🚀 Parallel Romberg Integration: HPC Implementation

High-performance parallel implementation of Romberg numerical integration using MPI and hybrid MPI+OpenMP strategies

Developed as part of the High Performance Computing for Data Science (HPC4DS) course at the University of Trento (2025-2026).

Authors: Lewis Ndambiri & Mehrab Fajar

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📋 Table of Contents

• Overview
• Key Features
• Performance Highlights
• Algorithm Background
• Implementation Strategies
• Project Structure
• Getting Started
• Running Experiments
• Results Visualization
• Performance Analysis
• Technical Details
• Future Work
• Acknowledgments
• License

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🎯 Overview

This project implements parallel Romberg integration, a sophisticated numerical method combining the trapezoidal rule with Richardson extrapolation. We developed three implementations:

1. Serial baseline – Pure sequential implementation for benchmarking
2. Pure MPI – Distributed memory parallelization using message passing
3. Hybrid MPI+OpenMP – Two-level parallelism exploiting distributed + shared memory

The implementations were tested on the University of Trento HPC cluster (6,092 CPU cores, Infiniband/Omnipath interconnect) with comprehensive scalability analysis.

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✨ Key Features

• Production-grade implementation with proper error handling and timing
• Multiple parallelization strategies (MPI, Hybrid MPI+OpenMP)
• Heavy computational workload simulation (10⁶ iterations per function evaluation)
• Automated benchmarking suite with PBS job submission scripts
• Comprehensive scalability analysis (strong, weak, placement strategies)
• Visualization tools for performance metrics and speedup curves
• Clean, documented codebase following HPC best practices

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🏆 Performance Highlights

Strong Scaling (Fixed Problem Size)

Configuration	Speedup	Efficiency	Problem Size
32 cores	31.77×	99.3%	Level 16 (65K intervals)
32 cores	21.70×	67.8%	Level 20 (1M intervals)
16 cores	18.12×	113.2%	Level 16 (super-linear!)

Key Findings

• 🎯 Near-perfect linear scaling up to 32 cores for medium-sized problems
• 🚀 Super-linear speedup observed due to improved cache utilization
• ⚠️ Communication overhead dominates beyond 64 cores for large problems
• 📊 Hybrid MPI+OpenMP underperformed pure MPI for this algorithm class

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📚 Algorithm Background

What is Romberg Integration?

Romberg integration is an extrapolation method that:

1. Computes successive trapezoidal approximations with increasing refinement
2. Applies Richardson extrapolation to eliminate low-order error terms
3. Achieves higher-order accuracy with fewer function evaluations than basic methods

Mathematical formulation:

R[i][0] = Trapezoidal rule with n = 2^i intervals
R[i][j] = R[i][j-1] + (R[i][j-1] - R[i-1][j-1]) / (4^j - 1)

Test Problem

We integrate:

f(x) = sin(x) * exp(-x²)  over [0, π]

To simulate expensive HPC workloads (PDE solvers, physics simulations), each function evaluation is repeated 1 million times.

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🔧 Implementation Strategies

1️⃣ Pure MPI (Distributed Memory)

Parallelization approach:

• Domain decomposition: Distribute trapezoidal subintervals across processes
• Load balancing: Block partitioning with remainder distribution
• Communication: Single MPI_Reduce per Romberg level
• Extrapolation: Sequential on rank 0 (negligible overhead)

// Block partitioning with balanced remainder distribution
int base = n / comm_sz;
int remainder = n % comm_sz;
int local_n = base + (my_rank < remainder);

Why MPI_Reduce instead of MPI_Allreduce?

• Only rank 0 needs the global sum for Romberg table construction
• Eliminates unnecessary communication overhead
• Other processes remain idle during extrapolation (acceptable given fast sequential computation)

2️⃣ Hybrid MPI+OpenMP

Two-level parallelism:

• MPI: Distributes subintervals across nodes (coarse-grained parallelism)
• OpenMP: Parallelizes local function evaluations within nodes (fine-grained parallelism)

#pragma omp parallel for reduction(+: local_sum) schedule(static)
for (int i = 1; i < local_n; i++) {
    local_sum += f(local_a + i * h);
}

Hybrid configuration tested:

• MPI ranks: {2, 4, 8}
• OpenMP threads per rank: 2
• Total cores: {4, 8, 16}

Finding: Pure MPI outperformed hybrid for this algorithm due to thread management overhead and NUMA effects.

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🔧 Project Archictecture

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📁 Project Structure

romberg-hpc4ds/
├── src/
│   ├── romberg_serial.c          # Serial baseline
│   ├── romberg_mpi.c              # Pure MPI implementation
│   └── romberg_mpi_openmp.c       # Hybrid MPI+OpenMP version
│
├── scripts/
│   ├── romberg_serial.pbs         # PBS job template (serial)
│   ├── romberg_mpi.pbs            # PBS job template (MPI)
│   ├── romberg_hybrid.pbs         # PBS job template (hybrid)
│   └── run_experiments.sh         # Automated benchmarking suite
│
├── analysis/
│   └── analyze_romberg_logs.py    # Performance visualization tools
│
├── results/
│   ├── *.out                      # Job output files
│   ├── *.csv                      # Performance data
│   └── *.png                      # Generated plots
│
├── Makefile                       # Build automation
├── README.md                      # This file
└── LICENSE                        # MIT License

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🚀 Getting Started

Prerequisites

• C compiler with MPI support (mpicc)
• MPI implementation (MPICH, OpenMPI)
• OpenMP support (for hybrid version)
• Python 3.x with matplotlib, numpy, scipy (for analysis)
• PBS/Torque job scheduler (for cluster execution)

Installation

1. Clone the repository:

   git clone https://github.com/lewisndambiri.io/romberg-hpc4ds.git
   cd romberg-hpc4ds
   

2. Build all versions:

   make clean
   make all
   

This compiles:

• romberg_serial – Serial baseline
• romberg_mpi – Pure MPI version
• romberg_mpi_openmp – Hybrid version

1. Verify compilation:

   ./romberg_serial 10      # Run serial with 10 Romberg levels
   mpiexec -n 4 ./romberg_mpi 10  # Run MPI with 4 processes
   

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🧪 Running Experiments

Local Testing (Single Machine)

Serial execution:

./romberg_serial 16

MPI parallel execution:

mpiexec -n 4 ./romberg_mpi 16

Hybrid execution:

export OMP_NUM_THREADS=2
mpiexec -n 4 ./romberg_mpi_openmp 16

Cluster Deployment (PBS)

Submit individual jobs:

qsub scripts/romberg_serial.pbs
qsub scripts/romberg_mpi.pbs
qsub scripts/romberg_hybrid.pbs

Run complete benchmark suite:

./scripts/run_experiments.sh

This automated script submits jobs for:

• Serial baselines (L16, L20)
• Strong scaling (p ∈ {2, 4, 8, 16, 32, 64})
• Weak scaling (work per processor constant)
• Placement strategies (pack, scatter, excl)
• Hybrid MPI+OpenMP configurations

Monitor job status:

qstat -u $USER

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📊 Results Visualization

Generate Performance Plots

After jobs complete, run the analysis script:

cd analysis/
python3 analyze_romberg_logs.py

This generates:

1. Strong scaling speedup (L16 vs L20)
2. Strong scaling efficiency (L16 vs L20)
3. Weak scaling efficiency
4. Hybrid scaling speedup & efficiency
5. Placement strategy comparison
6. Amdahl’s Law fit

Plots are saved to results/*.png with CSV data exported for further analysis.

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📈 Performance Analysis

Strong Scaling Results

Level 16 (65,536 intervals)

Cores	Time (s)	Speedup	Efficiency
1	176.17	1.00	100.0%
2	74.07	2.38	118.9%
4	39.97	4.41	110.2%
8	21.22	8.30	103.8%
16	9.72	18.12	113.2%
32	5.55	31.77	99.3%

Analysis: Super-linear speedup up to 16 cores due to improved cache locality. Near-perfect efficiency at 32 cores.

Level 20 (1,048,576 intervals)

Cores	Time (s)	Speedup	Efficiency
1	2482.75	1.00	100.0%
2	1250.90	1.98	99.2%
4	795.84	3.12	78.0%
8	389.62	6.37	79.7%
16	168.07	14.77	92.3%
32	114.42	21.70	67.8%
64	306.86	8.09	12.6%

Analysis: Efficiency degrades at 64 cores due to communication overhead exceeding computation benefits.

Weak Scaling Results

Cores	Level	Intervals	Time (s)	Efficiency
1	12	4,096	9.99	100.0%
2	13	8,192	11.51	43.4%
4	14	16,384	15.87	15.7%
8	15	32,768	10.61	11.8%
16	16	65,536	104.27	0.6%

Analysis: Weak scaling degrades because Richardson extrapolation overhead grows as O(L²), and each level requires completing all previous levels.

Placement Strategy Impact

Configuration: p=4, Level 20

Placement	Time (s)	Speedup
pack	640.97	0.84
scatter	767.95	0.70
pack:excl	536.17	1.00
scatter:excl	658.30	0.81

Conclusion: Minimal impact (<2% variation) for communication-light algorithms. pack:excl slightly optimal due to eliminating resource contention.

Hybrid MPI+OpenMP Results

Total Cores	MPI Ranks	OMP Threads	Time (s)	Speedup	Efficiency
4	2	2	1211.02	2.05	51.3%
8	4	2	1781.68	1.39	17.4%
16	8	2	305.26	8.13	50.8%

Conclusion: Hybrid model underperforms pure MPI due to thread management overhead and suboptimal load balancing between parallelism levels.

Amdahl’s Law Analysis

Empirical fitting to Level 20 data estimates serial fraction f ≈ 4.8%, yielding:

Maximum theoretical speedup = 1/0.048 ≈ 20.8×

Observed speedup at p=32 (21.7×) slightly exceeds this due to cache effects. Performance degradation at p=64 confirms communication overhead dominance.

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🔬 Technical Details

Computational Complexity

• Trapezoidal evaluations: O(2^L) function calls across all L levels
• Richardson extrapolation: O(L²) operations (negligible)
• Memory: O(L²) for Romberg table

Communication Pattern

• MPI_Reduce: One collective per Romberg level
• Bandwidth requirements: O(p) for reduction tree
• Latency hiding: None (blocking reduction acceptable given fast extrapolation)

Load Balancing Strategy

Block partitioning with remainder distribution:

base = n / comm_sz;
remainder = n % comm_sz;

// First 'remainder' processes get base+1 intervals
// Remaining processes get base intervals
local_n = base + (my_rank < remainder ? 1 : 0);

Example: n=10 intervals, p=3 processes

• Process 0: 4 intervals (indices 0-3)
• Process 1: 3 intervals (indices 4-6)
• Process 2: 3 intervals (indices 7-9)

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🛠️ Future Work

Potential improvements and extensions:

1. Adaptive quadrature: Focus computation on high-error regions
2. Pipeline parallelism: Overlap extrapolation with next level’s computation
3. GPU acceleration: Offload heavy integrand evaluations to accelerators
4. Non-blocking communication: Use MPI_Iallreduce for overlapping computation/communication
5. Dynamic load balancing: Adjust partitioning based on runtime performance
6. Fault tolerance: Implement checkpointing for long-running jobs

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🙏 Acknowledgments

This project was developed as part of the HPC4DS course (2025-2026) at the University of Trento, taught by Prof. Sandro Fiore.

Resources:

• University of Trento HPC Cluster (6,092 cores, Infiniband/Omnipath interconnect)
• Course materials: An Introduction to Parallel Programming by Peter Pacheco
• MPI implementation: MPICH 3.2

Special thanks to the DISI (Department of Information Engineering and Computer Science) system administrators for cluster support.

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📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

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📧 Contact

Authors:

• Lewis Ndambiri - GitHub
• Mehrab Fajar - GitHub

Course Instructor:

• Prof. Sandro Fiore - University of Trento

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Made with ❤️ at University of Trento