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rbmatlab
1.16.09
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rbmatlab
1.16.09
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IDetailedModel implementation for a non linear evolution problem.
This is a detailed model for a problem of this 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 (non-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
\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*}
\[ u_h(\cdot;t^0,\mu) = {\cal M}_h[u_0], \]
where the equation\[ F[u_h(\cdot;t^{k+1},\mu), u_h(\cdot;t^k,\mu)] =0 \]
is solved subsequently for the operator \(F:V_h\times V_h \to V_h\) defined by\[ F[u_h;v_h] := \left(\text{Id} + \Delta t^k {\cal L}_{h, I}(t^{k+1},\mu)\right)[u_h] - \left(\text{Id} + \Delta t^k {\cal L}_{h, E}(t^{k},\mu)\right)[v_h] \]
with a Newton scheme, minimizing defects \(\| \delta_h^{k+1,\nu+1}\| \leq R_{\text{Newton}}\) by solving\[ D_1 F|_{u_h^{k+1,\nu}}[\delta_h^{k+1,\nu+1};u_h^k] = -F[u_h^{k+1,\nu};u_h^k] \]
with \(u_h^{k+1,\nu+1} := u_h^{k+1,\nu}\), \(u_h^{k,0}:=u_h^k\), \(u_h^{k+1}:=u_h^{k+1, \nu_{\max}(k)}\) and \(\nu_{\max}(k) = \min_{\nu} \| \delta_h^{k+1,\nu} \| \leq R_{\text{Newton}}\)Furthermore the discretization consists of
Definition at line 18 of file DetailedModel.m.
Public Member Functions | |
| DetailedModel (ModelDescr descr) | |
| constructor 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... | |
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static function
snapshot = | get_dofs_at_time (sim_data, 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... | |
Public Attributes | |
| t | |
| current time instance More... | |
| tstep | |
| current time step More... | |
| compute_conditions = false | |
| boolean flag indicating whether conditions of system matrices shall be computed. More... | |
| 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... | |
Static Public Attributes | |
| static const | model_checks |
| This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check() More... | |
| static const | newton_checks |
| This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.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... | |
| NonlinEvol.DetailedModel.DetailedModel | ( | ModelDescr | descr | ) |
constructor
This checks and stores the model description structure
| descr | structure |
newton_solver — boolean flag indicating whether we are using a Newton scheme for the discretization. Definition at line 171 of file DetailedModel.m.
| function [ sim_data , fl ] = NonlinEvol.DetailedModel.detailed_simulation | ( | ModelData | model_data | ) |
executes a detailed simulation for a given parameter
| model_data | Matlab structure storing (possibly) high dimensional data needed by IDetailedModel.detailed_simulation(). |
| sim_data | structure holding the \(H\)-dimensional simulation data. |
| fl | debug output holding the flux evaluations returned by the L_E_local_ptr operator. |
T — final time nt — number of time-steps \(0 = t_0 \leq t_1 \leq \cdots \leq t^K+1 = T\), i.e. \(K+1\). init_values_algorithm — function pointer for computing the initvalues-DOF with arguments (model, model_data)L_E_local_ptr — function pointer to the local space-discretization operator evaluation with syntax *_local indicates results on a set of elements, *_local_ext includes information including their neighbours, i.e. the grid must be known also for the neighbours, the values of the previous timstep must be known on the neighbours. The subset of elements, which are to be computed are given in ind_local and the values produced are INC_local. A time-step can then be realized by data_const_in_time — if this optional field is 1, the time evolution is performed with constant operators, i.e. only the initial-time-operators are computed and used throughout the time simulation.U — matrix of size \(H \times K+1\) holding for each time step the solution dof vector. Unewton — matrix of size \(H \times K+1+\sum_{k=0}^K n_{\text{Newton}}(k)\) holding for each time step and each Newton iteration the intermediate solution dof vectors. conds.LI — conds.LI conds.gradLI — conds.gradLI Definition at line 19 of file detailed_simulation.m.
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virtual |
method which produces all \(H\)-dependent data, that is required for a detailed simulations and is independent of the parameter \(\mu\).
| model_data | model data |
W — inner product weighting matrix grid — a structure containing geometry information of a mesh used for discretizations Implements IDetailedModel.
Definition at line 19 of file gen_model_data.m.
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staticvirtual |
extracts the \(H\) dimensional Dof vector from the sim_data structure
| sim_data | matlab struct with simulation data generated e.g. by detailed_simulation() or rb_reconstruction() methods. |
| U | \(H\) dimensional Dof vector |
Implements Greedy.User.IDetailedModel.
Definition at line 204 of file DetailedModel.m.
| function p = NonlinEvol.DetailedModel.plot_sim_data | ( | ModelData | model_data, |
| sim_data, | |||
| plot_params | |||
| ) |
plots the simulation data as returned by detailed_simulation()
function plotting simulation results
| 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 |
| p | Matlab GUI handle to figure window |
U — DOF vector of solution snapshotsgrid — a structure containing geometry information of a mesh used for discretizationsbind_to_model — Sometimes the plot can adapted according to the parameters. For example if the domain of the simulation changes with the parameter. In this case this boolean flag can be set, and the copy_model_descr_to_plot_params() function is called to copy important parameters to the plot_params (default = false) plot — function pointer to a function plotting a single snapshot for a specific time instant. (default = descr.plot) Definition at line 19 of file plot_sim_data.m.
| NonlinEvol.DetailedModel.compute_conditions = false |
boolean flag indicating whether conditions of system matrices shall be computed.
Default: false
Definition at line 160 of file DetailedModel.m.
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static |
This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()
Default: struct(" \ 'fv_impl_diff_weight', {{@isscalar}}, \ 'fv_expl_diff_weight', {{@isscalar}}, \ 'fv_impl_conv_weight', {{@isscalar}}, \ 'fv_expl_conv_weight', {{@isscalar}}, \ 'newton_solver', {{@isscalar}}, \ 'implicit_nonlinear', {{@isscalar}}, \ 'L_E_local_ptr', {{@(x) isequal(class(x), 'function_handler')}}, \ 'L_I_local_ptr', {{@(x) isequal(class(x), 'function_handler')}}... ")
Definition at line 82 of file DetailedModel.m.
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static |
This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()
This check is only executed if we use a Newton scheme
Default: struct(" \ 'newton_steps', {{@isscalar}}, \ 'implicit_operators_algorithm', {{@(x) isequal(class(x), 'function_handler')}}, \ 'implicit_gradient_operators_algorithm', \ {{@(x) isequal(class(x), 'function_handler')}}, \ 'newton_epsilon', {{@isscalar}}... ")
Definition at line 110 of file DetailedModel.m.
| NonlinEvol.DetailedModel.t |
current time instance
Dependent set to true. Definition at line 136 of file DetailedModel.m.
| NonlinEvol.DetailedModel.tstep |
current time step
Dependent set to true. Definition at line 147 of file DetailedModel.m.
1.8.8