Projects

Projects that I have done so far and enjoyed:

2025

• Universal Approximation Theorem for Geometry Aware Operator Transformer (GAOT)
  • Proved Universal Approximation property (UAP) for Geometry Aware Operator Transformer (GAOT). UAP is the density result that says: GAOT can approximate any continuous solution operator between \(L^p\) spaces with arbitrary accuracy.
  • GAOT is the deep learning-based PDE solver that uses operator learning and transformers to numerically solve PDEs.
  • GAOT learns the solution operators of PDEs on arbitrary domains accurately and efficiently for CFD simulations. Hence theoretical foundations for GAOT are of great importance.
  • I learned classical PDE analysis, measure theory, functional and Fourier analysis and applied techniques from these areas to prove universal approximation property.

2024

• Distributed memory parallelization of Lax-Wendroff Flux Reconstruction
  • Performed parallelisation of the code for Lax-Wendroff Flux Reconstruction method for solving hyperbolic conservation laws for 1D and 2D cases using Message Passing Interface (MPI).
  • Improved execution time of code by 13 times on a multicore architecture with efficiency of 82% on cartesian and curvilinear meshes
  • The project report can be accessed here.

2023

• MPI-based Finite Difference Method for Parallel Numerical Simulation of Linear ODEs in C++
  • Applied master-worker parallel algorithm in solving linear boundary value problems with Dirichlet, Neumann, and mixed boundary conditions.
  • Code for parallel implementation of ordinary differential equation solver can be accessed here.
• Serial Numerical Simulation of Linear ODEs using Finite Difference Method in C++
  • Studied Finite difference method, derivative approximations for boundary value problems with Dirichlet, Neumann and mixed boundary conditions.
  • Code for serial implementation of ordinary differential equation solver can be accessed here.
• Reading Project on Advance C++ and Message Passing Interface (MPI)
  • Studied point-to-point and collective communication as well as blocking and non-blocking communication routines in MPI.
  • Reading includes the book "Guide to Scientific Computing in C++" by Dr. Joe Pitt-Francis and Prof. Jonathan Whiteley and some online mpi-tutorial websites.
  • A parallel and serial implementation of Gauss Elimination using MPI can be found here and few other related codes can be found here.

Some Other Codes (2024)

• This repository here, contains code for Message Passing Interface (MPI) in Julia.
  • From simple implementations of MPI routines to code for Linear-advection equation in serial and parallel (MPI) are there.
  • Codes for parallelization using Remote Memory Access (RMA), which is included in MPI 3.0, are also there for Linear Advection equation (1D and 2D).

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