x,rowvec< double > t,colvec< double > mu)
	This is the template method that actually evaluates the components at given points and values. More...

Protected Member Functions inherited from KerMorObject
function	checkType (obj, type)
	Object typechecker. More...

Protected Member Functions inherited from DPCMObject
function	registerProps (varargin)
	Call this method at any class that defines DPCM observed properties. More...

function	registerProps (varargin)

Additional Inherited Members
Public Attributes inherited from dscomponents.ACompEvalCoreFun
	PointSets

Public Attributes inherited from dscomponents.ACoreFun
logical	TimeDependent = true
	Flag that indicates if the ACoreFun is (truly) time-dependent. More...

	CustomProjection = false
	Set this property if the projection process is customized by overriding the default project method. More...

sparse< logical >	JSparsityPattern = "[]"
	Sparsity pattern for the jacobian matrix. More...

integer	xDim = "[]"
	The current state space dimension of the function's argument \(x\). More...

integer	fDim = "[]"
	The current output dimension of the function. More...

models.BaseFirstOrderSystem	System
	The system associated with the current ACoreFun. More...

colvec< double >	mu = "[]"
	The current model parameter mu for evaluations. Will not be persisted as only valid for runtime during simulations. More...

	Vcache

	Wcache

Public Attributes inherited from DPCMObject
	WorkspaceVariableName = ""
	The workspace variable name of this class. Optional. More...

	ID = "[]"
	An ID that allows to uniquely identify this DPCMObject (at least within the current MatLab session/context). More...

	PropertiesChanged = "[]"
	The Dictionary containing all the property settings as key/value pairs. More...

Public Attributes inherited from handle
	addlistener
	Creates a listener for the specified event and assigns a callback function to execute when the event occurs. More...

	notify
	Broadcast a notice that a specific event is occurring on a specified handle object or array of handle objects. More...

	delete
	Handle object destructor method that is called when the object's lifecycle ends. More...

	disp
	Handle object disp method which is called by the display method. See the MATLAB disp function. More...

	display
	Handle object display method called when MATLAB software interprets an expression returning a handle object that is not terminated by a semicolon. See the MATLAB display function. More...

	findobj
	Finds objects matching the specified conditions from the input array of handle objects. More...

	findprop
	Returns a meta.property objects associated with the specified property name. More...

	fields
	Returns a cell array of string containing the names of public properties. More...

	fieldnames
	Returns a cell array of string containing the names of public properties. See the MATLAB fieldnames function. More...

	isvalid
	Returns a logical array in which elements are true if the corresponding elements in the input array are valid handles. This method is Sealed so you cannot override it in a handle subclass. More...

	eq
	Relational functions example. See details for more information. More...

	transpose
	Transposes the elements of the handle object array. More...

	permute
	Rearranges the dimensions of the handle object array. See the MATLAB permute function. More...

	reshape
	hanges the dimensions of the handle object array to the specified dimensions. See the MATLAB reshape function. More...

	sort
	ort the handle objects in any array in ascending or descending order. More...

Public Attributes inherited from general.AProjectable
	V
	The \(V\) matrix of the biorthogonal pair \(V,W\). More...

	W
	The \(W\) matrix of the biorthogonal pair \(V,W\). More...

Static Protected Member Functions inherited from dscomponents.ACompEvalCoreFun
static function obj =	loadobj (obj, from)

Static Protected Member Functions inherited from dscomponents.ACoreFun
static function obj =	loadobj (obj, from)

Static Protected Member Functions inherited from DPCMObject
static function obj =	loadobj (obj, from)
	Re-register any registered change listeners! More...

static function obj =	loadobj (obj, from)

Static Protected Member Functions inherited from general.AProjectable
static function obj =	loadobj (obj, from)

Protected Attributes inherited from dscomponents.ACompEvalCoreFun
matrix< double >	S = {""}
	The x-component selection matrices (precomputed on setting PointSet/AltPointSet). �S� is passed to the function evaluating the components of �� or its derivatives. More...

Constructor & Destructor Documentation

testing.DEIM.DEIM	(	sys,
		dim
	)

Definition at line 47 of file DEIM.m.

References dscomponents.ACoreFun.fDim, dscomponents.ACoreFun.JSparsityPattern, and dscomponents.ACoreFun.xDim.

Member Function Documentation

static function testing.DEIM.analysis_DEIM_approx ( m )

static

Definition at line 112 of file DEIM.m.

References dscomponents.ACoreFun.mu, and t.

function [ e , aln , rowvec< integer > orders ] = testing.DEIM.compareDEIM_Full_Jacobian	(	models.BaseFullModel	m,
		data.ApproxTrainData	atd,
		rowvec< integer >	orders
	)

static

% Matrix DEIM approximation analysis Compares the MDEIM approximation with the full jacobian

Parameters

m	The full model
atd	The approximation training data
orders	The different orders \(m_J\) to try

Required fields of m:

Required fields of atd:

Definition at line 780 of file DEIM.m.

References models.BaseFullModel.Data, models.BaseFullModel.ErrorEstimator, models.BaseFirstOrderSystem.f, dscomponents.ACoreFun.getStateJacobian(), data.ModelData.JacobianTrainData, dscomponents.ACoreFun.JSparsityPattern, Norm.L2(), Utils.logNorm(), handle.reshape, models.BaseModel.System, and t.

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static function pm = testing.DEIM.compareDEIM_Full_Jacobian_plots	(	m,
		e,
		aln,
		orders,
		pm,
		atdsubset
	)

static

Definition at line 836 of file DEIM.m.

References LogPlot.logsurf(), handle.sort, X, and Y.

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function res = testing.DEIM.computeDEIMErrors	(	general.DEIM	deim,
		data.ApproxTrainData	atd,
		rowvec< integer >	orders,
		rowvec< integer >	errorders
	)

static

% DEIM approximation analysis over training data Computes the DEIM approximation error over the given training data for specified DEIM orders (M) and DEIM error orders (M').

The error over any training data point is measured L2 in state and Linf over all training points.

1: order 2: errororder 3: max abs true error (indep. of errororder) 4: max rel true error w.r.t true fxi norm (indep. of errororder) 5: mean abs true error (indep. of errororder) 6: mean rel true error w.r.t true fxi norm (indep. of errororder) 7: max abs estimated error 8: max rel estimated error w.r.t true fxi norm 9: mean abs estimated error 10: mean rel estimated error w.r.t true fxi norm 11: max rel error of (estimated-true error) w.r.t true error 12: mean rel error of (estimated-true error) w.r.t true error 13: max rel error of (estimated-maxorder error) w.r.t maxorder error 14: mean rel error of (estimated-maxorder error) w.r.t maxorder error

Parameters

deim	The DEIM instance
atd	The approximation training data
orders	The different orders \(1 \leq m\leq M\) to try
errorders	The different error orders \(1 \leq m'\leq M-m\) to try

Return values

res	A 12 x totalcombinations matrix containing the values in rows

Required fields of deim:

Required fields of atd:

Definition at line 146 of file DEIM.m.

References general.DEIM.evaluate(), data.ApproxTrainData.fxi, data.FileMatrix.getBlockPos(), general.DEIM.getEstimatedError(), k, Norm.L2(), data.FileMatrix.m, general.DEIM.MaxOrder, data.ApproxTrainData.mui, data.FileMatrix.nBlocks, general.DEIM.Order, t, data.ApproxTrainData.ti, general.AProjectable.V, and data.ApproxTrainData.xi.

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function pm = testing.DEIM.effectivityAnalysis	(	models.ReducedModel	r,
		colvec< double >	mu,
		integer	inputidx
	)

static

% Effectivity analysis of error estimators Plots an effectivity graph for the current error estimator.

Parameters

r	The reduced model
mu	The currently used parameter \(\vmu\). Set to \([]\) if not used.
inputidx	The index \(i\) of the currently used input function \(u_i(t)\). Set to \([]\) if not used.

Required fields of r:

Generated fields of pm:

Definition at line 889 of file DEIM.m.

References models.ReducedModel.ErrorEstimator, models.ReducedModel.FullModel, Norm.L2(), error.BaseEstimator.OutputError, models.BaseModel.simulate(), and t.

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function fx = testing.DEIM.evaluateComponents	(		pts,
			ends,
			idx,
			self,
		colvec< double >	x,
		double	t
	)

protected

Definition at line 67 of file DEIM.m.

References evaluateComponentsMulti(), and dscomponents.ACoreFun.mu.

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function fx = testing.DEIM.evaluateComponentsMulti	(		pts,
			ends,
			idx,
			self,
		colvec< double >	x,
		double	t,
		colvec< double >	mu
	)

protected

Definition at line 72 of file DEIM.m.

References dscomponents.ACoreFun.mu.

Referenced by evaluateComponents().

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function fx = testing.DEIM.evaluateCoreFun	(		unused1,
		double	t
	)

Definition at line 60 of file DEIM.m.

References dscomponents.ACoreFun.fDim, and dscomponents.ACoreFun.mu.

static function [efull , ered , fxno ] = testing.DEIM.getApproxErrorFullRed	(		r,
			xr,
		double	t,
		colvec< double >	mu,
			V
	)

static

Definition at line 446 of file DEIM.m.

References Norm.L2().

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function [ matrix< double > mui , fxi , afxi ] = testing.DEIM.getDEIMErrorsAtXForParams	(	models.BaseFullModel	m,
		colvec< double >	x,
		integer	numExtraSamples
	)

static

% DEIM approximation analysis over parameters for specific state space location

Only t=0 is used

Parameters

m	The full model
x	The state space location \(\vx\)
numExtraSamples	The number of extra samples to use. Default: 0

Required fields of m:

Definition at line 647 of file DEIM.m.

References models.BaseFullModel.Approx, models.BaseFullModel.Data, dscomponents.ACoreFun.evaluate(), models.BaseFirstOrderSystem.f, data.ModelData.ParamSamples, and models.BaseModel.System.

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static function pm = testing.DEIM.getDEIMErrorsAtXForParams_plots	(		m,
		matrix< double >	mui,
			fxi,
			afxi,
			pm
	)

static

Definition at line 676 of file DEIM.m.

References Norm.L2(), PlotManager.LeaveOpen, and LogPlot.logtrisurf().

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static function est = testing.DEIM.getDEIMEstimators_ErrOrders	(	models.ReducedModel	rmodel,
			est,
			errororders
	)

static

Definition at line 965 of file DEIM.m.

References error.BaseEstimator.clone(), and models.ReducedModel.ErrorEstimator.

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function struct est = testing.DEIM.getDEIMEstimators_MDEIM_ST	(	models.ReducedModel	rmodel,
		struct	est,
		rowvec< integer >	jdorders,
		rowvec< integer >	stsizes
	)

static

% Error estimator struct compilation Computes a set of DEIMEstimators for given MDEIM orders and simtrans-sizes.

Parameters

rmodel	The reduced model
est	The estimator settings struct
jdorders	The MDEIM orders \(1\leq m_J \leq M_J\) to use
stsizes	The similarity transformation sizes \(1\leq k \leq d\) to use

Return values

est	An estimator struct usable by the EstimatorAnalyzer

Required fields of rmodel:

Definition at line 917 of file DEIM.m.

References error.BaseEstimator.clone(), models.ReducedModel.ErrorEstimator, and k.

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function [ errs , relerrs , times , rowvec< integer > deim_orders ] = testing.DEIM.getDEIMReducedModelErrors	(	models.ReducedModel	r,
		colvec< double >	mu,
		integer	inidx,
		rowvec< integer >	deim_orders
	)

static

% Model DEIM reduction quality assessment pics

Parameters

r	The reduced model
mu	The current parameter \(\vmu\)
inidx	The input index \(i\) for the input function \(\vu_i(t)\)
deim_orders	The DEIM orders \(1 \leq m \leq M\) to use

Required fields of r:

Definition at line 706 of file DEIM.m.

References models.ReducedModel.ErrorEstimator, models.BaseFirstOrderSystem.f, models.ReducedModel.FullModel, Norm.L2(), error.BaseEstimator.OutputError, models.BaseModel.simulate(), models.BaseModel.System, and models.BaseModel.Times.

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static function pm = testing.DEIM.getDEIMReducedModelErrors_plots	(	r,
		errs,
		relerrs,
		times,
		deim_orders,
		pm,
		tag
	)

static

Definition at line 744 of file DEIM.m.

References LogPlot.logsurf(), X, and Y.

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function [ double t , values ] = testing.DEIM.getMinReqErrorOrdersTable	(	errordata,
		relerrs,
		tsize,
		maxorder
	)

static

Return values: t: A PrintTable containing the results. If not specified as a nargout argument, the table will be printed instead.

values: a value matrix with rows for orders and columns for relerrs

Generated fields of t:

Definition at line 587 of file DEIM.m.

References t.

function [ etrue , EE , ED , pm ] = testing.DEIM.getTrajApproxErrorDEIMEstimates	(	models.ReducedModel	r,
		colvec< double >	mu,
		integer	inputidx
	)

static

[etrue, EE, ED, pm] = getTrajApproxErrorDEIMEstimates(this, mu, inputidx) Computes the DEIM approximation errors for given mu and input.

Depending on the current Order \(o\) of the DEIM approximation, all possible ErrorOrders \(eo = o+1 ... N\) to the approx.DEIM.MaxOrder \(N\) are used estimations for the given trajectory computed.

Parameters

r	The reduced model
mu	The currently used parameter \(\vmu\). Set to \([]\) if not used.
inputidx	The index \(i\) of the currently used input function \(u_i(t)\). Set to \([]\) if not used.

Return values

etrue	The true approximation error using the full trajectory.
EE	The matrix of estimated errors, containing the estimated state-space L2 error (columns) for each possible ErrorOrder (rows). time step.
ED	The absolute difference between true and estimated error for each ErrorOrder (rows) and timestep (columns)
pm	The PlotManager used to create the result plots. If not requested as output, no plotting will be done.