### Install Dependencies

Source: https://github.com/aurion-polito/pypolydim/blob/main/README.md

Installs all required Python packages listed in the 'requirements.txt' file. This command should be run after activating the virtual environment.

```bash
pip install -r requirements.txt
```

--------------------------------

### Activate Python Virtual Environment

Source: https://github.com/aurion-polito/pypolydim/blob/main/README.md

Activates the previously created Python virtual environment. This ensures that subsequent commands use the environment's isolated Python installation and packages.

```bash
source .venv/bin/activate
```

--------------------------------

### Initialize Mesh and DOFs Data Structures

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Loads mesh connectivity data, creates reference elements for trial and test spaces, and initializes a DOFs manager. It then sets up mesh DOFs information and creates DOFs data for both trial and test spaces.

```python
mesh_connectivity_data = polydim.pde_tools.mesh.MeshMatricesDAO_mesh_connectivity_data(mesh)

trial_reference_element_data = polydim.pde_tools.local_space_pcc_2_d.create_reference_element(method_type, method_order)
test_reference_element_data = polydim.pde_tools.local_space_pcc_2_d.create_reference_element(method_type, method_order)

dof_manager = polydim.pde_tools.do_fs.DOFsManager()

trial_mesh_dofs_info = polydim.pde_tools.local_space_pcc_2_d.set_mesh_do_fs_info(trial_reference_element_data, mesh, boundary_info)
trial_dofs_data = dof_manager.create_do_fs_2_d(trial_mesh_dofs_info, mesh_connectivity_data)
test_mesh_dofs_info = polydim.pde_tools.local_space_pcc_2_d.set_mesh_do_fs_info(trial_reference_element_data, mesh, boundary_info)
test_dofs_data = dof_manager.create_do_fs_2_d(test_mesh_dofs_info, mesh_connectivity_data)
```

--------------------------------

### Create Snapshot Matrix

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Creates a snapshot matrix by solving a system for each parameter set in the training data. Uses sparse linear system solver.

```python
#### snapshot matrix creation
thetaA1 = 1.
snapshot_matrix = []

tol = 1. - 1e-7  
N_max = 10

for mu in training_set:
    thetaA2 = mu[0]
    thetaf1 = mu[1]
    
    A = thetaA1 * A_1 + thetaA2 * A_2
    f = thetaf1 * weak_term

    snapshot = scipy.sparse.linalg.spsolve(A, f)
    snapshot_matrix.append(np.copy(snapshot))

snapshot_matrix = np.array(snapshot_matrix) 

print(snapshot_matrix.shape)
```

--------------------------------

### Setting Up Export Folders and Mesh Parameters

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Configures the export paths for simulation results and mesh files, creating directories if they do not exist. Defines mesh type, method type, import folder, and method order for mesh processing.

```python
# Export folder
export_file_path = "./Export/ROM_POD"
if not os.path.exists(export_file_path):
    os.makedirs(export_file_path)

# Mesh file path
export_mesh_path = export_file_path + "/Mesh"
if not os.path.exists(export_mesh_path):
    os.makedirs(export_mesh_path)

# Solution file path
export_solution_path = export_file_path + "/Solution"
if not os.path.exists(export_solution_path):
    os.makedirs(export_solution_path)

mesh_type = polydim.pde_tools.mesh.pde_mesh_utilities.MeshGenerator_Types_2D.triangular_simple_importer
method_type = polydim.pde_tools.local_space_pcc_2_d.MethodTypes.fem_pcc
import_mesh_folder = "./ROM_POD/Mesh3"
method_order = 1
```

--------------------------------

### Initializing Geometry and Mesh Utilities

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Initializes configuration and utility objects for geometry and mesh processing. Sets tolerances for 2D geometry calculations.

```python
geometry_utilities_config = gedim.GeometryUtilitiesConfig()
geometry_utilities_config.tolerance1_d = 1.0e-6
geometry_utilities_config.tolerance2_d = 1.0e-12
geometry_utilities = gedim.GeometryUtilities(geometry_utilities_config)
mesh_utilities = gedim.MeshUtilities()
vtk_utilities = ExportVTKUtilities()
```

--------------------------------

### Assembling Reduced Linear System (Offline Phase)

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Assembles the reduced matrices A1, A2, and the reduced vector 'weak_term' for the offline phase of the ROM. These are used to form the reduced linear system.

```python
########## ASSEMBLE THE LINEAR SYSTEM ##### STILL OFFLINE
reduced_A1 = np.transpose(basis_functions) @ A_1 @ basis_functions
reduced_A2 = np.transpose(basis_functions) @ A_2 @ basis_functions
reduced_f =  np.transpose(basis_functions) @ weak_term

### shape? ##### 
```

--------------------------------

### Importing Libraries for POD

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Imports necessary libraries for performing Proper Orthogonal Decomposition, including numerical computation, argument parsing, file system operations, sparse matrices, VTK, and PyPolyDim modules.

```python
import numpy as np
import argparse
import os.path
import scipy.sparse
import vtk
from pypolydim import polydim, gedim
from pypolydim.export_vtk_utilities import ExportVTKUtilities
from pypolydim.assembler_utilities import assembler_utilities

import sys
sys.path.insert(1, './ROM_POD')
import ROM_POD.other_utilities as other_ut
```

--------------------------------

### Solving Reduced Linear System (Online Phase)

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Solves the reduced linear system for a new parameter in the online phase. It combines the pre-assembled reduced matrices and vectors based on parameter values and then solves for the reduced solution.

```python
thetaA2 = 2.
thetaf1 = 0.8

reduced_rhs = thetaA1*reduced_A1 + thetaA2*reduced_A2
reduced_lhs = thetaf1*reduced_f

##### solve ######### 

reduced_solution = np.linalg.solve(reduced_rhs, reduced_lhs)
proj_reduced_solution = basis_functions @ reduced_solution
```

--------------------------------

### Constructing Basis Functions for ROM

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Constructs a set of basis functions using the eigenvectors obtained from the eigenvalue decomposition. Each basis function is normalized using an inner product matrix.

```python
# Create the basis function matrix
basis_functions = []
for n in range(N):
  eigenvector =  eigenvectors[n]
  
  # basis = (1/np.sqrt(snapshot_num))*np.transpose(snapshot_matrix)@eigenvector  (This is the one of the book!!)
  
  basis = np.transpose(snapshot_matrix)@eigenvector
  norm = np.sqrt(np.transpose(basis) @ inner_product @ basis) 
  
  basis /= norm
  basis_functions.append(np.copy(basis))

basis_functions = np.transpose(np.array(basis_functions))
```

--------------------------------

### Define Training Set for Parameters

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Generates a random uniform training set for model parameters. Sets the number of snapshots and parameter ranges.

```python
### define the training set

snapshot_num = 100 # (M)
mu1_range = [0.1, 10.]
mu2_range = [-1., 1.]
P = np.array([mu1_range, mu2_range])

training_set = np.random.uniform(low=P[:, 0], high=P[:, 1], size=(snapshot_num, P.shape[0]))
```

--------------------------------

### Assemble Reaction Operator

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Assembles the reaction operator for a 2D problem. Requires geometry, mesh, and dofs data.

```python
def inner_react(x, y, z):
    return 1.0

A_l_2 = polydim.pde_tools.assembler_utilities.pcc_2_d.assemble_reaction_operator(geometry_utilities,
                                                                                      mesh,
                                                                                      mesh_geometric_data,
                                                                                      trial_dofs_data,
                                                                                      test_dofs_data,
                                                                                      trial_reference_element_data,
                                                                                      test_reference_element_data,
                                                                                      inner_react)
A_L2 = other_ut.make_np_sparse(A_l_2.operator_dofs)
```

--------------------------------

### Create Python Virtual Environment

Source: https://github.com/aurion-polito/pypolydim/blob/main/README.md

Creates a new Python virtual environment named '.venv'. This is a standard practice for managing project dependencies.

```bash
python3 -m venv .venv
```

--------------------------------

### Solving Full-Order Model (FOM)

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Solves the full-order model's linear system using sparse matrix solver from SciPy. This is used for comparison with the ROM solution.

```python
A = thetaA1*A_1 + thetaA2*A_2
f = thetaf1*weak_term

full_solution = scipy.sparse.linalg.spsolve(A, f)
```

--------------------------------

### Exporting and Plotting Full-Order Solution

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Extracts the full-order solution onto the cell0Ds of the mesh and exports it to a VTK file. It also plots the full-order solution for comparison.

```python
proj_full_on_cell0Ds = polydim.pde_tools.assembler_utilities.pcc_2_d.extract_solution_on_cell0_ds(mesh,
                                                                                          trial_dofs_data,
                                                                                          full_solution,
                                                                                          u_D)

vtk_utilities.export_solution_2(export_solution_path + '/u_full',
                                mesh,
                                proj_full_on_cell0Ds.numeric_solution)
other_ut.plot_solution(mesh, proj_full_on_cell0Ds.numeric_solution)
```

--------------------------------

### Assemble Strong Solution

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Assembles the strong form of the solution for a 2D PDE. Used to define boundary conditions or specific solution values.

```python
def strong_solution_function(marker, x, y, z):  
    return 0.0

u_D = polydim.pde_tools.assembler_utilities.pcc_2_d.assemble_strong_solution(geometry_utilities,
                                                                       mesh,
                                                                       mesh_geometric_data,
                                                                       trial_mesh_dofs_info,
                                                                       trial_dofs_data,
                                                                       trial_reference_element_data,
                                                                       strong_solution_function)
```

--------------------------------

### Compute Error and Speedup Analysis

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

This snippet iterates through a testing set to compute the full-order model (FOM) and reduced-order model (ROM) solutions. It calculates the time taken for each, the speedup achieved by the ROM, and the absolute and relative errors between the solutions. Requires numpy, scipy.sparse.linalg, and time modules.

```python
import time

abs_err = []
rel_err = []
testing_set = np.random.uniform(low=P[:, 0], high=P[:, 1], size=(100, P.shape[0]))
speed_up = []

print("Computing error and speedup analysis")

for mu in testing_set:
    
    thetaA2 = mu[0]
    thetaf1 = mu[1]
        
    ##### full #####
    A = thetaA1*A_1 + thetaA2*A_2
    f = thetaf1*weak_term
  
    start_fom = time.time()
    full_solution = scipy.sparse.linalg.spsolve(A, f)
    time_fom = time.time() - start_fom
    
    #### reduced #####
    
    reduced_rhs = thetaA1*reduced_A1 + thetaA2*reduced_A2
    reduced_lhs = thetaf1*reduced_f
    
    start_rom = time.time()
    reduced_solution = np.linalg.solve(reduced_rhs, reduced_lhs)
    time_rom = time.time() - start_rom
    
    speed_up.append(time_fom/time_rom)
    
    proj_reduced_solution = basis_functions@reduced_solution
    
    ### computing error
    
    error_function = full_solution - proj_reduced_solution
    error_norm_squared_component = np.transpose(error_function) @ inner_product @ error_function
    absolute_error = np.sqrt(abs(error_norm_squared_component))
    abs_err.append(absolute_error)
    
    full_solution_norm_squared_component = np.transpose(full_solution) @  inner_product @ full_solution
    relative_error = absolute_error/np.sqrt(abs(full_solution_norm_squared_component))
    rel_err.append(relative_error)

print("average relative error = ", np.mean(rel_err) )
print("average absolute error = ", np.mean(abs_err) )
print("average speed_up = ", np.mean(speed_up) )
```

--------------------------------

### Exporting and Plotting Reduced Solution

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Extracts the reduced solution onto the cell0Ds of the mesh and exports it to a VTK file. It also plots the projected reduced solution.

```python
proj_reduced_on_cell0Ds = polydim.pde_tools.assembler_utilities.pcc_2_d.extract_solution_on_cell0_ds(mesh,
                                                                                          trial_dofs_data,
                                                                                          proj_reduced_solution,
                                                                                          u_D)

vtk_utilities.export_solution_2(export_solution_path + '/u_reduced',
                                mesh,
                                proj_reduced_on_cell0Ds.numeric_solution)
other_ut.plot_solution(mesh, proj_reduced_on_cell0Ds.numeric_solution)
```

--------------------------------

### Loading and Processing Mesh Data

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Initializes mesh data structures and imports a 2D mesh from a specified folder. Computes geometric data associated with the mesh.

```python
mesh_data = gedim.MeshMatrices()
mesh = gedim.MeshMatricesDAO(mesh_data)

polydim.pde_tools.mesh.pde_mesh_utilities.import_mesh_2_d(geometry_utilities,
                                                          mesh_utilities,
                                                          mesh_type,
                                                          import_mesh_folder,
                                                          mesh)
mesh_geometric_data = polydim.pde_tools.mesh.pde_mesh_utilities.compute_mesh_2_d_geometry_data(geometry_utilities, mesh_utilities, mesh)
```

--------------------------------

### Assemble Diffusion Operator for Exterior Domain

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Defines a radial function 'omega_2' to represent the exterior of a circular domain and assembles the diffusion operator for this region. The resulting operator matrices are converted to NumPy sparse format.

```python
def omega_2(x, y, z):
    if (x * x + y * y) <= (R * R + 1.0e-13):
        return 0.0
    return 1.0

A_omega_2 = polydim.pde_tools.assembler_utilities.pcc_2_d.assemble_diffusion_operator(geometry_utilities,
                                                                                      mesh,
                                                                                      mesh_geometric_data,
                                                                                      trial_dofs_data,
                                                                                      test_dofs_data,
                                                                                      trial_reference_element_data,
                                                                                      test_reference_element_data,
                                                                                      omega_2)
A_2 = other_ut.make_np_sparse(A_omega_2.operator_dofs)
A_2_D = other_ut.make_np_sparse(A_omega_2.operator_strong)
```

--------------------------------

### Assemble Weak Term

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Assembles a weak term for a 2D PDE. The function handles different markers to define the term's behavior.

```python
def weak_term_function(marker, x, y, z):
    match marker:
        case 1:
            return 1.0
        case _:
            raise ValueError("not valid marker", marker)

weak_term = polydim.pde_tools.assembler_utilities.pcc_2_d.assemble_weak_term(geometry_utilities,
                                                                             mesh,
                                                                             mesh_geometric_data,
                                                                             trial_mesh_dofs_info,
                                                                             test_dofs_data,
                                                                             trial_reference_element_data,
                                                                             test_reference_element_data,
                                                                             weak_term_function)
```

--------------------------------

### Define Boundary Information for Mesh DOFs

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Sets up boundary information objects for different boundary types (internal, Dirichlet, Neumann) and assigns markers. This is used to configure the DOFs manager for mesh processing.

```python
info_internal = polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo(polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo.BoundaryTypes.none)
info_internal.marker = 0

info_dirichlet_up = polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo(polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo.BoundaryTypes.strong)
info_dirichlet_up.marker = 3

info_neumann_down = polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo(polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo.BoundaryTypes.weak)
info_neumann_down.marker = 1

info_neumann_none = polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo(polydim.pde_tools.do_fs.DOFsManager.MeshDOFsInfo.BoundaryInfo.BoundaryTypes.none)
info_neumann_none.marker = 2

boundary_info = {
    0: info_internal,
    1: info_neumann_down,
    2: info_neumann_none,
    3: info_dirichlet_up
}
```

--------------------------------

### Exporting and Plotting Mesh

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Exports the processed mesh data to a VTK file and generates a plot of the mesh using utility functions.

```python
vtk_utilities.export_mesh(export_mesh_path, mesh)
other_ut.plot_mesh(mesh)
```

--------------------------------

### Assemble Diffusion Operator for a Circular Domain

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Defines a radial function 'omega_1' to represent a circular domain and assembles the diffusion operator for this domain. The resulting operator matrices are converted to NumPy sparse format.

```python
R = 0.5

def omega_1(x, y, z):
    if (x * x + y * y) <= (R * R + 1.0e-13):
        return 1.0
    return 0.0

A_omega_1 = polydim.pde_tools.assembler_utilities.pcc_2_d.assemble_diffusion_operator(geometry_utilities,
                                                                                      mesh,
                                                                                      mesh_geometric_data,
                                                                                      trial_dofs_data,
                                                                                      test_dofs_data,
                                                                                      trial_reference_element_data,
                                                                                      test_reference_element_data,
                                                                                      omega_1)
A_1 = other_ut.make_np_sparse(A_omega_1.operator_dofs)
A_1_D = other_ut.make_np_sparse(A_omega_1.operator_strong)
```

--------------------------------

### Eigenvalue Decomposition and Dimension Reduction

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Performs eigenvalue decomposition on a matrix C to obtain eigenvalues and eigenvectors. It then determines the reduced dimension N based on a tolerance 'tol' for retained energy.

```python
# ALTERNATIVE:  VM, L, VMt = np.linalg.svd((C))

L_e, VM_e = np.linalg.eig(C)
eigenvalues = []
eigenvectors = []


#### check


for i in range(len(L_e)):
  eig_real = L_e[i].real
  eig_complex = L_e[i].imag
  assert np.isclose(eig_complex, 0.)
  eigenvalues.append(eig_real)
  eigenvectors.append(VM_e[i].real)


total_energy = sum(eigenvalues)
retained_energy_vector = np.cumsum(eigenvalues)
relative_retained_energy = retained_energy_vector/total_energy


if all(flag==False for flag in relative_retained_energy>= tol):
  N = N_max
else:
  N = np.argmax(relative_retained_energy >= tol) + 1

print("The reduced dimension is", N)
```

--------------------------------

### Calculate Covariance Matrix for POD

Source: https://github.com/aurion-polito/pypolydim/blob/main/ROM_POD.ipynb

Calculates the covariance matrix C used in the Proper Orthogonal Decomposition (POD). This matrix is essential for solving the eigenvalue problem.

```python
### covariance matrix

C = snapshot_matrix @ inner_product @ np.transpose(snapshot_matrix) 

###### shape?? ############
```

--------------------------------

### Deactivate Python Virtual Environment

Source: https://github.com/aurion-polito/pypolydim/blob/main/README.md

Deactivates the currently active Python virtual environment, returning the shell to the system's default Python environment.

```bash
deactivate
```

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