Detailed Description

Detailed model for a linear evolution problem with a Finite volume discretization.

This is a detailed model for a problem of the type

\begin{align} \partial_t u(x;t,\mu) - {\cal L}(t,\mu)[u(x,t;\mu)] &= 0 \qquad &\text{in }& \Omega \times [0, T], \\\ u(x;0,\mu) &= u_0(x;\mu) \qquad &\text{in }& \Omega \times \{0\} \end{align}

with suitable boundary conditions and linear operators \({\cal L}(t,\mu) : {\cal W} \to {\cal W}\) parametrized over the time-parameter domain \([0,T] \times {\cal M}\).

The numerical scheme for the numerical solution of this problem is assumed to be of the following type:

\begin{align*} u_h(\cdot;t^0,\mu) &= {\cal M}_h[u_0] \\\ \left(\text{Id} - \Delta t^k {\cal L}_{h, I}(t^{k+1},\mu)\right)[u_h(\cdot;t^{k+1},\mu)] &= \left(\text{Id} + \Delta t^k {\cal L}_{h, E}(t^{k},\mu)\right)[u_h(\cdot;t^{k},\mu)] \end{align*}

with the following ingredients:

a time discretization \(0 = t^0 \leq t^1 = t^0 + \Delta t^0 \leq \ldots \leq t^{K+1} = t^{K} + \Delta t^{K} = T\),
solution snapshots \(u_h(\cdot;t,\mu) \in { cal W }_h \) for all parameters and time instances \((t,\mu) \in [0,T] \times { \cal M }\)
separable operators for the discretization \({\cal L}_{h, I}(t,\mu):{\cal W}_h \to {\cal W}_h\), \({\cal L}_{h, E}(t,\mu):{\cal W}_h \to {\cal W}_h\) and separable initial data \(u_0\) in the parameter argument. That is these must be decomposable like
\begin{align*} {\cal L}_{h, *}(t,\mu) &= \sum_{q=1}^{q_*} \sigma_*^q(\mu) {\cal L}^q_{h, *}(t) \\\ u_0(x;\mu) &= \sum_{q=1}^{u_0} \sigma_{u_0}^q(\mu) u_0(x) \end{align*}
and
a projection operator \({\cal M}_h : {\cal W} \to {\cal W}_h\)

Definition at line 18 of file DetailedModel.m.

Inheritance diagram for LinEvol.DetailedModel:

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Public Member Functions
	DetailedModel (descr)
	constructor based on problem description More...

function sim_data =	detailed_simulation (ModelData model_data)
	executes a detailed simulation for a given parameter More...

function ModelData model_data =	gen_model_data ()
	method which produces all \(H\)-dependent data, that is required for a detailed simulations and is independent of the parameter \(\mu\). More...

function p =	plot_sim_data (ModelData model_data, sim_data, plot_params)
	plots the simulation data as returned by detailed_simulation() More...

Public Member Functions inherited from Greedy.User.FVDetailedModelDefault
	FVDetailedModelDefault (ModelDescr descr)
	Constructor for this class. More...

function UON =	orthonormalize (ModelData model_data, U)
	orthonormalizes a set of discrete functions \(\{ v_h^l \}_{l=1}^L\) More...

function U0 =	init_values_algorithm (ModelData model_data)
	computes the Dof vectors for the initial value function \(u_h^0(\cdot;t\mu) = {\cal P}_h[u_0(\mu)]\) More...

function UV =	inner_product (ModelData model_data, U, V)
	computes the inner project between the Dof vectors of two discrete functions More...

Public Member Functions inherited from Greedy.User.IDetailedModel
	IDetailedModel (descr)
	constructor More...

Public Member Functions inherited from IDetailedModel
	IDetailedModel (descr)
	constructor More...

function isequal =	eq (IDetailedModel other)
	overloaded equality (`==`) operator, by default comparing the descr variables More...

function IDetailedModel this =	set_mu (mu)
	sets the active parameter vector \(\mu \in {\cal M}\) More...

function mu =	get_mu ()
	returns the active parameter vector \(\boldsymbol\mu \in {\cal M}\) More...

virtual function sim_data =	detailed_simulation (model_data)
	executes a detailed simulation for a given parameter More...

virtual function p =	plot_sim_data (model_data, sim_data, plot_params)
	plots the simulation data as returned by detailed_simulation() More...

Static Public Member Functions
static function U =	get_dofs_from_sim_data (sim_data)
	extracts the \(H\) dimensional Dof vector from the `sim_data` structure More...

static function snapshot =	get_dofs_at_time (U, time_index)

Static Public Member Functions inherited from Greedy.User.FVDetailedModelDefault
static function errs =	l2_error_sequence_algorithm (U, Uapprox,ModelData model_data)
	computes the \(L^2(\Omega)\) error for each "snapshot" of a trajectories and returns the sequence of these errors. More...

static function errs =	linfty_error_sequence_algorithm (U, Uapprox,ModelData model_data)
	computes the \(L^{\infty}(\Omega)\) error for each "snapshot" of a trajectories and returns the sequence of these errors. More...

static function W =	get_inner_product_matrix (ModelData model_data)
	returns the inner product matrix for efficient computation of inner products on \({\cal W}_h\) More...

Static Public Member Functions inherited from IModel
static function ok =	struct_check (descr, checks)
	executes checks on the fields of a structure object More...

Static Public Attributes
static const	ddescr_checks
	This constant is used for a consistency check of the model descr with help of Greedy.Remove.IDetailedModelBase.struct_check() More...

Static Public Attributes inherited from IDetailedModel
static const	base_checks
	this structure holds variable names that need to be defined by the descr structure. More...

Static Public Attributes inherited from IModel
static const	time_checks
	This constant can be used for a consistency check of time evolution members in the ModelDescr with help of IModel.struct_check() More...

Additional Inherited Members
Public Attributes inherited from IDetailedModel
	descr
	The description structure holding information about the analytical parametrized problem and its discretization. More...

	decomp_mode

	mu_names
	cell array of strings describing the parameters of the model More...

	mu_ranges
	cell array of vectors of size two defining the allowed interval range for the parameter components More...

	verbose
	an integer defining the verbosity level of information output during basis generation More...

	debug
	an integer defining the debugging level controlling error output and extra tests during basis generation More...

Public Attributes inherited from IModel
	num_cpus = 4
	The number of CPUs used for parallel sessions. More...

	decomp_mode
	Decomposition operation mode. More...

	mu_names
	cell array of strings describing the parameters of the model More...

	mu_ranges
	cell array of vectors of size two defining the allowed interval range for the parameter components More...

	verbose
	an integer defining the verbosity level of information output during basis generation More...

	debug
	an integer defining the debugging level controlling error output and extra tests during basis generation More...

Constructor & Destructor Documentation

LinEvol.DetailedModel.DetailedModel ( descr )

constructor based on problem description

Parameters

descr structure describing the problem and the descretization

Definition at line 78 of file DetailedModel.m.

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Member Function Documentation

function sim_data = LinEvol.DetailedModel.detailed_simulation ( ModelData model_data )

executes a detailed simulation for a given parameter

This function computes a numerical scheme defined by the descr for the parameter set via the set_mu() method.

Parameters

model_data Matlab structure storing (possibly) high dimensional data needed by IDetailedModel.detailed_simulation().

Return values

sim_data structure holding the \(H\)-dimensional simulation data.

Generated fields of sim_data:

U — Dof vectors for the solution snapshots \(u_h(\cdot, t^{k_i})\) where \((k_i)_{i=1}^{\hat{K}}\) are the time_indices.
time_indices — This is a vector of stored time indices \((k_i)_{i=1}^{\hat{K}}\)
time — This is a vector of stored time instances \((t^{k_i})_{i=1}^{\hat{K}}\)
y — Vector of output functional values (if activated)
mu — mu

Definition at line 19 of file detailed_simulation.m.

function ModelData model_data = LinEvol.DetailedModel.gen_model_data ( )

virtual

method which produces all \(H\)-dependent data, that is required for a detailed simulations and is independent of the parameter \(\mu\).

Return values

model_data model data

Generated fields of model_data:

W — inner product weighting matrix
grid — a structure containing geometry information of a mesh used for discretizations

Documentation Update:

Implements IDetailedModel.

Definition at line 19 of file gen_model_data.m.

static function U = LinEvol.DetailedModel.get_dofs_from_sim_data ( sim_data )

staticvirtual

extracts the \(H\) dimensional Dof vector from the sim_data structure

Parameters

sim_data matlab struct with simulation data generated e.g. by detailed_simulation() or rb_reconstruction() methods.

Return values

U	\(H\) dimensional Dof vector

Implements Greedy.User.IDetailedModel.

Definition at line 102 of file DetailedModel.m.

function p = LinEvol.DetailedModel.plot_sim_data	(	ModelData	model_data,
			sim_data,
			plot_params
	)

plots the simulation data as returned by detailed_simulation()

Parameters

model_data	Matlab structure storing (possibly) high dimensional data needed by IDetailedModel.detailed_simulation().
sim_data	simulation data structure as returned by detailed_simulation()
plot_params	structure which controls the plot output

Return values

p	Matlab GUI handle to figure window

Required fields of sim_data:

U — DOF vector of solution snapshots
y — DOF vector of output function
s — s

Required fields of model_data:

grid — a structure containing geometry information of a mesh used for discretizations

Optional fields of plot_params:

plot_title — title of the plot (default = trajectory)
plot_output_title — title of the output functional plot (default = output)
axis_equal — flag equalizing the axes ratio

Definition at line 19 of file plot_sim_data.m.

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Member Data Documentation

LinEvol.DetailedModel.ddescr_checks

static

Initial value:

= struct("   \
      'mass_matrix',           {{\@(x) isequal(class(x), 'function_handle')}}   \
      ")

This constant is used for a consistency check of the model descr with help of Greedy.Remove.IDetailedModelBase.struct_check()

Default: struct(" \ 'mass_matrix', {{@(x) isequal(class(x), 'function_handle')}} \ ")

Definition at line 63 of file DetailedModel.m.

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

rbasis/problem_types/+LinEvol/@DetailedModel/DetailedModel.m
rbasis/problem_types/+LinEvol/@DetailedModel/detailed_simulation.m
rbasis/problem_types/+LinEvol/@DetailedModel/gen_model_data.m
rbasis/problem_types/+LinEvol/@DetailedModel/plot_sim_data.m

Detailed Description

Public Member Functions

Static Public Member Functions

Static Public Attributes

Additional Inherited Members

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation