Building a Parallel Poisson Solver with OpenMP, MPI, and CUDA

Mon, 15 Dec 2025 00:00:00 +0000

How This Coursework Is Graded in Practice

Exam exemption, CUDA grading, and the final report submission.

What Are We Asked to Build?

The SM25 coursework follows one numerical problem through several stages of parallelization. We begin with a two-dimensional Poisson equation with Dirichlet boundary conditions, construct a finite-difference approximation, and solve the resulting linear system with a preconditioned conjugate-gradient method. From that common numerical core, the two assignments together gradually introduce shared-memory parallelism with OpenMP, distributed-memory execution with MPI, a hybrid MPI+OpenMP implementation, and finally GPU acceleration with MPI+CUDA.

The important part is to keep these stages connected. The OpenMP version should grow from a verified sequential solver; the MPI version needs a two-dimensional decomposition of the same computational grid; and the CUDA stage continues from the distributed implementation by moving suitable local operations to the GPU. Along the way, the two assignments also require correctness checks, timing breakdowns, speedup and efficiency measurements, cluster job configuration, and a final report that explains the observed performance.

This walkthrough follows that development path from beginning to end. Each stage focuses on the new responsibility introduced by the next execution model: first getting the numerical method right, then parallelizing local loops, distributing the grid across processes, coordinating MPI and OpenMP, moving computational kernels to CUDA, and finally running and evaluating the program on IBM Polus. The accompanying SM25 repository provides a working reference implementation covering this development path, including the solver code, build scripts, cluster configuration, numerical outputs, plots, and the course report.

Steps to Follow

Step 1 — From the Mathematical Problem to the Solver

Derive the numerical scheme, the fictitious-domain operator, and the PCG core.

Step 2 — Build the Sequential Reference Solver

Assemble the sequential program, run the baseline cases, and validate it.

Step 3 — Parallelize the Solver with OpenMP

Parallelize the pointwise, stencil, and reduction kernels with OpenMP, then measure speedup and efficiency.

Step 4 — Decompose the Domain with MPI

Split the grid into MPI subdomains, exchange halo values, and combine reductions across processes.

Step 5 — Combine MPI and OpenMP

Add OpenMP threads inside each MPI process to build the hybrid MPI+OpenMP solver.

Step 6 — Extend the MPI Solver to CUDA

Move the local numerical kernels to the GPU while keeping the MPI decomposition.

Step 7 — Run the Solver on IBM Polus

Configure, submit, and run every execution mode on the IBM Polus cluster.

Step 8 — Verify Correctness and Measure Performance

Validate numerical correctness across all modes and report speedup and efficiency.

Sources

This walkthrough follows the SM25 coursework of the course Supercomputer Modeling and Technologies (MSU, 2025) and summarizes its requirements in its own words. The official course materials, including the assignment statements, submission deadlines, and the final-report format, are available in the course’s shared folder:

MPI | Hailin Liu

Building a Parallel Poisson Solver with OpenMP, MPI, and CUDA

What Are We Asked to Build?

Steps to Follow

Sources