Collection of MATLAB prototypes for forward uncertainty quantification (UQ) of a Poisson diffusion problem with random material properties. Each folder contains a self‐contained workflow that assembles finite‐element operators, propagates lognormal/gaussian uncertainty via stochastic spectral finite element methods (SSFEM), and compares those solutions to Monte Carlo surrogates.

• Intrusive — Intrusive SSFEM for random variables. Key scripts: ssfem_Intrusive.m, AssembleMatrix_ss.m, AssembleVector_ss.m, Poisson_ss.m. Includes pre/post-processing data (*.mat, .fig) plus gmsh meshes under mac/ and source/.
• NISP — Non-intrusive spectral projection (Monte Carlo and quadrature variants). Core drivers are ssfem_NISP.m, ssfem_NISP_quadrature.m, and Normalized_ssfem_NISP_quadrature.m; helper PDE definitions live beside gmsh utilities (ExtractGmsh.m, square_refined.m).
• NISP_Process — NISP workflow for lognormal random processes driven by Karhunen–Loève expansions. Uses NISP_process.m, GetKLEterms.m, GetLambdaSymmetric_2D.m, and quadrature data from UQTk.
• NISP_analytical — Analytical/non-intrusive reference solutions for verification (NISP_analytical.m, NISP_quadrature.m, PC_std_quadrature.m) plus saved Monte Carlo statistics (MCS_*.mat).
• ssfem_LNP_Intrusive — Intrusive SSFEM for lognormal random processes (ssfem_Intrusive.m, ssfem_Intrusive_Normalized.m, GetKLEterms.m, Cijk.m). Contains meshes (mac/, source/) and comparison figures.
• ssfem_MCS_Process — Monte Carlo sampling of random processes (sfem_MCS_Process.m) using the same assembly utilities as the intrusive lognormal process workflow.
• ssfem_MCS_RV — Monte Carlo sampling for single random variables (ssfem_MCS.m) that mirrors the non-intrusive RV setup.

Shared helper files include:

• AssembleMatrix.m, AssembleMatrix_ss.m, AssembleVector.m, and AssembleVector_ss.m – generic stiffness/load assemblers for deterministic vs. stochastic forms.
• Poisson.m, Poisson_ss.m, Poisson_ss_Normalized.m – PDE weak forms with configurable random diffusivity.
• Cijk.m – generates Hermite polynomial triple products (Cijk*.mat) and norm-squared data via UQTk (gen_mi).
• GetKLEterms.m, GetLambdaSymmetric_2D.m, GetKLEterms_Normalized.m – build KLE eigenpairs and evaluate lognormal scaling terms.
• Mesh utilities such as ExtractGmsh.m, square.m, square_refined.m, and gmsh sources under each {module}/mac or {module}/source directory.

• MATLAB R2017a or newer (scripts rely on base language plus pdesurf from PDE Toolbox).
• Gmsh to generate .msh files from the .geo geometries stored under mac/ or source/.
• UQTk (tested with v3.0.4) for gen_mi and generate_quad. Update uqtk_path inside the scripts to match your install.
• Access to the misc/cijk_ord_dim directory referenced by several scripts for precomputed Cijk*.mat, norm_squared*.mat, and mindex*.mat/.dat files. Recreate these files with Cijk.m and UQTk if they are not present.

1. Prepare a mesh
   • Edit or reuse the .geo files in */source or */mac.
   • Run Gmsh to export a version 2 ASCII .msh.
   • Convert the mesh with ExtractGmsh.m, which writes points.txt, edges.txt, and triangles.txt. Scripts such as square_refined.m can also be run directly in MATLAB to load a hard-coded mesh into p, e, t.
2. Generate spectral data (if needed)
   • Use Cijk.m to compute multiplication tensors for the desired polynomial order and dimensionality. The script writes CijkXXXXX.mat and norm_squaredXXXXX.mat files whose suffixes encode order/dimension (e.g., Cijk030003.mat).
   • Create multi-index tables with UQTk’s gen_mi and place the resulting mindex*.dat alongside the solver that needs it.
   • For random processes, run GetLambdaSymmetric_2D.m to sample the covariance eigenpairs and reuse them through GetKLEterms.m.
3. Run a solver
   • Intrusive RV – from Intrusive/, set ord_in, ord_out, dim, and num_spectral in ssfem_Intrusive.m, ensure the correct Cijk/norm_squared paths, then run the script to obtain polynomial chaos coefficients, means, and standard deviations.
   • Non-intrusive RV (NISP) – use ssfem_NISP.m for Monte Carlo projection or ssfem_NISP_quadrature.m / Normalized_ssfem_NISP_quadrature.m for Gauss–Hermite quadrature. Configure sample counts or quadrature orders and Hermite coefficients (kappa(*)).
   • Intrusive/NISP lognormal process – use the drivers in ssfem_LNP_Intrusive/ or NISP_Process/. These scripts expect lambda_2D.mat, mindex*.dat, and norm_squared*.mat, and they call UQTk’s generate_quad internally; update the uqtk_path variable and normalize inputs to match your workspace.
   • Monte Carlo baselines – ssfem_MCS_RV/ssfem_MCS.m and ssfem_MCS_Process/sfem_MCS_Process.m sample random inputs directly to compare against the spectral solvers.
4. Post-process
   • Most scripts plot pdesurf surfaces for the mean field, higher-order coefficients, and standard deviations while also saving .fig, .mat, or KDE data (MCS_50000_*.mat, compare_*.fig, ssfem_intrsv.mat, etc.).
   • NISP_analytical/ holds scripts for closed-form coefficient evaluations and PDF reconstruction at specific nodes to benchmark the numerical schemes.

• Add the desired module folder to the MATLAB path before invoking any driver script so that shared assemblers and PDE definitions resolve correctly.
• Several scripts contain hard-coded cd statements (for example, to /Users/sudhipv/...) around UQTk calls or result exports; replace those paths with locations on your machine to avoid runtime failures.
• ExtractGmsh.m assumes a 2D mesh saved in Gmsh 2 ASCII format where edges precede triangle elements. Confirm that your .msh files follow the same layout or update the reader accordingly.
• Normalized vs. non-normalized polynomial chaos: the "Normalized" scripts expect Hermite norms loaded from norm_squared*.mat. Be sure that the file suffix (_O11_D1, 030003, etc.) matches the (order, dimension) pair configured at the top of the solver.
• The Monte Carlo and NISP scripts often loop over ~10k–50k samples. Adjust n, n_sample, or num_qd for quicker smoke tests when validating new setups.

Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems

Authors: Vasudevan, Padillath and Sharma, Sudhi
Institution: Carleton University (2023)
DOI: 10.22215/etd/2023-15817

@phdthesis{vasudevan2023scalable,
 title={Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems},
 author={Vasudevan, Padillath and Sharma, Sudhi},
 year={2023},
 school={Carleton University},
 doi={10.22215/etd/2023-15817}
}
\```
</details>



## Questions?
Contact : Sudhi Sharma P V 
Email: sudhisharmapadillath@gmail.com