ROM_LPIS

A fast ROM for Linear Parabolic Inverse Source problems

For comparision purpose, we implemented FEM+CG and ROM+CG pipelines.

• FEM/ROM is used to perform forward operations (computing the solution to the heat equations)
• And CG is used to perform backward operations (computing the reconstructed source term)

Created by: Yuxuan Huang

Required python packages: numpy, scipy, pandas, matplotlib, NGSolve(for mesh initialization and FEM solver)
To download NGSolve: https://ngsolve.org/downloads

- Data folder

This folder contains test data that stores the FEM and ROM evaluation results for different parameter sets and organized into csv files. (data includes 1/mesh_stepsize, runtime, iteration number, etc.)

- Letter-images folder

This folder contains images for true source terms, final time observations, and reconstructed final results via FEM/ROM.
The source terms are all letters.
Naming rule: [1/mesh_stepsize]_[letter or pattern name]_[type(source or observation or (reconstuciton)method]

- Letters folder

This folder contains primitive square images with basic Arial font letters of different (pixel)sizes.
They are used as source terms.
Naming rule: [letter name]_[image side length:number of pixels]

- Utils folder

This folder contains all the utility functions for the numerical methods, input (image) conversion, and plotting.

- animations folder

This folder contains some animated processes (forward and backward).

- non-Letters folder

This folder contains primitive and processed images for true source terms, final time observations, and reconstructed final results via FEM/ROM.
The source terms are all non-letter patterns.
Naming rule: [pattern name]_[type(source or observation or (reconstuciton)method][1/mesh_stepsize]

Notebooks:

Single_Run

• this is the main testing notebook.
• conducts a single reconstruction process for the final observation data of a "unknown" source pattern using FEM+CG and/or ROM+CG.

Runtime_plot

• plots the runtime and/or iteration comparison for the FEM+CG and ROM+CG method.

Comparision_Error_plots

• plots L-curve graphs for FEM and ROM results and their differences in L_2 errors in terms of different hyper parameters.

Gen_Data_tocsv

• conducts a series of reconstruction trials over one pattern under different hyper parameter combination and organized the results into one csv file.

Prototype_[something]

• prototype notebooks to verify pipeline feasibility and conduct miscellaneous tests.